BT0006 · Bertrand theorem

mul_comm

Stable checked-use theorem · independently kernel verified

Multiplication is commutative.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

forall n m. n * m = m * n

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

none

0 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n m. n * m = m * n

Proof neighborhood

Direct theorem prerequisites

BT0004 mul_zero_left BT0005 mul_succ_left

Direct theorem dependents

BT000B add_mul BT002B multiple_mul_left BT0036 balanced_combination_scale_right BT003T mod_eq_mul_left BT003X mod_eq_to_remainder_decomposition BT0041 beta_at_exists BT0042 beta_at_unique BT004F binary_crt BT004I beta_modulus_coprime_base BT004K beta_moduli_coprime_of_gap_dvd BT004M bounded_common_multiple_step BT004V right_factor_divides_product BT0069 beta_factor_divides_product BT00BG coprime_product_is_lcm BT00QJ mul_shuffle_four BT00QU two_mul_eq_add_self BT00QV pow_mul_base BT00R4 square_six_shift_identity BT00RE three_mul_eq_two_mul_add_self BT00TW factorial_weighted_product_combine BT00TY mul_lt_mul_right_nonzero BT00U0 four_power_central_recurrence_step BT00VK central_binom_strong_upper_step BT00VP central_binom_odd_middle_le_four_pow BT00VX central_binom_prime_divisor_le_double BT00WC bertrand_scaled_budget_root_36 BT00WD bertrand_scaled_budget_root_37 BT00WU bertrand_h_root_37_from_total BT00X4 bertrand_floor_power_product_le_h_from_total BT00X5 bertrand_four_power_product_le_of_sum_from_total BT0118 nonprime_has_small_prime_divisor_below_square BT011C scaled_remainder_lift BT0122 bertrand_cover_three_hundred_seventeen_five_hundred_twenty_one BT0123 bertrand_cutoff_lt_final_prime

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

4 script commands · 2 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–1

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
02Induction on mL2–4

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L2
    induction m
  2. L3
    simp [mul_zero_left]
  3. L4
    simp [IH, mul_succ_left]

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001intro n
  2. 0002induction m
  3. 0003simp [mul_zero_left]
  4. 0004simp [IH, mul_succ_left]