Theorem nnmcom 8590

Description: Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
nnmcom	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))

Proof of Theorem nnmcom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7398	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·o 𝐵) = (𝐴 ·o 𝐵))
2		oveq2 7399	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐴))
3	1, 2	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
4	3	imbi2d 342	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))))
5		oveq1 7398	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·o 𝐵) = (∅ ·o 𝐵))
6		oveq2 7399	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
7	5, 6	eqeq12d 2777	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (∅ ·o 𝐵) = (𝐵 ·o ∅)))
8		oveq1 7398	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐵) = (𝑦 ·o 𝐵))
9		oveq2 7399	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
10	8, 9	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝑦 ·o 𝐵) = (𝐵 ·o 𝑦)))
11		oveq1 7398	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·o 𝐵) = (suc 𝑦 ·o 𝐵))
12		oveq2 7399	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
13	11, 12	eqeq12d 2777	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
14		nnm0r 8574	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
15		nnm0 8569	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅)
16	14, 15	eqtr4d 2799	. . . 4 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = (𝐵 ·o ∅))
17		oveq1 7398	. . . . . 6 ⊢ ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵))
18		nnmsucr 8589	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·o 𝐵) = ((𝑦 ·o 𝐵) +o 𝐵))
19		nnmsuc 8571	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
20	19	ancoms 462	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
21	18, 20	eqeq12d 2777	. . . . . 6 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦) ↔ ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵)))
22	17, 21	imbitrrid 248	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
23	22	ex 416	. . . 4 ⊢ (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))))
24	7, 10, 13, 16, 23	finds2 7874	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)))
25	4, 24	vtoclga 3540	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
26	25	imp 410	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∅c0 4283  suc csuc 6343  (class class class)co 7391  ωcom 7841   +_o coa 8428   ·_o comu 8429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-oadd 8435  df-omul 8436

This theorem is referenced by:  nnmwordri  8600  nn2m  8618  omopthlem1  8623  mulcompi  10848