Theorem nnmcom 8632

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 7419	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·o 𝐵) = (𝐴 ·o 𝐵))
2		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐴))
3	1, 2	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
4	3	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))))
5		oveq1 7419	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ·o 𝐵) = (∅ ·o 𝐵))
6		oveq2 7420	. . . . 5 ⊢ (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
7	5, 6	eqeq12d 2747	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (∅ ·o 𝐵) = (𝐵 ·o ∅)))
8		oveq1 7419	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ·o 𝐵) = (𝑦 ·o 𝐵))
9		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
10	8, 9	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝑦 ·o 𝐵) = (𝐵 ·o 𝑦)))
11		oveq1 7419	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝑥 ·o 𝐵) = (suc 𝑦 ·o 𝐵))
12		oveq2 7420	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
13	11, 12	eqeq12d 2747	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
14		nnm0r 8616	. . . . 5 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
15		nnm0 8611	. . . . 5 ⊢ (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅)
16	14, 15	eqtr4d 2774	. . . 4 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = (𝐵 ·o ∅))
17		oveq1 7419	. . . . . 6 ⊢ ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵))
18		nnmsucr 8631	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·o 𝐵) = ((𝑦 ·o 𝐵) +o 𝐵))
19		nnmsuc 8613	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
20	19	ancoms 458	. . . . . . 7 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
21	18, 20	eqeq12d 2747	. . . . . 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 245	. . . . 5 ⊢ ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
23	22	ex 412	. . . 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 7895	. . 3 ⊢ (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)))
25	4, 24	vtoclga 3566	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
26	25	imp 406	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∅c0 4322  suc csuc 6366  (class class class)co 7412  ωcom 7859   +_o coa 8469   ·_o comu 8470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-oadd 8476  df-omul 8477

This theorem is referenced by:  nnmwordri  8642  nn2m  8659  omopthlem1  8664  mulcompi  10897