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Theorem nnmcom 6099
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.
StepHypRef Expression
1 oveq1 5547 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐵))
2 oveq2 5548 . . . . 5 (𝑥 = 𝐴 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐴))
31, 2eqeq12d 2070 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)))
43imbi2d 223 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))))
5 oveq1 5547 . . . . 5 (𝑥 = ∅ → (𝑥 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
6 oveq2 5548 . . . . 5 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
75, 6eqeq12d 2070 . . . 4 (𝑥 = ∅ → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅)))
8 oveq1 5547 . . . . 5 (𝑥 = 𝑦 → (𝑥 ·𝑜 𝐵) = (𝑦 ·𝑜 𝐵))
9 oveq2 5548 . . . . 5 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
108, 9eqeq12d 2070 . . . 4 (𝑥 = 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦)))
11 oveq1 5547 . . . . 5 (𝑥 = suc 𝑦 → (𝑥 ·𝑜 𝐵) = (suc 𝑦 ·𝑜 𝐵))
12 oveq2 5548 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
1311, 12eqeq12d 2070 . . . 4 (𝑥 = suc 𝑦 → ((𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥) ↔ (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦)))
14 nnm0r 6089 . . . . 5 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
15 nnm0 6085 . . . . 5 (𝐵 ∈ ω → (𝐵 ·𝑜 ∅) = ∅)
1614, 15eqtr4d 2091 . . . 4 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = (𝐵 ·𝑜 ∅))
17 oveq1 5547 . . . . . 6 ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → ((𝑦 ·𝑜 𝐵) +𝑜 𝐵) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
18 nnmsucr 6098 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·𝑜 𝐵) = ((𝑦 ·𝑜 𝐵) +𝑜 𝐵))
19 nnmsuc 6087 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
2019ancoms 259 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
2118, 20eqeq12d 2070 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦) ↔ ((𝑦 ·𝑜 𝐵) +𝑜 𝐵) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2217, 21syl5ibr 149 . . . . 5 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦)))
2322ex 112 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑦) → (suc 𝑦 ·𝑜 𝐵) = (𝐵 ·𝑜 suc 𝑦))))
247, 10, 13, 16, 23finds2 4352 . . 3 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·𝑜 𝐵) = (𝐵 ·𝑜 𝑥)))
254, 24vtoclga 2636 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴)))
2625imp 119 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  c0 3252  suc csuc 4130  ωcom 4341  (class class class)co 5540   +𝑜 coa 6029   ·𝑜 comu 6030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037
This theorem is referenced by:  nndir  6100  nn2m  6130  mulcompig  6487  enq0sym  6588  enq0ref  6589  enq0tr  6590  addcmpblnq0  6599  mulcmpblnq0  6600  mulcanenq0ec  6601  nnanq0  6614  distrnq0  6615  mulcomnq0  6616  addassnq0  6618  nq02m  6621
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