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Theorem nnmcom 6492
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 5884 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·o 𝐵) = (𝐴 ·o 𝐵))
2 oveq2 5885 . . . . 5 (𝑥 = 𝐴 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐴))
31, 2eqeq12d 2192 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
43imbi2d 230 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))))
5 oveq1 5884 . . . . 5 (𝑥 = ∅ → (𝑥 ·o 𝐵) = (∅ ·o 𝐵))
6 oveq2 5885 . . . . 5 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
75, 6eqeq12d 2192 . . . 4 (𝑥 = ∅ → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (∅ ·o 𝐵) = (𝐵 ·o ∅)))
8 oveq1 5884 . . . . 5 (𝑥 = 𝑦 → (𝑥 ·o 𝐵) = (𝑦 ·o 𝐵))
9 oveq2 5885 . . . . 5 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
108, 9eqeq12d 2192 . . . 4 (𝑥 = 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝑦 ·o 𝐵) = (𝐵 ·o 𝑦)))
11 oveq1 5884 . . . . 5 (𝑥 = suc 𝑦 → (𝑥 ·o 𝐵) = (suc 𝑦 ·o 𝐵))
12 oveq2 5885 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1311, 12eqeq12d 2192 . . . 4 (𝑥 = suc 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
14 nnm0r 6482 . . . . 5 (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
15 nnm0 6478 . . . . 5 (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅)
1614, 15eqtr4d 2213 . . . 4 (𝐵 ∈ ω → (∅ ·o 𝐵) = (𝐵 ·o ∅))
17 oveq1 5884 . . . . . 6 ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵))
18 nnmsucr 6491 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·o 𝐵) = ((𝑦 ·o 𝐵) +o 𝐵))
19 nnmsuc 6480 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
2019ancoms 268 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
2118, 20eqeq12d 2192 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦) ↔ ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵)))
2217, 21imbitrrid 156 . . . . 5 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
2322ex 115 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))))
247, 10, 13, 16, 23finds2 4602 . . 3 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)))
254, 24vtoclga 2805 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
2625imp 124 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  c0 3424  suc csuc 4367  ωcom 4591  (class class class)co 5877   +o coa 6416   ·o comu 6417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424
This theorem is referenced by:  nndir  6493  nn2m  6530  mulcompig  7332  enq0sym  7433  enq0ref  7434  enq0tr  7435  addcmpblnq0  7444  mulcmpblnq0  7445  mulcanenq0ec  7446  nnanq0  7459  distrnq0  7460  mulcomnq0  7461  addassnq0  7463  nq02m  7466
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