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Theorem nnmcom 6468
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 5860 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·o 𝐵) = (𝐴 ·o 𝐵))
2 oveq2 5861 . . . . 5 (𝑥 = 𝐴 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝐴))
31, 2eqeq12d 2185 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
43imbi2d 229 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)) ↔ (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))))
5 oveq1 5860 . . . . 5 (𝑥 = ∅ → (𝑥 ·o 𝐵) = (∅ ·o 𝐵))
6 oveq2 5861 . . . . 5 (𝑥 = ∅ → (𝐵 ·o 𝑥) = (𝐵 ·o ∅))
75, 6eqeq12d 2185 . . . 4 (𝑥 = ∅ → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (∅ ·o 𝐵) = (𝐵 ·o ∅)))
8 oveq1 5860 . . . . 5 (𝑥 = 𝑦 → (𝑥 ·o 𝐵) = (𝑦 ·o 𝐵))
9 oveq2 5861 . . . . 5 (𝑥 = 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o 𝑦))
108, 9eqeq12d 2185 . . . 4 (𝑥 = 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (𝑦 ·o 𝐵) = (𝐵 ·o 𝑦)))
11 oveq1 5860 . . . . 5 (𝑥 = suc 𝑦 → (𝑥 ·o 𝐵) = (suc 𝑦 ·o 𝐵))
12 oveq2 5861 . . . . 5 (𝑥 = suc 𝑦 → (𝐵 ·o 𝑥) = (𝐵 ·o suc 𝑦))
1311, 12eqeq12d 2185 . . . 4 (𝑥 = suc 𝑦 → ((𝑥 ·o 𝐵) = (𝐵 ·o 𝑥) ↔ (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
14 nnm0r 6458 . . . . 5 (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
15 nnm0 6454 . . . . 5 (𝐵 ∈ ω → (𝐵 ·o ∅) = ∅)
1614, 15eqtr4d 2206 . . . 4 (𝐵 ∈ ω → (∅ ·o 𝐵) = (𝐵 ·o ∅))
17 oveq1 5860 . . . . . 6 ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → ((𝑦 ·o 𝐵) +o 𝐵) = ((𝐵 ·o 𝑦) +o 𝐵))
18 nnmsucr 6467 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝑦 ·o 𝐵) = ((𝑦 ·o 𝐵) +o 𝐵))
19 nnmsuc 6456 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
2019ancoms 266 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ·o suc 𝑦) = ((𝐵 ·o 𝑦) +o 𝐵))
2118, 20eqeq12d 2185 . . . . . 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 155 . . . . 5 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦)))
2322ex 114 . . . 4 (𝑦 ∈ ω → (𝐵 ∈ ω → ((𝑦 ·o 𝐵) = (𝐵 ·o 𝑦) → (suc 𝑦 ·o 𝐵) = (𝐵 ·o suc 𝑦))))
247, 10, 13, 16, 23finds2 4585 . . 3 (𝑥 ∈ ω → (𝐵 ∈ ω → (𝑥 ·o 𝐵) = (𝐵 ·o 𝑥)))
254, 24vtoclga 2796 . 2 (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)))
2625imp 123 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  c0 3414  suc csuc 4350  ωcom 4574  (class class class)co 5853   +o coa 6392   ·o comu 6393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400
This theorem is referenced by:  nndir  6469  nn2m  6506  mulcompig  7293  enq0sym  7394  enq0ref  7395  enq0tr  7396  addcmpblnq0  7405  mulcmpblnq0  7406  mulcanenq0ec  7407  nnanq0  7420  distrnq0  7421  mulcomnq0  7422  addassnq0  7424  nq02m  7427
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