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Theorem nnmcom 6007
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 5462 . . . . 5 (x = A → (x ·𝑜 B) = (A ·𝑜 B))
2 oveq2 5463 . . . . 5 (x = A → (B ·𝑜 x) = (B ·𝑜 A))
31, 2eqeq12d 2051 . . . 4 (x = A → ((x ·𝑜 B) = (B ·𝑜 x) ↔ (A ·𝑜 B) = (B ·𝑜 A)))
43imbi2d 219 . . 3 (x = A → ((B 𝜔 → (x ·𝑜 B) = (B ·𝑜 x)) ↔ (B 𝜔 → (A ·𝑜 B) = (B ·𝑜 A))))
5 oveq1 5462 . . . . 5 (x = ∅ → (x ·𝑜 B) = (∅ ·𝑜 B))
6 oveq2 5463 . . . . 5 (x = ∅ → (B ·𝑜 x) = (B ·𝑜 ∅))
75, 6eqeq12d 2051 . . . 4 (x = ∅ → ((x ·𝑜 B) = (B ·𝑜 x) ↔ (∅ ·𝑜 B) = (B ·𝑜 ∅)))
8 oveq1 5462 . . . . 5 (x = y → (x ·𝑜 B) = (y ·𝑜 B))
9 oveq2 5463 . . . . 5 (x = y → (B ·𝑜 x) = (B ·𝑜 y))
108, 9eqeq12d 2051 . . . 4 (x = y → ((x ·𝑜 B) = (B ·𝑜 x) ↔ (y ·𝑜 B) = (B ·𝑜 y)))
11 oveq1 5462 . . . . 5 (x = suc y → (x ·𝑜 B) = (suc y ·𝑜 B))
12 oveq2 5463 . . . . 5 (x = suc y → (B ·𝑜 x) = (B ·𝑜 suc y))
1311, 12eqeq12d 2051 . . . 4 (x = suc y → ((x ·𝑜 B) = (B ·𝑜 x) ↔ (suc y ·𝑜 B) = (B ·𝑜 suc y)))
14 nnm0r 5997 . . . . 5 (B 𝜔 → (∅ ·𝑜 B) = ∅)
15 nnm0 5993 . . . . 5 (B 𝜔 → (B ·𝑜 ∅) = ∅)
1614, 15eqtr4d 2072 . . . 4 (B 𝜔 → (∅ ·𝑜 B) = (B ·𝑜 ∅))
17 oveq1 5462 . . . . . 6 ((y ·𝑜 B) = (B ·𝑜 y) → ((y ·𝑜 B) +𝑜 B) = ((B ·𝑜 y) +𝑜 B))
18 nnmsucr 6006 . . . . . . 7 ((y 𝜔 B 𝜔) → (suc y ·𝑜 B) = ((y ·𝑜 B) +𝑜 B))
19 nnmsuc 5995 . . . . . . . 8 ((B 𝜔 y 𝜔) → (B ·𝑜 suc y) = ((B ·𝑜 y) +𝑜 B))
2019ancoms 255 . . . . . . 7 ((y 𝜔 B 𝜔) → (B ·𝑜 suc y) = ((B ·𝑜 y) +𝑜 B))
2118, 20eqeq12d 2051 . . . . . 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 145 . . . . 5 ((y 𝜔 B 𝜔) → ((y ·𝑜 B) = (B ·𝑜 y) → (suc y ·𝑜 B) = (B ·𝑜 suc y)))
2322ex 108 . . . 4 (y 𝜔 → (B 𝜔 → ((y ·𝑜 B) = (B ·𝑜 y) → (suc y ·𝑜 B) = (B ·𝑜 suc y))))
247, 10, 13, 16, 23finds2 4267 . . 3 (x 𝜔 → (B 𝜔 → (x ·𝑜 B) = (B ·𝑜 x)))
254, 24vtoclga 2613 . 2 (A 𝜔 → (B 𝜔 → (A ·𝑜 B) = (B ·𝑜 A)))
2625imp 115 1 ((A 𝜔 B 𝜔) → (A ·𝑜 B) = (B ·𝑜 A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  c0 3218  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +𝑜 coa 5937   ·𝑜 comu 5938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945
This theorem is referenced by:  nndir  6008  nn2m  6035  mulcompig  6315  enq0sym  6415  enq0ref  6416  enq0tr  6417  addcmpblnq0  6426  mulcmpblnq0  6427  mulcanenq0ec  6428  nnanq0  6441  distrnq0  6442  mulcomnq0  6443  addassnq0  6445  nq02m  6448
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