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Theorem nnmass 6005
Description: Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnmass ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶)))

Proof of Theorem nnmass
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5463 . . . . . 6 (x = 𝐶 → ((A ·𝑜 B) ·𝑜 x) = ((A ·𝑜 B) ·𝑜 𝐶))
2 oveq2 5463 . . . . . . 7 (x = 𝐶 → (B ·𝑜 x) = (B ·𝑜 𝐶))
32oveq2d 5471 . . . . . 6 (x = 𝐶 → (A ·𝑜 (B ·𝑜 x)) = (A ·𝑜 (B ·𝑜 𝐶)))
41, 3eqeq12d 2051 . . . . 5 (x = 𝐶 → (((A ·𝑜 B) ·𝑜 x) = (A ·𝑜 (B ·𝑜 x)) ↔ ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶))))
54imbi2d 219 . . . 4 (x = 𝐶 → (((A 𝜔 B 𝜔) → ((A ·𝑜 B) ·𝑜 x) = (A ·𝑜 (B ·𝑜 x))) ↔ ((A 𝜔 B 𝜔) → ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶)))))
6 oveq2 5463 . . . . . 6 (x = ∅ → ((A ·𝑜 B) ·𝑜 x) = ((A ·𝑜 B) ·𝑜 ∅))
7 oveq2 5463 . . . . . . 7 (x = ∅ → (B ·𝑜 x) = (B ·𝑜 ∅))
87oveq2d 5471 . . . . . 6 (x = ∅ → (A ·𝑜 (B ·𝑜 x)) = (A ·𝑜 (B ·𝑜 ∅)))
96, 8eqeq12d 2051 . . . . 5 (x = ∅ → (((A ·𝑜 B) ·𝑜 x) = (A ·𝑜 (B ·𝑜 x)) ↔ ((A ·𝑜 B) ·𝑜 ∅) = (A ·𝑜 (B ·𝑜 ∅))))
10 oveq2 5463 . . . . . 6 (x = y → ((A ·𝑜 B) ·𝑜 x) = ((A ·𝑜 B) ·𝑜 y))
11 oveq2 5463 . . . . . . 7 (x = y → (B ·𝑜 x) = (B ·𝑜 y))
1211oveq2d 5471 . . . . . 6 (x = y → (A ·𝑜 (B ·𝑜 x)) = (A ·𝑜 (B ·𝑜 y)))
1310, 12eqeq12d 2051 . . . . 5 (x = y → (((A ·𝑜 B) ·𝑜 x) = (A ·𝑜 (B ·𝑜 x)) ↔ ((A ·𝑜 B) ·𝑜 y) = (A ·𝑜 (B ·𝑜 y))))
14 oveq2 5463 . . . . . 6 (x = suc y → ((A ·𝑜 B) ·𝑜 x) = ((A ·𝑜 B) ·𝑜 suc y))
15 oveq2 5463 . . . . . . 7 (x = suc y → (B ·𝑜 x) = (B ·𝑜 suc y))
1615oveq2d 5471 . . . . . 6 (x = suc y → (A ·𝑜 (B ·𝑜 x)) = (A ·𝑜 (B ·𝑜 suc y)))
1714, 16eqeq12d 2051 . . . . 5 (x = suc y → (((A ·𝑜 B) ·𝑜 x) = (A ·𝑜 (B ·𝑜 x)) ↔ ((A ·𝑜 B) ·𝑜 suc y) = (A ·𝑜 (B ·𝑜 suc y))))
18 nnmcl 5999 . . . . . . 7 ((A 𝜔 B 𝜔) → (A ·𝑜 B) 𝜔)
19 nnm0 5993 . . . . . . 7 ((A ·𝑜 B) 𝜔 → ((A ·𝑜 B) ·𝑜 ∅) = ∅)
2018, 19syl 14 . . . . . 6 ((A 𝜔 B 𝜔) → ((A ·𝑜 B) ·𝑜 ∅) = ∅)
21 nnm0 5993 . . . . . . . 8 (B 𝜔 → (B ·𝑜 ∅) = ∅)
2221oveq2d 5471 . . . . . . 7 (B 𝜔 → (A ·𝑜 (B ·𝑜 ∅)) = (A ·𝑜 ∅))
23 nnm0 5993 . . . . . . 7 (A 𝜔 → (A ·𝑜 ∅) = ∅)
2422, 23sylan9eqr 2091 . . . . . 6 ((A 𝜔 B 𝜔) → (A ·𝑜 (B ·𝑜 ∅)) = ∅)
2520, 24eqtr4d 2072 . . . . 5 ((A 𝜔 B 𝜔) → ((A ·𝑜 B) ·𝑜 ∅) = (A ·𝑜 (B ·𝑜 ∅)))
26 oveq1 5462 . . . . . . . . 9 (((A ·𝑜 B) ·𝑜 y) = (A ·𝑜 (B ·𝑜 y)) → (((A ·𝑜 B) ·𝑜 y) +𝑜 (A ·𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))
27 nnmsuc 5995 . . . . . . . . . . . 12 (((A ·𝑜 B) 𝜔 y 𝜔) → ((A ·𝑜 B) ·𝑜 suc y) = (((A ·𝑜 B) ·𝑜 y) +𝑜 (A ·𝑜 B)))
2818, 27sylan 267 . . . . . . . . . . 11 (((A 𝜔 B 𝜔) y 𝜔) → ((A ·𝑜 B) ·𝑜 suc y) = (((A ·𝑜 B) ·𝑜 y) +𝑜 (A ·𝑜 B)))
29283impa 1098 . . . . . . . . . 10 ((A 𝜔 B 𝜔 y 𝜔) → ((A ·𝑜 B) ·𝑜 suc y) = (((A ·𝑜 B) ·𝑜 y) +𝑜 (A ·𝑜 B)))
30 nnmsuc 5995 . . . . . . . . . . . . 13 ((B 𝜔 y 𝜔) → (B ·𝑜 suc y) = ((B ·𝑜 y) +𝑜 B))
31303adant1 921 . . . . . . . . . . . 12 ((A 𝜔 B 𝜔 y 𝜔) → (B ·𝑜 suc y) = ((B ·𝑜 y) +𝑜 B))
3231oveq2d 5471 . . . . . . . . . . 11 ((A 𝜔 B 𝜔 y 𝜔) → (A ·𝑜 (B ·𝑜 suc y)) = (A ·𝑜 ((B ·𝑜 y) +𝑜 B)))
33 nnmcl 5999 . . . . . . . . . . . . . . . . 17 ((B 𝜔 y 𝜔) → (B ·𝑜 y) 𝜔)
34 nndi 6004 . . . . . . . . . . . . . . . . 17 ((A 𝜔 (B ·𝑜 y) 𝜔 B 𝜔) → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))
3533, 34syl3an2 1168 . . . . . . . . . . . . . . . 16 ((A 𝜔 (B 𝜔 y 𝜔) B 𝜔) → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))
36353exp 1102 . . . . . . . . . . . . . . 15 (A 𝜔 → ((B 𝜔 y 𝜔) → (B 𝜔 → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))))
3736expd 245 . . . . . . . . . . . . . 14 (A 𝜔 → (B 𝜔 → (y 𝜔 → (B 𝜔 → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B))))))
3837com34 77 . . . . . . . . . . . . 13 (A 𝜔 → (B 𝜔 → (B 𝜔 → (y 𝜔 → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B))))))
3938pm2.43d 44 . . . . . . . . . . . 12 (A 𝜔 → (B 𝜔 → (y 𝜔 → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))))
40393imp 1097 . . . . . . . . . . 11 ((A 𝜔 B 𝜔 y 𝜔) → (A ·𝑜 ((B ·𝑜 y) +𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))
4132, 40eqtrd 2069 . . . . . . . . . 10 ((A 𝜔 B 𝜔 y 𝜔) → (A ·𝑜 (B ·𝑜 suc y)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B)))
4229, 41eqeq12d 2051 . . . . . . . . 9 ((A 𝜔 B 𝜔 y 𝜔) → (((A ·𝑜 B) ·𝑜 suc y) = (A ·𝑜 (B ·𝑜 suc y)) ↔ (((A ·𝑜 B) ·𝑜 y) +𝑜 (A ·𝑜 B)) = ((A ·𝑜 (B ·𝑜 y)) +𝑜 (A ·𝑜 B))))
4326, 42syl5ibr 145 . . . . . . . 8 ((A 𝜔 B 𝜔 y 𝜔) → (((A ·𝑜 B) ·𝑜 y) = (A ·𝑜 (B ·𝑜 y)) → ((A ·𝑜 B) ·𝑜 suc y) = (A ·𝑜 (B ·𝑜 suc y))))
44433exp 1102 . . . . . . 7 (A 𝜔 → (B 𝜔 → (y 𝜔 → (((A ·𝑜 B) ·𝑜 y) = (A ·𝑜 (B ·𝑜 y)) → ((A ·𝑜 B) ·𝑜 suc y) = (A ·𝑜 (B ·𝑜 suc y))))))
4544com3r 73 . . . . . 6 (y 𝜔 → (A 𝜔 → (B 𝜔 → (((A ·𝑜 B) ·𝑜 y) = (A ·𝑜 (B ·𝑜 y)) → ((A ·𝑜 B) ·𝑜 suc y) = (A ·𝑜 (B ·𝑜 suc y))))))
4645impd 242 . . . . 5 (y 𝜔 → ((A 𝜔 B 𝜔) → (((A ·𝑜 B) ·𝑜 y) = (A ·𝑜 (B ·𝑜 y)) → ((A ·𝑜 B) ·𝑜 suc y) = (A ·𝑜 (B ·𝑜 suc y)))))
479, 13, 17, 25, 46finds2 4267 . . . 4 (x 𝜔 → ((A 𝜔 B 𝜔) → ((A ·𝑜 B) ·𝑜 x) = (A ·𝑜 (B ·𝑜 x))))
485, 47vtoclga 2613 . . 3 (𝐶 𝜔 → ((A 𝜔 B 𝜔) → ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶))))
4948expdcom 1328 . 2 (A 𝜔 → (B 𝜔 → (𝐶 𝜔 → ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶)))))
50493imp 1097 1 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A ·𝑜 B) ·𝑜 𝐶) = (A ·𝑜 (B ·𝑜 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = 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-bnd 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:  mulasspig  6316  enq0tr  6416  addcmpblnq0  6425  mulcmpblnq0  6426  mulcanenq0ec  6427  distrnq0  6441  addassnq0  6444
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