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Theorem mulasspi 9679
Description: Multiplication of positive integers is associative. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulasspi ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))

Proof of Theorem mulasspi
StepHypRef Expression
1 pinn 9660 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 9660 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 9660 . . . 4 (𝐶N𝐶 ∈ ω)
4 nnmass 7664 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
51, 2, 3, 4syl3an 1365 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
6 mulclpi 9675 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
7 mulpiord 9667 . . . . . 6 (((𝐴 ·N 𝐵) ∈ N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·𝑜 𝐶))
86, 7sylan 488 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·𝑜 𝐶))
9 mulpiord 9667 . . . . . . 7 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
109oveq1d 6630 . . . . . 6 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
1110adantr 481 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
128, 11eqtrd 2655 . . . 4 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
13123impa 1256 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶))
14 mulclpi 9675 . . . . . 6 ((𝐵N𝐶N) → (𝐵 ·N 𝐶) ∈ N)
15 mulpiord 9667 . . . . . 6 ((𝐴N ∧ (𝐵 ·N 𝐶) ∈ N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·𝑜 (𝐵 ·N 𝐶)))
1614, 15sylan2 491 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·𝑜 (𝐵 ·N 𝐶)))
17 mulpiord 9667 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 ·N 𝐶) = (𝐵 ·𝑜 𝐶))
1817oveq2d 6631 . . . . . 6 ((𝐵N𝐶N) → (𝐴 ·𝑜 (𝐵 ·N 𝐶)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
1918adantl 482 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·𝑜 (𝐵 ·N 𝐶)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
2016, 19eqtrd 2655 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
21203impb 1257 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·𝑜 (𝐵 ·𝑜 𝐶)))
225, 13, 213eqtr4d 2665 . 2 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
23 dmmulpi 9673 . . 3 dom ·N = (N × N)
24 0npi 9664 . . 3 ¬ ∅ ∈ N
2523, 24ndmovass 6787 . 2 (¬ (𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
2622, 25pm2.61i 176 1 ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1036   = wceq 1480  wcel 1987  (class class class)co 6615  ωcom 7027   ·𝑜 comu 7518  Ncnpi 9626   ·N cmi 9628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-omul 7525  df-ni 9654  df-mi 9656
This theorem is referenced by:  enqer  9703  adderpqlem  9736  mulerpqlem  9737  addassnq  9740  mulassnq  9741  mulcanenq  9742  distrnq  9743  ltsonq  9751  lterpq  9752  ltanq  9753  ltmnq  9754  ltexnq  9757
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