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Theorem mulasspi 10371
 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 10352 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 10352 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 10352 . . . 4 (𝐶N𝐶 ∈ ω)
4 nnmass 8267 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
51, 2, 3, 4syl3an 1158 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
6 mulclpi 10367 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
7 mulpiord 10359 . . . . . 6 (((𝐴 ·N 𝐵) ∈ N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·o 𝐶))
86, 7sylan 583 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·o 𝐶))
9 mulpiord 10359 . . . . . . 7 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
109oveq1d 7172 . . . . . 6 ((𝐴N𝐵N) → ((𝐴 ·N 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
1110adantr 484 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
128, 11eqtrd 2794 . . . 4 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
13123impa 1108 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
14 mulclpi 10367 . . . . . 6 ((𝐵N𝐶N) → (𝐵 ·N 𝐶) ∈ N)
15 mulpiord 10359 . . . . . 6 ((𝐴N ∧ (𝐵 ·N 𝐶) ∈ N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·N 𝐶)))
1614, 15sylan2 595 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·N 𝐶)))
17 mulpiord 10359 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 ·N 𝐶) = (𝐵 ·o 𝐶))
1817oveq2d 7173 . . . . . 6 ((𝐵N𝐶N) → (𝐴 ·o (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
1918adantl 485 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·o (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
2016, 19eqtrd 2794 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
21203impb 1113 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
225, 13, 213eqtr4d 2804 . 2 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
23 dmmulpi 10365 . . 3 dom ·N = (N × N)
24 0npi 10356 . . 3 ¬ ∅ ∈ N
2523, 24ndmovass 7339 . 2 (¬ (𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
2622, 25pm2.61i 185 1 ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  (class class class)co 7157  ωcom 7586   ·o comu 8117  Ncnpi 10318   ·N cmi 10320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-oadd 8123  df-omul 8124  df-ni 10346  df-mi 10348 This theorem is referenced by:  enqer  10395  adderpqlem  10428  mulerpqlem  10429  addassnq  10432  mulassnq  10433  mulcanenq  10434  distrnq  10435  ltsonq  10443  lterpq  10444  ltanq  10445  ltmnq  10446  ltexnq  10449
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