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Theorem mulasspi 10172
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 10153 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 10153 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 10153 . . . 4 (𝐶N𝐶 ∈ ω)
4 nnmass 8107 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
51, 2, 3, 4syl3an 1153 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
6 mulclpi 10168 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
7 mulpiord 10160 . . . . . 6 (((𝐴 ·N 𝐵) ∈ N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·o 𝐶))
86, 7sylan 580 . . . . 5 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·o 𝐶))
9 mulpiord 10160 . . . . . . 7 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
109oveq1d 7038 . . . . . 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 2833 . . . 4 (((𝐴N𝐵N) ∧ 𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
13123impa 1103 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
14 mulclpi 10168 . . . . . 6 ((𝐵N𝐶N) → (𝐵 ·N 𝐶) ∈ N)
15 mulpiord 10160 . . . . . 6 ((𝐴N ∧ (𝐵 ·N 𝐶) ∈ N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·N 𝐶)))
1614, 15sylan2 592 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·N 𝐶)))
17 mulpiord 10160 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 ·N 𝐶) = (𝐵 ·o 𝐶))
1817oveq2d 7039 . . . . . 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 2833 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
21203impb 1108 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
225, 13, 213eqtr4d 2843 . 2 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
23 dmmulpi 10166 . . 3 dom ·N = (N × N)
24 0npi 10157 . . 3 ¬ ∅ ∈ N
2523, 24ndmovass 7199 . 2 (¬ (𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
2622, 25pm2.61i 183 1 ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1080   = wceq 1525  wcel 2083  (class class class)co 7023  ωcom 7443   ·o comu 7958  Ncnpi 10119   ·N cmi 10121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-oadd 7964  df-omul 7965  df-ni 10147  df-mi 10149
This theorem is referenced by:  enqer  10196  adderpqlem  10229  mulerpqlem  10230  addassnq  10233  mulassnq  10234  mulcanenq  10235  distrnq  10236  ltsonq  10244  lterpq  10245  ltanq  10246  ltmnq  10247  ltexnq  10250
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