Theorem mulasspi 10881

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

Step	Hyp	Ref	Expression
1		pinn 10862	. . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		pinn 10862	. . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
3		pinn 10862	. . . 4 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
4		nnmass 8609	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
5	1, 2, 3, 4	syl3an 1176	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶)))
6		mulclpi 10877	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
7		mulpiord 10869	. . . . . 6 ⊢ (((𝐴 ·N 𝐵) ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·o 𝐶))
8	6, 7	sylan 591	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·N 𝐵) ·o 𝐶))
9		mulpiord 10869	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
10	9	oveq1d 7425	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
11	10	adantr 485	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
12	8, 11	eqtrd 2796	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
13	12	3impa 1125	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = ((𝐴 ·o 𝐵) ·o 𝐶))
14		mulclpi 10877	. . . . . 6 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 ·N 𝐶) ∈ N)
15		mulpiord 10869	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ (𝐵 ·N 𝐶) ∈ N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·N 𝐶)))
16	14, 15	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·N 𝐶)))
17		mulpiord 10869	. . . . . . 7 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 ·N 𝐶) = (𝐵 ·o 𝐶))
18	17	oveq2d 7426	. . . . . 6 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·o (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
19	18	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·o (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
20	16, 19	eqtrd 2796	. . . 4 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
21	20	3impb 1130	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 ·N 𝐶)) = (𝐴 ·o (𝐵 ·o 𝐶)))
22	5, 13, 21	3eqtr4d 2806	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
23		dmmulpi 10875	. . 3 ⊢ dom ·N = (N × N)
24		0npi 10866	. . 3 ⊢ ¬ ∅ ∈ N
25	23, 24	ndmovass 7598	. 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶)))
26	22, 25	pm2.61i 184	1 ⊢ ((𝐴 ·N 𝐵) ·N 𝐶) = (𝐴 ·N (𝐵 ·N 𝐶))

Syntax hints:   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  (class class class)co 7410  ωcom 7861   ·_o comu 8450  Ncnpi 10828   ·_N cmi 10830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456  df-omul 8457  df-ni 10856  df-mi 10858

This theorem is referenced by:  enqer  10905  adderpqlem  10938  mulerpqlem  10939  addassnq  10942  mulassnq  10943  mulcanenq  10944  distrnq  10945  ltsonq  10953  lterpq  10954  ltanq  10955  ltmnq  10956  ltexnq  10959