Theorem mulasspi 3819

Description: Multiplication of positive integers is associative.

Hypotheses

Ref	Expression
mulasspi.1	⊢ B ∈ V
mulasspi.2	⊢ C ∈ V

Assertion

Ref	Expression
mulasspi	⊢ ((A ·N B) ·N C) = (A ·N (B ·N C))

Proof of Theorem mulasspi

Step	Hyp	Ref	Expression
1		nnmass 3181	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → ((A ·o B) ·o C) = (A ·o (B ·o C)))
2		pinn 3800	. . . 4 ⊢ (A ∈ N → A ∈ ω)
3		pinn 3800	. . . 4 ⊢ (B ∈ N → B ∈ ω)
4		pinn 3800	. . . 4 ⊢ (C ∈ N → C ∈ ω)
5	1, 2, 3, 4	syl3an 628	. . 3 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → ((A ·o B) ·o C) = (A ·o (B ·o C)))
6		mulpiord 3807	. . . . . 6 ⊢ (((A ·N B) ∈ N ∧ C ∈ N) → ((A ·N B) ·N C) = ((A ·N B) ·o C))
7		mulclpi 3815	. . . . . 6 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) ∈ N)
8	6, 7	sylan 343	. . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ C ∈ N) → ((A ·N B) ·N C) = ((A ·N B) ·o C))
9		mulpiord 3807	. . . . . . 7 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) = (A ·o B))
10	9	opreq1d 3012	. . . . . 6 ⊢ ((A ∈ N ∧ B ∈ N) → ((A ·N B) ·o C) = ((A ·o B) ·o C))
11	10	adantr 306	. . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ C ∈ N) → ((A ·N B) ·o C) = ((A ·o B) ·o C))
12	8, 11	eqtrd 1128	. . . 4 ⊢ (((A ∈ N ∧ B ∈ N) ∧ C ∈ N) → ((A ·N B) ·N C) = ((A ·o B) ·o C))
13	12	3impa 609	. . 3 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → ((A ·N B) ·N C) = ((A ·o B) ·o C))
14		mulpiord 3807	. . . . . 6 ⊢ ((A ∈ N ∧ (B ·N C) ∈ N) → (A ·N (B ·N C)) = (A ·o (B ·N C)))
15		mulclpi 3815	. . . . . 6 ⊢ ((B ∈ N ∧ C ∈ N) → (B ·N C) ∈ N)
16	14, 15	sylan2 346	. . . . 5 ⊢ ((A ∈ N ∧ (B ∈ N ∧ C ∈ N)) → (A ·N (B ·N C)) = (A ·o (B ·N C)))
17		mulpiord 3807	. . . . . . 7 ⊢ ((B ∈ N ∧ C ∈ N) → (B ·N C) = (B ·o C))
18	17	opreq2d 3013	. . . . . 6 ⊢ ((B ∈ N ∧ C ∈ N) → (A ·o (B ·N C)) = (A ·o (B ·o C)))
19	18	adantl 305	. . . . 5 ⊢ ((A ∈ N ∧ (B ∈ N ∧ C ∈ N)) → (A ·o (B ·N C)) = (A ·o (B ·o C)))
20	16, 19	eqtrd 1128	. . . 4 ⊢ ((A ∈ N ∧ (B ∈ N ∧ C ∈ N)) → (A ·N (B ·N C)) = (A ·o (B ·o C)))
21	20	3impb 610	. . 3 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → (A ·N (B ·N C)) = (A ·o (B ·o C)))
22	5, 13, 21	3eqtr4d 1134	. 2 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → ((A ·N B) ·N C) = (A ·N (B ·N C)))
23		mulasspi.1	. . 3 ⊢ B ∈ V
24		dmmulpi 3813	. . 3 ⊢ dom ·N = (N × N)
25		mulasspi.2	. . 3 ⊢ C ∈ V
26		0npi 3804	. . 3 ⊢ ¬ ∅ ∈ N
27	23, 24, 25, 26	ndmoprass 3062	. 2 ⊢ (¬ (A ∈ N ∧ B ∈ N ∧ C ∈ N) → ((A ·N B) ·N C) = (A ·N (B ·N C)))
28	22, 27	pm2.61i 110	1 ⊢ ((A ·N B) ·N C) = (A ·N (B ·N C))

Syntax hints:   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ωcom 2372  (class class class)co 3001   ·_o comu 3102  Ncnpi 3766   ·_N cmi 3768

This theorem is referenced by:  enqer 3840  addcmpblnq 3846  mulcmpblnq 3847  ordpipq 3850  addasspq 3857  mulasspq 3859  distrpqlem 3860  distrpq 3861  ltsopq 3869  ltapq 3870  ltmpq 3871  ltexpq 3874  prlem934b 3932

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-omul 3107  df-ni 3794  df-mi 3796

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