Theorem distrpi 3820

Description: Multiplication of positive integers is distributive.

Hypotheses

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

Assertion

Ref	Expression
distrpi	⊢ (A ·N (B +N C)) = ((A ·N B) +N (A ·N C))

Proof of Theorem distrpi

Step	Hyp	Ref	Expression
1		nndi 3180	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ ω ∧ C ∈ ω) → (A ·o (B +o C)) = ((A ·o B) +o (A ·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 (A ·o C)))
6		mulpiord 3807	. . . . . 6 ⊢ ((A ∈ N ∧ (B +N C) ∈ N) → (A ·N (B +N C)) = (A ·o (B +N C)))
7		addclpi 3814	. . . . . 6 ⊢ ((B ∈ N ∧ C ∈ N) → (B +N C) ∈ N)
8	6, 7	sylan2 346	. . . . 5 ⊢ ((A ∈ N ∧ (B ∈ N ∧ C ∈ N)) → (A ·N (B +N C)) = (A ·o (B +N C)))
9		addpiord 3806	. . . . . . 7 ⊢ ((B ∈ N ∧ C ∈ N) → (B +N C) = (B +o C))
10	9	opreq2d 3013	. . . . . 6 ⊢ ((B ∈ N ∧ C ∈ N) → (A ·o (B +N C)) = (A ·o (B +o C)))
11	10	adantl 305	. . . . 5 ⊢ ((A ∈ N ∧ (B ∈ N ∧ C ∈ N)) → (A ·o (B +N 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	3impb 610	. . 3 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → (A ·N (B +N C)) = (A ·o (B +o C)))
14		addpiord 3806	. . . . . 6 ⊢ (((A ·N B) ∈ N ∧ (A ·N C) ∈ N) → ((A ·N B) +N (A ·N C)) = ((A ·N B) +o (A ·N C)))
15		mulclpi 3815	. . . . . 6 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) ∈ N)
16		mulclpi 3815	. . . . . 6 ⊢ ((A ∈ N ∧ C ∈ N) → (A ·N C) ∈ N)
17	14, 15, 16	syl2an 349	. . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (A ∈ N ∧ C ∈ N)) → ((A ·N B) +N (A ·N C)) = ((A ·N B) +o (A ·N C)))
18		mulpiord 3807	. . . . . 6 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) = (A ·o B))
19		mulpiord 3807	. . . . . 6 ⊢ ((A ∈ N ∧ C ∈ N) → (A ·N C) = (A ·o C))
20	18, 19	opreqan12d 3015	. . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (A ∈ N ∧ C ∈ N)) → ((A ·N B) +o (A ·N C)) = ((A ·o B) +o (A ·o C)))
21	17, 20	eqtrd 1128	. . . 4 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (A ∈ N ∧ C ∈ N)) → ((A ·N B) +N (A ·N C)) = ((A ·o B) +o (A ·o C)))
22	21	3impdi 630	. . 3 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → ((A ·N B) +N (A ·N C)) = ((A ·o B) +o (A ·o C)))
23	5, 13, 22	3eqtr4d 1134	. 2 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → (A ·N (B +N C)) = ((A ·N B) +N (A ·N C)))
24		distrpi.1	. . 3 ⊢ B ∈ V
25		dmaddpi 3812	. . 3 ⊢ dom +N = (N × N)
26		distrpi.2	. . 3 ⊢ C ∈ V
27		0npi 3804	. . 3 ⊢ ¬ ∅ ∈ N
28		dmmulpi 3813	. . 3 ⊢ dom ·N = (N × N)
29	24, 25, 26, 27, 28	ndmoprdistr 3063	. 2 ⊢ (¬ (A ∈ N ∧ B ∈ N ∧ C ∈ N) → (A ·N (B +N C)) = ((A ·N B) +N (A ·N C)))
30	23, 29	pm2.61i 110	1 ⊢ (A ·N (B +N C)) = ((A ·N B) +N (A ·N C))

Syntax hints:   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ωcom 2372  (class class class)co 3001   +_o coa 3101   ·_o comu 3102  Ncnpi 3766   +_N cpli 3767   ·_N cmi 3768

This theorem is referenced by:  addcmpblnq 3846  addasspq 3857  distrpq 3861  ltapq 3870  ltexpq 3874  halfpq 3876

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-pli 3795  df-mi 3796

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