Theorem distrpi 9720

Description: Multiplication of positive integers is distributive. (Contributed by NM, 21-Sep-1995.) (New usage is discouraged.)

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		pinn 9700	. . . 4 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)
2		pinn 9700	. . . 4 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
3		pinn 9700	. . . 4 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω)
4		nndi 7703	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
5	1, 2, 3, 4	syl3an 1368	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
6		addclpi 9714	. . . . . 6 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 +N 𝐶) ∈ N)
7		mulpiord 9707	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ (𝐵 +N 𝐶) ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +N 𝐶)))
8	6, 7	sylan2 491	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +N 𝐶)))
9		addpiord 9706	. . . . . . 7 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 +N 𝐶) = (𝐵 +𝑜 𝐶))
10	9	oveq2d 6666	. . . . . 6 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·𝑜 (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
11	10	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·𝑜 (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
12	8, 11	eqtrd 2656	. . . 4 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
13	12	3impb 1260	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
14		mulclpi 9715	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N)
15		mulclpi 9715	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N)
16		addpiord 9706	. . . . . 6 ⊢ (((𝐴 ·N 𝐵) ∈ N ∧ (𝐴 ·N 𝐶) ∈ N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)))
17	14, 15, 16	syl2an 494	. . . . 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 9707	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
19		mulpiord 9707	. . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
20	18, 19	oveqan12d 6669	. . . . 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 2656	. . . 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 1381	. . 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 2666	. 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
24		dmaddpi 9712	. . 3 ⊢ dom +N = (N × N)
25		0npi 9704	. . 3 ⊢ ¬ ∅ ∈ N
26		dmmulpi 9713	. . 3 ⊢ dom ·N = (N × N)
27	24, 25, 26	ndmovdistr 6823	. 2 ⊢ (¬ (𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
28	23, 27	pm2.61i 176	1 ⊢ (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))

Syntax hints:   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557   ·_𝑜 comu 7558  Ncnpi 9666   +_N cpli 9667   ·_N cmi 9668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565  df-ni 9694  df-pli 9695  df-mi 9696

This theorem is referenced by:  adderpqlem  9776  addassnq  9780  distrnq  9783  ltanq  9793  ltexnq  9797