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Theorem distrpi 9664
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
StepHypRef Expression
1 pinn 9644 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 9644 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 9644 . . . 4 (𝐶N𝐶 ∈ ω)
4 nndi 7648 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
51, 2, 3, 4syl3an 1365 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
6 addclpi 9658 . . . . . 6 ((𝐵N𝐶N) → (𝐵 +N 𝐶) ∈ N)
7 mulpiord 9651 . . . . . 6 ((𝐴N ∧ (𝐵 +N 𝐶) ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +N 𝐶)))
86, 7sylan2 491 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +N 𝐶)))
9 addpiord 9650 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 +N 𝐶) = (𝐵 +𝑜 𝐶))
109oveq2d 6620 . . . . . 6 ((𝐵N𝐶N) → (𝐴 ·𝑜 (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
1110adantl 482 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·𝑜 (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
128, 11eqtrd 2655 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
13123impb 1257 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
14 mulclpi 9659 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
15 mulclpi 9659 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
16 addpiord 9650 . . . . . 6 (((𝐴 ·N 𝐵) ∈ N ∧ (𝐴 ·N 𝐶) ∈ N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)))
1714, 15, 16syl2an 494 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)))
18 mulpiord 9651 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
19 mulpiord 9651 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
2018, 19oveqan12d 6623 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
2117, 20eqtrd 2655 . . . 4 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
22213impdi 1378 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
235, 13, 223eqtr4d 2665 . 2 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
24 dmaddpi 9656 . . 3 dom +N = (N × N)
25 0npi 9648 . . 3 ¬ ∅ ∈ N
26 dmmulpi 9657 . . 3 dom ·N = (N × N)
2724, 25, 26ndmovdistr 6776 . 2 (¬ (𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
2823, 27pm2.61i 176 1 (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1036   = wceq 1480  wcel 1987  (class class class)co 6604  ωcom 7012   +𝑜 coa 7502   ·𝑜 comu 7503  Ncnpi 9610   +N cpli 9611   ·N cmi 9612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510  df-ni 9638  df-pli 9639  df-mi 9640
This theorem is referenced by:  adderpqlem  9720  addassnq  9724  distrnq  9727  ltanq  9737  ltexnq  9741
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