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Theorem distrpi 9599
 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 9579 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 9579 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 9579 . . . 4 (𝐶N𝐶 ∈ ω)
4 nndi 7590 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
51, 2, 3, 4syl3an 1360 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
6 addclpi 9593 . . . . . 6 ((𝐵N𝐶N) → (𝐵 +N 𝐶) ∈ N)
7 mulpiord 9586 . . . . . 6 ((𝐴N ∧ (𝐵 +N 𝐶) ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +N 𝐶)))
86, 7sylan2 490 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +N 𝐶)))
9 addpiord 9585 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 +N 𝐶) = (𝐵 +𝑜 𝐶))
109oveq2d 6565 . . . . . 6 ((𝐵N𝐶N) → (𝐴 ·𝑜 (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
1110adantl 481 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·𝑜 (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
128, 11eqtrd 2644 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
13123impb 1252 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
14 mulclpi 9594 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
15 mulclpi 9594 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
16 addpiord 9585 . . . . . 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 493 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)))
18 mulpiord 9586 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
19 mulpiord 9586 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
2018, 19oveqan12d 6568 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
2117, 20eqtrd 2644 . . . 4 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
22213impdi 1373 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
235, 13, 223eqtr4d 2654 . 2 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
24 dmaddpi 9591 . . 3 dom +N = (N × N)
25 0npi 9583 . . 3 ¬ ∅ ∈ N
26 dmmulpi 9592 . . 3 dom ·N = (N × N)
2724, 25, 26ndmovdistr 6721 . 2 (¬ (𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
2823, 27pm2.61i 175 1 (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  ωcom 6957   +𝑜 coa 7444   ·𝑜 comu 7445  Ncnpi 9545   +N cpli 9546   ·N cmi 9547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-ni 9573  df-pli 9574  df-mi 9575 This theorem is referenced by:  adderpqlem  9655  addassnq  9659  distrnq  9662  ltanq  9672  ltexnq  9676
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