MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  distrpi Structured version   Visualization version   GIF version

Theorem distrpi 9758
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 9738 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 9738 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 9738 . . . 4 (𝐶N𝐶 ∈ ω)
4 nndi 7748 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
51, 2, 3, 4syl3an 1408 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
6 addclpi 9752 . . . . . 6 ((𝐵N𝐶N) → (𝐵 +N 𝐶) ∈ N)
7 mulpiord 9745 . . . . . 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 9744 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 +N 𝐶) = (𝐵 +𝑜 𝐶))
109oveq2d 6706 . . . . . 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 2685 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
13123impb 1279 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
14 mulclpi 9753 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
15 mulclpi 9753 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
16 addpiord 9744 . . . . . 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 9745 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵))
19 mulpiord 9745 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
2018, 19oveqan12d 6709 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +𝑜 (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
2117, 20eqtrd 2685 . . . 4 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
22213impdi 1421 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
235, 13, 223eqtr4d 2695 . 2 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
24 dmaddpi 9750 . . 3 dom +N = (N × N)
25 0npi 9742 . . 3 ¬ ∅ ∈ N
26 dmmulpi 9751 . . 3 dom ·N = (N × N)
2724, 25, 26ndmovdistr 6865 . 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 383  w3a 1054   = wceq 1523  wcel 2030  (class class class)co 6690  ωcom 7107   +𝑜 coa 7602   ·𝑜 comu 7603  Ncnpi 9704   +N cpli 9705   ·N cmi 9706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610  df-ni 9732  df-pli 9733  df-mi 9734
This theorem is referenced by:  adderpqlem  9814  addassnq  9818  distrnq  9821  ltanq  9831  ltexnq  9835
  Copyright terms: Public domain W3C validator