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Theorem distrpi 10309
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 10289 . . . 4 (𝐴N𝐴 ∈ ω)
2 pinn 10289 . . . 4 (𝐵N𝐵 ∈ ω)
3 pinn 10289 . . . 4 (𝐶N𝐶 ∈ ω)
4 nndi 8239 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
51, 2, 3, 4syl3an 1154 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
6 addclpi 10303 . . . . . 6 ((𝐵N𝐶N) → (𝐵 +N 𝐶) ∈ N)
7 mulpiord 10296 . . . . . 6 ((𝐴N ∧ (𝐵 +N 𝐶) ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +N 𝐶)))
86, 7sylan2 592 . . . . 5 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +N 𝐶)))
9 addpiord 10295 . . . . . . 7 ((𝐵N𝐶N) → (𝐵 +N 𝐶) = (𝐵 +o 𝐶))
109oveq2d 7164 . . . . . 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 2861 . . . 4 ((𝐴N ∧ (𝐵N𝐶N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +o 𝐶)))
13123impb 1109 . . 3 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +o 𝐶)))
14 mulclpi 10304 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) ∈ N)
15 mulclpi 10304 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
16 addpiord 10295 . . . . . 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 595 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +o (𝐴 ·N 𝐶)))
18 mulpiord 10296 . . . . . 6 ((𝐴N𝐵N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵))
19 mulpiord 10296 . . . . . 6 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶))
2018, 19oveqan12d 7167 . . . . 5 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +o (𝐴 ·N 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
2117, 20eqtrd 2861 . . . 4 (((𝐴N𝐵N) ∧ (𝐴N𝐶N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
22213impdi 1344 . . 3 ((𝐴N𝐵N𝐶N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
235, 13, 223eqtr4d 2871 . 2 ((𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
24 dmaddpi 10301 . . 3 dom +N = (N × N)
25 0npi 10293 . . 3 ¬ ∅ ∈ N
26 dmmulpi 10302 . . 3 dom ·N = (N × N)
2724, 25, 26ndmovdistr 7327 . 2 (¬ (𝐴N𝐵N𝐶N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)))
2823, 27pm2.61i 183 1 (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1081   = wceq 1530  wcel 2107  (class class class)co 7148  ωcom 7568   +o coa 8090   ·o comu 8091  Ncnpi 10255   +N cpli 10256   ·N cmi 10257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-oadd 8097  df-omul 8098  df-ni 10283  df-pli 10284  df-mi 10285
This theorem is referenced by:  adderpqlem  10365  addassnq  10369  distrnq  10372  ltanq  10382  ltexnq  10386
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