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Theorem nndir 5984
Description: Distributive law for natural numbers (right-distributivity). (Contributed by Jim Kingdon, 3-Dec-2019.)
Assertion
Ref Expression
nndir ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A +𝑜 B) ·𝑜 𝐶) = ((A ·𝑜 𝐶) +𝑜 (B ·𝑜 𝐶)))

Proof of Theorem nndir
StepHypRef Expression
1 nndi 5980 . . 3 ((𝐶 𝜔 A 𝜔 B 𝜔) → (𝐶 ·𝑜 (A +𝑜 B)) = ((𝐶 ·𝑜 A) +𝑜 (𝐶 ·𝑜 B)))
213coml 1097 . 2 ((A 𝜔 B 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 (A +𝑜 B)) = ((𝐶 ·𝑜 A) +𝑜 (𝐶 ·𝑜 B)))
3 nnacl 5974 . . . . 5 ((A 𝜔 B 𝜔) → (A +𝑜 B) 𝜔)
4 nnmcom 5983 . . . . 5 ((𝐶 𝜔 (A +𝑜 B) 𝜔) → (𝐶 ·𝑜 (A +𝑜 B)) = ((A +𝑜 B) ·𝑜 𝐶))
53, 4sylan2 270 . . . 4 ((𝐶 𝜔 (A 𝜔 B 𝜔)) → (𝐶 ·𝑜 (A +𝑜 B)) = ((A +𝑜 B) ·𝑜 𝐶))
65ancoms 255 . . 3 (((A 𝜔 B 𝜔) 𝐶 𝜔) → (𝐶 ·𝑜 (A +𝑜 B)) = ((A +𝑜 B) ·𝑜 𝐶))
763impa 1085 . 2 ((A 𝜔 B 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 (A +𝑜 B)) = ((A +𝑜 B) ·𝑜 𝐶))
8 nnmcom 5983 . . . . 5 ((𝐶 𝜔 A 𝜔) → (𝐶 ·𝑜 A) = (A ·𝑜 𝐶))
98ancoms 255 . . . 4 ((A 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 A) = (A ·𝑜 𝐶))
1093adant2 911 . . 3 ((A 𝜔 B 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 A) = (A ·𝑜 𝐶))
11 nnmcom 5983 . . . . 5 ((𝐶 𝜔 B 𝜔) → (𝐶 ·𝑜 B) = (B ·𝑜 𝐶))
1211ancoms 255 . . . 4 ((B 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 B) = (B ·𝑜 𝐶))
13123adant1 910 . . 3 ((A 𝜔 B 𝜔 𝐶 𝜔) → (𝐶 ·𝑜 B) = (B ·𝑜 𝐶))
1410, 13oveq12d 5454 . 2 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((𝐶 ·𝑜 A) +𝑜 (𝐶 ·𝑜 B)) = ((A ·𝑜 𝐶) +𝑜 (B ·𝑜 𝐶)))
152, 7, 143eqtr3d 2062 1 ((A 𝜔 B 𝜔 𝐶 𝜔) → ((A +𝑜 B) ·𝑜 𝐶) = ((A ·𝑜 𝐶) +𝑜 (B ·𝑜 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   = wceq 1228   wcel 1374  𝜔com 4240  (class class class)co 5436   +𝑜 coa 5913   ·𝑜 comu 5914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921
This theorem is referenced by:  addassnq0  6317
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