Theorem nndir 6069

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

Step	Hyp	Ref	Expression
1		nndi 6065	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 ·𝑜 𝐴) +𝑜 (𝐶 ·𝑜 𝐵)))
2	1	3coml 1111	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 ·𝑜 𝐴) +𝑜 (𝐶 ·𝑜 𝐵)))
3		nnacl 6059	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
4		nnmcom 6068	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐴 +𝑜 𝐵) ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
5	3, 4	sylan2 270	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
6	5	ancoms 255	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
7	6	3impa 1099	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
8		nnmcom 6068	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) = (𝐴 ·𝑜 𝐶))
9	8	ancoms 255	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐴) = (𝐴 ·𝑜 𝐶))
10	9	3adant2 923	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐴) = (𝐴 ·𝑜 𝐶))
11		nnmcom 6068	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐶))
12	11	ancoms 255	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐶))
13	12	3adant1 922	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐶))
14	10, 13	oveq12d 5530	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) +𝑜 (𝐶 ·𝑜 𝐵)) = ((𝐴 ·𝑜 𝐶) +𝑜 (𝐵 ·𝑜 𝐶)))
15	2, 7, 14	3eqtr3d 2080	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) +𝑜 (𝐵 ·𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ωcom 4313  (class class class)co 5512   +_𝑜 coa 5998   ·_𝑜 comu 5999

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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006

This theorem is referenced by:  addassnq0  6560