Theorem nndir 6134

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 6130	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 ·𝑜 𝐴) +𝑜 (𝐶 ·𝑜 𝐵)))
2	1	3coml 1146	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐶 ·𝑜 𝐴) +𝑜 (𝐶 ·𝑜 𝐵)))
3		nnacl 6124	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +𝑜 𝐵) ∈ ω)
4		nnmcom 6133	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐴 +𝑜 𝐵) ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
5	3, 4	sylan2 280	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
6	5	ancoms 264	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
7	6	3impa 1134	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 (𝐴 +𝑜 𝐵)) = ((𝐴 +𝑜 𝐵) ·𝑜 𝐶))
8		nnmcom 6133	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) = (𝐴 ·𝑜 𝐶))
9	8	ancoms 264	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐴) = (𝐴 ·𝑜 𝐶))
10	9	3adant2 958	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐴) = (𝐴 ·𝑜 𝐶))
11		nnmcom 6133	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐶))
12	11	ancoms 264	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐶))
13	12	3adant1 957	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·𝑜 𝐵) = (𝐵 ·𝑜 𝐶))
14	10, 13	oveq12d 5561	. 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 2122	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) +𝑜 (𝐵 ·𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ωcom 4339  (class class class)co 5543   +_𝑜 coa 6062   ·_𝑜 comu 6063

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070

This theorem is referenced by:  addassnq0  6714