Theorem nndir 6491

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 6487	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
2	1	3coml 1210	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
3		nnacl 6481	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
4		nnmcom 6490	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐴 +o 𝐵) ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
5	3, 4	sylan2 286	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
6	5	ancoms 268	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
7	6	3impa 1194	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
8		nnmcom 6490	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
9	8	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
10	9	3adant2 1016	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
11		nnmcom 6490	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
12	11	ancoms 268	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
13	12	3adant1 1015	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
14	10, 13	oveq12d 5893	. 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 2218	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) +o (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ωcom 4590  (class class class)co 5875   +_o coa 6414   ·_o comu 6415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-oadd 6421  df-omul 6422

This theorem is referenced by:  addassnq0  7461