Theorem nndir 6599

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 6595	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
2	1	3coml 1213	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
3		nnacl 6589	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
4		nnmcom 6598	. . . . 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 1197	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
8		nnmcom 6598	. . . . 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 1019	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
11		nnmcom 6598	. . . . 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 1018	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
14	10, 13	oveq12d 5985	. 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 2248	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 981   = wceq 1373   ∈ wcel 2178  ωcom 4656  (class class class)co 5967   +_o coa 6522   ·_o comu 6523

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-oadd 6529  df-omul 6530

This theorem is referenced by:  addassnq0  7610