Theorem nndir 6469

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 6465	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
2	1	3coml 1205	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
3		nnacl 6459	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
4		nnmcom 6468	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ (𝐴 +o 𝐵) ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
5	3, 4	sylan2 284	. . . 4 ⊢ ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
6	5	ancoms 266	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
7	6	3impa 1189	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
8		nnmcom 6468	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
9	8	ancoms 266	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
10	9	3adant2 1011	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
11		nnmcom 6468	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
12	11	ancoms 266	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
13	12	3adant1 1010	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
14	10, 13	oveq12d 5871	. 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 2211	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) ·o 𝐶) = ((𝐴 ·o 𝐶) +o (𝐵 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ωcom 4574  (class class class)co 5853   +_o coa 6392   ·_o comu 6393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400

This theorem is referenced by:  addassnq0  7424