Theorem nndir 6458

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 6454	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
2	1	3coml 1200	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐶 ·o 𝐴) +o (𝐶 ·o 𝐵)))
3		nnacl 6448	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω)
4		nnmcom 6457	. . . . 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 1184	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o (𝐴 +o 𝐵)) = ((𝐴 +o 𝐵) ·o 𝐶))
8		nnmcom 6457	. . . . 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 1006	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐴) = (𝐴 ·o 𝐶))
11		nnmcom 6457	. . . . 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 1005	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 ·o 𝐵) = (𝐵 ·o 𝐶))
14	10, 13	oveq12d 5860	. 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 2206	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 968   = wceq 1343   ∈ wcel 2136  ωcom 4567  (class class class)co 5842   +_o coa 6381   ·_o comu 6382

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389

This theorem is referenced by:  addassnq0  7403