Theorem nndi 6749

Description: Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nndi	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Proof of Theorem nndi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
2	1	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝐶)))
3		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
4	3	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
5	2, 4	eqeq12d 2253	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
6	5	imbi2d 230	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))))
7		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
8	7	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o ∅)))
9		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
10	9	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
11	8, 10	eqeq12d 2253	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅))))
12		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
13	12	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝑦)))
14		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
15	14	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))
16	13, 15	eqeq12d 2253	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦))))
17		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
18	17	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o suc 𝑦)))
19		oveq2 6083	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
20	19	oveq2d 6091	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))
21	18, 20	eqeq12d 2253	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) ↔ (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))
22		nna0 6737	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
23	22	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o ∅) = 𝐵)
24	23	oveq2d 6091	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = (𝐴 ·o 𝐵))
25		nnmcl 6744	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
26		nna0 6737	. . . . . . . 8 ⊢ ((𝐴 ·o 𝐵) ∈ ω → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
27	25, 26	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
28	24, 27	eqtr4d 2274	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
29		nnm0 6738	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
30	29	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
31	30	oveq2d 6091	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
32	28, 31	eqtr4d 2274	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
33		oveq1 6082	. . . . . . . . 9 ⊢ ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴))
34		nnasuc 6739	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
35	34	3adant1 1046	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
36	35	oveq2d 6091	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = (𝐴 ·o suc (𝐵 +o 𝑦)))
37		nnacl 6743	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
38		nnmsuc 6740	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ (𝐵 +o 𝑦) ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
39	37, 38	sylan2 286	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
40	39	3impb 1230	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
41	36, 40	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
42		nnmsuc 6740	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
43	42	3adant2 1047	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
44	43	oveq2d 6091	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
45		nnmcl 6744	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
46		nnaass 6748	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
47	25, 46	syl3an1 1311	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝑦) ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
48	45, 47	syl3an2 1312	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝐴 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
49	48	3exp 1233	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))
50	49	exp4b 367	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐴 ∈ ω → (𝑦 ∈ ω → (𝐴 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))))
51	50	pm2.43a 51	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (𝐴 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))
52	51	com4r 86	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴))))))
53	52	pm2.43i 49	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))))
54	53	3imp 1224	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴) = ((𝐴 ·o 𝐵) +o ((𝐴 ·o 𝑦) +o 𝐴)))
55	44, 54	eqtr4d 2274	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴))
56	41, 55	eqeq12d 2253	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)) ↔ ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴) = (((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) +o 𝐴)))
57	33, 56	imbitrrid 156	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))
58	57	3exp 1233	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝑦 ∈ ω → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))))
59	58	com3r 79	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → (𝐵 ∈ ω → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦))))))
60	59	impd 254	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o (𝐵 +o 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))))
61	11, 16, 21, 32, 60	finds2 4743	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
62	6, 61	vtoclga 2889	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
63	62	expdcom 1492	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))))
64	63	3imp 1224	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∅c0 3520  suc csuc 4505  ωcom 4732  (class class class)co 6075   +_o coa 6674   ·_o comu 6675

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682

This theorem is referenced by:  nnmass  6750  nndir  6753  distrpig  7690  addcmpblnq0  7800  nnanq0  7815  distrnq0  7816  addassnq0  7819