Theorem nndi 6489

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 5885	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
2	1	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝐶)))
3		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
4	3	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
5	2, 4	eqeq12d 2192	. . . . 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 5885	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
8	7	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o ∅)))
9		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
10	9	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
11	8, 10	eqeq12d 2192	. . . . 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 5885	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
13	12	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝑦)))
14		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
15	14	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))
16	13, 15	eqeq12d 2192	. . . . 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 5885	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
18	17	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o suc 𝑦)))
19		oveq2 5885	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
20	19	oveq2d 5893	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))
21	18, 20	eqeq12d 2192	. . . . 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 6477	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
23	22	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o ∅) = 𝐵)
24	23	oveq2d 5893	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = (𝐴 ·o 𝐵))
25		nnmcl 6484	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
26		nna0 6477	. . . . . . . 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 2213	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
29		nnm0 6478	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
30	29	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
31	30	oveq2d 5893	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
32	28, 31	eqtr4d 2213	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
33		oveq1 5884	. . . . . . . . 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 6479	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
35	34	3adant1 1015	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
36	35	oveq2d 5893	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = (𝐴 ·o suc (𝐵 +o 𝑦)))
37		nnacl 6483	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
38		nnmsuc 6480	. . . . . . . . . . . . 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 1199	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
41	36, 40	eqtrd 2210	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
42		nnmsuc 6480	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
43	42	3adant2 1016	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
44	43	oveq2d 5893	. . . . . . . . . . 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 6484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
46		nnaass 6488	. . . . . . . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . . . . . . 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 1272	. . . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . 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 2213	. . . . . . . . . 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 2192	. . . . . . . . 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 1202	. . . . . . 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 4602	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
62	6, 61	vtoclga 2805	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
63	62	expdcom 1442	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))))
64	63	3imp 1193	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 978   = wceq 1353   ∈ wcel 2148  ∅c0 3424  suc csuc 4367  ωcom 4591  (class class class)co 5877   +_o coa 6416   ·_o comu 6417

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424

This theorem is referenced by:  nnmass  6490  nndir  6493  distrpig  7334  addcmpblnq0  7444  nnanq0  7459  distrnq0  7460  addassnq0  7463