Theorem nndi 8577

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 7389	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
2	1	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝐶)))
3		oveq2 7389	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐶))
4	3	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))
5	2, 4	eqeq12d 2768	. . . . 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 342	. . . 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 7389	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
8	7	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o ∅)))
9		oveq2 7389	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
10	9	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
11	8, 10	eqeq12d 2768	. . . . 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 7389	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
13	12	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o 𝑦)))
14		oveq2 7389	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
15	14	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑦)))
16	13, 15	eqeq12d 2768	. . . . 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 7389	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
18	17	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (𝐵 +o 𝑥)) = (𝐴 ·o (𝐵 +o suc 𝑦)))
19		oveq2 7389	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
20	19	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o suc 𝑦)))
21	18, 20	eqeq12d 2768	. . . . 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 8558	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
23	22	adantl 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o ∅) = 𝐵)
24	23	oveq2d 7397	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = (𝐴 ·o 𝐵))
25		nnmcl 8566	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω)
26		nna0 8558	. . . . . . . 8 ⊢ ((𝐴 ·o 𝐵) ∈ ω → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
27	25, 26	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o ∅) = (𝐴 ·o 𝐵))
28	24, 27	eqtr4d 2790	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
29		nnm0 8559	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
30	29	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
31	30	oveq2d 7397	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)) = ((𝐴 ·o 𝐵) +o ∅))
32	28, 31	eqtr4d 2790	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o ∅)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o ∅)))
33		oveq1 7388	. . . . . . . . 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 8560	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
35	34	3adant1 1139	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
36	35	oveq2d 7397	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = (𝐴 ·o suc (𝐵 +o 𝑦)))
37		nnacl 8565	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
38		nnmsuc 8561	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ (𝐵 +o 𝑦) ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
39	37, 38	sylan2 601	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
40	39	3impb 1123	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc (𝐵 +o 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
41	36, 40	eqtrd 2787	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o (𝐵 +o suc 𝑦)) = ((𝐴 ·o (𝐵 +o 𝑦)) +o 𝐴))
42		nnmsuc 8561	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
43	42	3adant2 1140	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
44	43	oveq2d 7397	. . . . . . . . . . 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 8566	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
46		nnaass 8576	. . . . . . . . . . . . . . . . . . 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 1172	. . . . . . . . . . . . . . . . . 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 1173	. . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . 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 54	. . . . . . . . . . . . . 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 94	. . . . . . . . . . . . 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 52	. . . . . . . . . . . 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 1119	. . . . . . . . . . 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 2790	. . . . . . . . . 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 2768	. . . . . . . . 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 248	. . . . . . . 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 1128	. . . . . . 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 87	. . . . . 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 413	. . . . 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 7864	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝑥)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝑥))))
62	6, 61	vtoclga 3532	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))))
63	62	expdcom 417	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))))
64	63	3imp 1119	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  ∅c0 4276  suc csuc 6333  (class class class)co 7381  ωcom 7831   +_o coa 8418   ·_o comu 8419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-oadd 8425  df-omul 8426

This theorem is referenced by:  nnmass  8578  distrpi  10842