Theorem nndi 6088

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 5540	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
2	1	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)))
3		oveq2 5540	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
4	3	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))
5	2, 4	eqeq12d 2095	. . . . 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 228	. . . 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 5540	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
8	7	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 ∅)))
9		oveq2 5540	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
10	9	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
11	8, 10	eqeq12d 2095	. . . . 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 5540	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
13	12	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 𝑦)))
14		oveq2 5540	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
15	14	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑦)))
16	13, 15	eqeq12d 2095	. . . . 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 5540	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
18	17	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)))
19		oveq2 5540	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
20	19	oveq2d 5548	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 suc 𝑦)))
21	18, 20	eqeq12d 2095	. . . . 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 6076	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
23	22	adantl 271	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 ∅) = 𝐵)
24	23	oveq2d 5548	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = (𝐴 ·𝑜 𝐵))
25		nnmcl 6083	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
26		nna0 6076	. . . . . . . 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 2116	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
29		nnm0 6077	. . . . . . . 8 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
30	29	adantr 270	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ∅) = ∅)
31	30	oveq2d 5548	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 ∅))
32	28, 31	eqtr4d 2116	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 ∅)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 ∅)))
33		oveq1 5539	. . . . . . . . 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 6078	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
35	34	3adant1 956	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
36	35	oveq2d 5548	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)))
37		nnacl 6082	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
38		nnmsuc 6079	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ (𝐵 +𝑜 𝑦) ∈ ω) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
39	37, 38	sylan2 280	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝑦 ∈ ω)) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
40	39	3impb 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc (𝐵 +𝑜 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
41	36, 40	eqtrd 2113	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (𝐵 +𝑜 𝑦)) +𝑜 𝐴))
42		nnmsuc 6079	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
43	42	3adant2 957	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
44	43	oveq2d 5548	. . . . . . . . . . 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 6083	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
46		nnaass 6087	. . . . . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . 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 359	. . . . . . . . . . . . . . 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 50	. . . . . . . . . . . . . 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 85	. . . . . . . . . . . . 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 48	. . . . . . . . . . . 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 1132	. . . . . . . . . . 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 2116	. . . . . . . . . 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 2095	. . . . . . . . 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	syl5ibr 154	. . . . . . . 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 1137	. . . . . . 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 78	. . . . . 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 251	. . . . 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 4342	. . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝑥)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝑥))))
62	6, 61	vtoclga 2664	. . 3 ⊢ (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶))))
63	62	expdcom 1371	. 2 ⊢ (𝐴 ∈ ω → (𝐵 ∈ ω → (𝐶 ∈ ω → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))))
64	63	3imp 1132	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 (𝐵 +𝑜 𝐶)) = ((𝐴 ·𝑜 𝐵) +𝑜 (𝐴 ·𝑜 𝐶)))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∅c0 3251  suc csuc 4120  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021   ·_𝑜 comu 6022

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029

This theorem is referenced by:  nnmass  6089  nndir  6092  distrpig  6523  addcmpblnq0  6633  nnanq0  6648  distrnq0  6649  addassnq0  6652