Theorem sadcp1 16386

Description: The carry sequence (which is a sequence of wffs, encoded as 1_o and ∅) is defined recursively as the carry operation applied to the previous carry and the two current inputs. (Contributed by Mario Carneiro, 5-Sep-2016.)

Hypotheses

Ref	Expression
sadval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ0)
sadval.b	⊢ (𝜑 → 𝐵 ⊆ ℕ0)
sadval.c	⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
sadcp1.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
sadcp1	⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))

Distinct variable groups:   𝑚,𝑐,𝑛   𝐴,𝑐,𝑚   𝐵,𝑐,𝑚   𝑛,𝑁

Allowed substitution hints:   𝜑(𝑚,𝑛,𝑐)   𝐴(𝑛)   𝐵(𝑛)   𝐶(𝑚,𝑛,𝑐)   𝑁(𝑚,𝑐)

Proof of Theorem sadcp1

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

Step	Hyp	Ref	Expression
1		sadcp1.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		nn0uz 12793	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2847	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 13943	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1)) = ((seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
6		sadval.c	. . . . . 6 ⊢ 𝐶 = seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
7	6	fveq1i 6836	. . . . 5 ⊢ (𝐶‘(𝑁 + 1)) = (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑁 + 1))
8	6	fveq1i 6836	. . . . . 6 ⊢ (𝐶‘𝑁) = (seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)
9	8	oveq1i 7370	. . . . 5 ⊢ ((𝐶‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((seq0((𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)))
10	5, 7, 9	3eqtr4g 2797	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑁 + 1)) = ((𝐶‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))))
11		peano2nn0 12445	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
12		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
13		oveq1 7367	. . . . . . . . 9 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
14	12, 13	ifbieq2d 4507	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
15		eqid 2737	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
16		0ex 5253	. . . . . . . . 9 ⊢ ∅ ∈ V
17		ovex 7393	. . . . . . . . 9 ⊢ ((𝑁 + 1) − 1) ∈ V
18	16, 17	ifex 4531	. . . . . . . 8 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
19	14, 15, 18	fvmpt 6942	. . . . . . 7 ⊢ ((𝑁 + 1) ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
20	1, 11, 19	3syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
21		nn0p1nn 12444	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
22	1, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
23	22	nnne0d 12199	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
24		ifnefalse 4492	. . . . . . 7 ⊢ ((𝑁 + 1) ≠ 0 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
25	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
26	1	nn0cnd 12468	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		1cnd 11131	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
28	26, 27	pncand 11497	. . . . . 6 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
29	20, 25, 28	3eqtrd 2776	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
30	29	oveq2d 7376	. . . 4 ⊢ (𝜑 → ((𝐶‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1))) = ((𝐶‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))𝑁))
31		sadval.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ0)
32		sadval.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ ℕ0)
33	31, 32, 6	sadcf 16384	. . . . . 6 ⊢ (𝜑 → 𝐶:ℕ0⟶2o)
34	33, 1	ffvelcdmd 7032	. . . . 5 ⊢ (𝜑 → (𝐶‘𝑁) ∈ 2o)
35		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
36	35	eleq1d 2822	. . . . . . . 8 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
37	35	eleq1d 2822	. . . . . . . 8 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐵 ↔ 𝑁 ∈ 𝐵))
38		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝐶‘𝑁))
39	38	eleq2d 2823	. . . . . . . 8 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (∅ ∈ 𝑥 ↔ ∅ ∈ (𝐶‘𝑁)))
40	36, 37, 39	cadbi123d 1612	. . . . . . 7 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))
41	40	ifbid 4504	. . . . . 6 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → if(cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅) = if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
42		biidd 262	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑚 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
43		biidd 262	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑚 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
44		eleq2w 2821	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑥))
45	42, 43, 44	cadbi123d 1612	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥)))
46	45	ifbid 4504	. . . . . . 7 ⊢ (𝑐 = 𝑥 → if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅))
47		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
48		eleq1w 2820	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
49		biidd 262	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑥))
50	47, 48, 49	cadbi123d 1612	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥) ↔ cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥)))
51	50	ifbid 4504	. . . . . . 7 ⊢ (𝑚 = 𝑦 → if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅) = if(cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅))
52	46, 51	cbvmpov 7455	. . . . . 6 ⊢ (𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅)) = (𝑥 ∈ 2o, 𝑦 ∈ ℕ0 ↦ if(cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅))
53		1oex 8409	. . . . . . 7 ⊢ 1o ∈ V
54	53, 16	ifex 4531	. . . . . 6 ⊢ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ∈ V
55	41, 52, 54	ovmpoa 7515	. . . . 5 ⊢ (((𝐶‘𝑁) ∈ 2o ∧ 𝑁 ∈ ℕ0) → ((𝐶‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))𝑁) = if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
56	34, 1, 55	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐶‘𝑁)(𝑐 ∈ 2o, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅))𝑁) = if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
57	10, 30, 56	3eqtrd 2776	. . 3 ⊢ (𝜑 → (𝐶‘(𝑁 + 1)) = if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
58	57	eleq2d 2823	. 2 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅)))
59		noel 4291	. . . . 5 ⊢ ¬ ∅ ∈ ∅
60		iffalse 4489	. . . . . 6 ⊢ (¬ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) = ∅)
61	60	eleq2d 2823	. . . . 5 ⊢ (¬ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ↔ ∅ ∈ ∅))
62	59, 61	mtbiri 327	. . . 4 ⊢ (¬ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → ¬ ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
63	62	con4i 114	. . 3 ⊢ (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) → cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))
64		0lt1o 8433	. . . 4 ⊢ ∅ ∈ 1o
65		iftrue 4486	. . . 4 ⊢ (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) = 1o)
66	64, 65	eleqtrrid 2844	. . 3 ⊢ (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
67	63, 66	impbii 209	. 2 ⊢ (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))
68	58, 67	bitrdi 287	1 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  caddwcad 1608   ∈ wcel 2114   ≠ wne 2933   ⊆ wss 3902  ∅c0 4286  ifcif 4480   ↦ cmpt 5180  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  1_oc1o 8392  2_oc2o 8393  0cc0 11030  1c1 11031   + caddc 11033   − cmin 11368  ℕcn 12149  ℕ₀cn0 12405  ℤ_≥cuz 12755  seqcseq 13928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-cad 1609  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-seq 13929

This theorem is referenced by:  sadcaddlem  16388  sadadd2lem  16390  saddisjlem  16395