Theorem sadcp1 16625

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 13003	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2870	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 14159	. . . . . 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 18	. . . . 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 6886	. . . . 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 6886	. . . . . 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 7430	. . . . 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 2820	. . . 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 12646	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
12		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
13		oveq1 7427	. . . . . . . . 9 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
14	12, 13	ifbieq2d 4508	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
15		eqid 2760	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
16		0ex 5260	. . . . . . . . 9 ⊢ ∅ ∈ V
17		ovex 7453	. . . . . . . . 9 ⊢ ((𝑁 + 1) − 1) ∈ V
18	16, 17	ifex 4532	. . . . . . . 8 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
19	14, 15, 18	fvmpt 6993	. . . . . . 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 19	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
21		nn0p1nn 12645	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
22	1, 21	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
23	22	nnne0d 12388	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
24		ifnefalse 4493	. . . . . . 7 ⊢ ((𝑁 + 1) ≠ 0 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
25	23, 24	syl 18	. . . . . 6 ⊢ (𝜑 → if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) = ((𝑁 + 1) − 1))
26	1	nn0cnd 12669	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		1cnd 11302	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
28	26, 27	pncand 11670	. . . . . 6 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
29	20, 25, 28	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
30	29	oveq2d 7436	. . . 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 16623	. . . . . 6 ⊢ (𝜑 → 𝐶:ℕ0⟶2o)
34	33, 1	ffvelcdmd 7085	. . . . 5 ⊢ (𝜑 → (𝐶‘𝑁) ∈ 2o)
35		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → 𝑦 = 𝑁)
36	35	eleq1d 2845	. . . . . . . 8 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐴 ↔ 𝑁 ∈ 𝐴))
37	35	eleq1d 2845	. . . . . . . 8 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (𝑦 ∈ 𝐵 ↔ 𝑁 ∈ 𝐵))
38		simpl 488	. . . . . . . . 9 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → 𝑥 = (𝐶‘𝑁))
39	38	eleq2d 2846	. . . . . . . 8 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (∅ ∈ 𝑥 ↔ ∅ ∈ (𝐶‘𝑁)))
40	36, 37, 39	cadbi123d 1642	. . . . . . 7 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))
41	40	ifbid 4505	. . . . . 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 265	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑚 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
43		biidd 265	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑚 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
44		eleq2w 2844	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑥))
45	42, 43, 44	cadbi123d 1642	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥)))
46	45	ifbid 4505	. . . . . . 7 ⊢ (𝑐 = 𝑥 → if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐), 1o, ∅) = if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅))
47		eleq1w 2843	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
48		eleq1w 2843	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (𝑚 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
49		biidd 265	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑥))
50	47, 48, 49	cadbi123d 1642	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥) ↔ cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥)))
51	50	ifbid 4505	. . . . . . 7 ⊢ (𝑚 = 𝑦 → if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅) = if(cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅))
52	46, 51	cbvmpov 7515	. . . . . 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 8486	. . . . . . 7 ⊢ 1o ∈ V
54	53, 16	ifex 4532	. . . . . 6 ⊢ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ∈ V
55	41, 52, 54	ovmpoa 7575	. . . . 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 596	. . . 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 2799	. . 3 ⊢ (𝜑 → (𝐶‘(𝑁 + 1)) = if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
58	57	eleq2d 2846	. 2 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅)))
59		noel 4283	. . . . 5 ⊢ ¬ ∅ ∈ ∅
60		iffalse 4490	. . . . . 6 ⊢ (¬ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) = ∅)
61	60	eleq2d 2846	. . . . 5 ⊢ (¬ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ↔ ∅ ∈ ∅))
62	59, 61	mtbiri 330	. . . 4 ⊢ (¬ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → ¬ ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
63	62	con4i 115	. . 3 ⊢ (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) → cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))
64		0lt1o 8512	. . . 4 ⊢ ∅ ∈ 1o
65		iftrue 4487	. . . 4 ⊢ (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) = 1o)
66	64, 65	eleqtrrid 2867	. . 3 ⊢ (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
67	63, 66	impbii 212	. 2 ⊢ (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))
68	58, 67	bitrdi 290	1 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  caddwcad 1638   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895  ∅c0 4278  ifcif 4481   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1_oc1o 8469  2_oc2o 8470  0cc0 11200  1c1 11201   + caddc 11203   − cmin 11541  ℕcn 12335  ℕ₀cn0 12606  ℤ_≥cuz 12965  seqcseq 14144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1541  df-tru 1572  df-fal 1582  df-cad 1639  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-seq 14145

This theorem is used by:  sadcaddlem  16627  sadadd2lem  16629  saddisjlem  16634