Theorem sadcp1 16335

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 12805	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2848	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
4		seqp1 13921	. . . . . 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 6843	. . . . 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 6843	. . . . . 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 7367	. . . . 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 2801	. . . 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 12453	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
12		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 = 0 ↔ (𝑁 + 1) = 0))
13		oveq1 7364	. . . . . . . . 9 ⊢ (𝑛 = (𝑁 + 1) → (𝑛 − 1) = ((𝑁 + 1) − 1))
14	12, 13	ifbieq2d 4512	. . . . . . . 8 ⊢ (𝑛 = (𝑁 + 1) → if(𝑛 = 0, ∅, (𝑛 − 1)) = if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)))
15		eqid 2736	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))
16		0ex 5264	. . . . . . . . 9 ⊢ ∅ ∈ V
17		ovex 7390	. . . . . . . . 9 ⊢ ((𝑁 + 1) − 1) ∈ V
18	16, 17	ifex 4536	. . . . . . . 8 ⊢ if((𝑁 + 1) = 0, ∅, ((𝑁 + 1) − 1)) ∈ V
19	14, 15, 18	fvmpt 6948	. . . . . . 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 12452	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
22	1, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
23	22	nnne0d 12203	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ≠ 0)
24		ifnefalse 4498	. . . . . . 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 12475	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
27		1cnd 11150	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
28	26, 27	pncand 11513	. . . . . 6 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
29	20, 25, 28	3eqtrd 2780	. . . . 5 ⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘(𝑁 + 1)) = 𝑁)
30	29	oveq2d 7373	. . . 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 16333	. . . . . 6 ⊢ (𝜑 → 𝐶:ℕ0⟶2o)
34	33, 1	ffvelcdmd 7036	. . . . 5 ⊢ (𝜑 → (𝐶‘𝑁) ∈ 2o)
35		simpr 485	. . . . . . . . 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 483	. . . . . . . . 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 1611	. . . . . . 7 ⊢ ((𝑥 = (𝐶‘𝑁) ∧ 𝑦 = 𝑁) → (cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))
41	40	ifbid 4509	. . . . . 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 261	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑚 ∈ 𝐴 ↔ 𝑚 ∈ 𝐴))
43		biidd 261	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (𝑚 ∈ 𝐵 ↔ 𝑚 ∈ 𝐵))
44		eleq2w 2821	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (∅ ∈ 𝑐 ↔ ∅ ∈ 𝑥))
45	42, 43, 44	cadbi123d 1611	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑐) ↔ cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥)))
46	45	ifbid 4509	. . . . . . 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 261	. . . . . . . . 9 ⊢ (𝑚 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑥))
50	47, 48, 49	cadbi123d 1611	. . . . . . . 8 ⊢ (𝑚 = 𝑦 → (cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥) ↔ cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥)))
51	50	ifbid 4509	. . . . . . 7 ⊢ (𝑚 = 𝑦 → if(cadd(𝑚 ∈ 𝐴, 𝑚 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅) = if(cadd(𝑦 ∈ 𝐴, 𝑦 ∈ 𝐵, ∅ ∈ 𝑥), 1o, ∅))
52	46, 51	cbvmpov 7452	. . . . . 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 8422	. . . . . . 7 ⊢ 1o ∈ V
54	53, 16	ifex 4536	. . . . . 6 ⊢ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ∈ V
55	41, 52, 54	ovmpoa 7510	. . . . 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 584	. . . 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 2780	. . 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 4290	. . . . 5 ⊢ ¬ ∅ ∈ ∅
60		iffalse 4495	. . . . . 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 326	. . . 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 8450	. . . 4 ⊢ ∅ ∈ 1o
65		iftrue 4492	. . . 4 ⊢ (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) = 1o)
66	64, 65	eleqtrrid 2845	. . 3 ⊢ (cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)) → ∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅))
67	63, 66	impbii 208	. 2 ⊢ (∅ ∈ if(cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)), 1o, ∅) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁)))
68	58, 67	bitrdi 286	1 ⊢ (𝜑 → (∅ ∈ (𝐶‘(𝑁 + 1)) ↔ cadd(𝑁 ∈ 𝐴, 𝑁 ∈ 𝐵, ∅ ∈ (𝐶‘𝑁))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  caddwcad 1607   ∈ wcel 2106   ≠ wne 2943   ⊆ wss 3910  ∅c0 4282  ifcif 4486   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1_oc1o 8405  2_oc2o 8406  0cc0 11051  1c1 11052   + caddc 11054   − cmin 11385  ℕcn 12153  ℕ₀cn0 12413  ℤ_≥cuz 12763  seqcseq 13906

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-cad 1608  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907

This theorem is referenced by:  sadcaddlem  16337  sadadd2lem  16339  saddisjlem  16344