Theorem splval2OLD 13874

Description: Obsolete proof of splval2 13873 as of 12-Oct-2022. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
splval2.a	⊢ (𝜑 → 𝐴 ∈ Word 𝑋)
splval2.b	⊢ (𝜑 → 𝐵 ∈ Word 𝑋)
splval2.c	⊢ (𝜑 → 𝐶 ∈ Word 𝑋)
splval2.r	⊢ (𝜑 → 𝑅 ∈ Word 𝑋)
splval2.s	⊢ (𝜑 → 𝑆 = ((𝐴 ++ 𝐵) ++ 𝐶))
splval2.f	⊢ (𝜑 → 𝐹 = (♯‘𝐴))
splval2.t	⊢ (𝜑 → 𝑇 = (𝐹 + (♯‘𝐵)))

Assertion

Ref	Expression
splval2OLD	⊢ (𝜑 → (𝑆 spliceOLD ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))

Proof of Theorem splval2OLD

Step	Hyp	Ref	Expression
1		splval2.s	. . . 4 ⊢ (𝜑 → 𝑆 = ((𝐴 ++ 𝐵) ++ 𝐶))
2		splval2.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Word 𝑋)
3		splval2.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Word 𝑋)
4		ccatcl 13635	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (𝐴 ++ 𝐵) ∈ Word 𝑋)
5	2, 3, 4	syl2anc 581	. . . . 5 ⊢ (𝜑 → (𝐴 ++ 𝐵) ∈ Word 𝑋)
6		splval2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Word 𝑋)
7		ccatcl 13635	. . . . 5 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) ++ 𝐶) ∈ Word 𝑋)
8	5, 6, 7	syl2anc 581	. . . 4 ⊢ (𝜑 → ((𝐴 ++ 𝐵) ++ 𝐶) ∈ Word 𝑋)
9	1, 8	eqeltrd 2907	. . 3 ⊢ (𝜑 → 𝑆 ∈ Word 𝑋)
10		splval2.f	. . . 4 ⊢ (𝜑 → 𝐹 = (♯‘𝐴))
11		lencl 13594	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
12	2, 11	syl 17	. . . 4 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0)
13	10, 12	eqeltrd 2907	. . 3 ⊢ (𝜑 → 𝐹 ∈ ℕ0)
14		splval2.t	. . . 4 ⊢ (𝜑 → 𝑇 = (𝐹 + (♯‘𝐵)))
15		lencl 13594	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑋 → (♯‘𝐵) ∈ ℕ0)
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0)
17	13, 16	nn0addcld 11683	. . . 4 ⊢ (𝜑 → (𝐹 + (♯‘𝐵)) ∈ ℕ0)
18	14, 17	eqeltrd 2907	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℕ0)
19		splval2.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Word 𝑋)
20		splvalOLD 13862	. . 3 ⊢ ((𝑆 ∈ Word 𝑋 ∧ (𝐹 ∈ ℕ0 ∧ 𝑇 ∈ ℕ0 ∧ 𝑅 ∈ Word 𝑋)) → (𝑆 spliceOLD ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
21	9, 13, 18, 19, 20	syl13anc 1497	. 2 ⊢ (𝜑 → (𝑆 spliceOLD ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
22		nn0uz 12005	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
23	13, 22	syl6eleq 2917	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℤ≥‘0))
24		eluzfz1 12642	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℤ≥‘0) → 0 ∈ (0...𝐹))
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝐹))
26	13	nn0zd 11809	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ ℤ)
27		uzid 11984	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ℤ → 𝐹 ∈ (ℤ≥‘𝐹))
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℤ≥‘𝐹))
29		uzaddcl 12027	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (ℤ≥‘𝐹) ∧ (♯‘𝐵) ∈ ℕ0) → (𝐹 + (♯‘𝐵)) ∈ (ℤ≥‘𝐹))
30	28, 16, 29	syl2anc 581	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 + (♯‘𝐵)) ∈ (ℤ≥‘𝐹))
31	14, 30	eqeltrd 2907	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (ℤ≥‘𝐹))
32		elfzuzb 12630	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝑇) ↔ (𝐹 ∈ (ℤ≥‘0) ∧ 𝑇 ∈ (ℤ≥‘𝐹)))
33	23, 31, 32	sylanbrc 580	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))
34	18, 22	syl6eleq 2917	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (ℤ≥‘0))
35		ccatlen 13636	. . . . . . . . . . . 12 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) → (♯‘((𝐴 ++ 𝐵) ++ 𝐶)) = ((♯‘(𝐴 ++ 𝐵)) + (♯‘𝐶)))
36	5, 6, 35	syl2anc 581	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘((𝐴 ++ 𝐵) ++ 𝐶)) = ((♯‘(𝐴 ++ 𝐵)) + (♯‘𝐶)))
37	1	fveq2d 6438	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑆) = (♯‘((𝐴 ++ 𝐵) ++ 𝐶)))
38	10	oveq1d 6921	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
39		ccatlen 13636	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
40	2, 3, 39	syl2anc 581	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
41	38, 14, 40	3eqtr4d 2872	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = (♯‘(𝐴 ++ 𝐵)))
42	41	oveq1d 6921	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 + (♯‘𝐶)) = ((♯‘(𝐴 ++ 𝐵)) + (♯‘𝐶)))
43	36, 37, 42	3eqtr4d 2872	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑆) = (𝑇 + (♯‘𝐶)))
44	18	nn0zd 11809	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℤ)
45		uzid 11984	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ℤ → 𝑇 ∈ (ℤ≥‘𝑇))
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ (ℤ≥‘𝑇))
47		lencl 13594	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Word 𝑋 → (♯‘𝐶) ∈ ℕ0)
48	6, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐶) ∈ ℕ0)
49		uzaddcl 12027	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (ℤ≥‘𝑇) ∧ (♯‘𝐶) ∈ ℕ0) → (𝑇 + (♯‘𝐶)) ∈ (ℤ≥‘𝑇))
50	46, 48, 49	syl2anc 581	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 + (♯‘𝐶)) ∈ (ℤ≥‘𝑇))
51	43, 50	eqeltrd 2907	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ≥‘𝑇))
52		elfzuzb 12630	. . . . . . . . 9 ⊢ (𝑇 ∈ (0...(♯‘𝑆)) ↔ (𝑇 ∈ (ℤ≥‘0) ∧ (♯‘𝑆) ∈ (ℤ≥‘𝑇)))
53	34, 51, 52	sylanbrc 580	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))
54		ccatswrd 13747	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝑋 ∧ (0 ∈ (0...𝐹) ∧ 𝐹 ∈ (0...𝑇) ∧ 𝑇 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝑆 substr ⟨0, 𝑇⟩))
55	9, 25, 33, 53, 54	syl13anc 1497	. . . . . . 7 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝑆 substr ⟨0, 𝑇⟩))
56		eluzfz1 12642	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑇))
57	34, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ (0...𝑇))
58		lencl 13594	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Word 𝑋 → (♯‘𝑆) ∈ ℕ0)
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ0)
60	59, 22	syl6eleq 2917	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ≥‘0))
61		eluzfz2 12643	. . . . . . . . . . . 12 ⊢ ((♯‘𝑆) ∈ (ℤ≥‘0) → (♯‘𝑆) ∈ (0...(♯‘𝑆)))
62	60, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑆) ∈ (0...(♯‘𝑆)))
63		ccatswrd 13747	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝑋 ∧ (0 ∈ (0...𝑇) ∧ 𝑇 ∈ (0...(♯‘𝑆)) ∧ (♯‘𝑆) ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) = (𝑆 substr ⟨0, (♯‘𝑆)⟩))
64	9, 57, 53, 62, 63	syl13anc 1497	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) = (𝑆 substr ⟨0, (♯‘𝑆)⟩))
65		swrdidOLD 13716	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑋 → (𝑆 substr ⟨0, (♯‘𝑆)⟩) = 𝑆)
66	9, 65	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 substr ⟨0, (♯‘𝑆)⟩) = 𝑆)
67	64, 66, 1	3eqtrd 2866	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) = ((𝐴 ++ 𝐵) ++ 𝐶))
68		swrdcl 13706	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑋 → (𝑆 substr ⟨0, 𝑇⟩) ∈ Word 𝑋)
69	9, 68	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝑇⟩) ∈ Word 𝑋)
70		swrdcl 13706	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑋 → (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) ∈ Word 𝑋)
71	9, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) ∈ Word 𝑋)
72		swrd0lenOLD 13709	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝑋 ∧ 𝑇 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨0, 𝑇⟩)) = 𝑇)
73	9, 53, 72	syl2anc 581	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑆 substr ⟨0, 𝑇⟩)) = 𝑇)
74	73, 41	eqtrd 2862	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑆 substr ⟨0, 𝑇⟩)) = (♯‘(𝐴 ++ 𝐵)))
75		ccatopth 13805	. . . . . . . . . 10 ⊢ ((((𝑆 substr ⟨0, 𝑇⟩) ∈ Word 𝑋 ∧ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) ∈ Word 𝑋) ∧ ((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) ∧ (♯‘(𝑆 substr ⟨0, 𝑇⟩)) = (♯‘(𝐴 ++ 𝐵))) → (((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) = ((𝐴 ++ 𝐵) ++ 𝐶) ↔ ((𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵) ∧ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) = 𝐶)))
76	69, 71, 5, 6, 74, 75	syl221anc 1506	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) = ((𝐴 ++ 𝐵) ++ 𝐶) ↔ ((𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵) ∧ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) = 𝐶)))
77	67, 76	mpbid 224	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵) ∧ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) = 𝐶))
78	77	simpld 490	. . . . . . 7 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵))
79	55, 78	eqtrd 2862	. . . . . 6 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝐴 ++ 𝐵))
80		swrdcl 13706	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝑋 → (𝑆 substr ⟨0, 𝐹⟩) ∈ Word 𝑋)
81	9, 80	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝐹⟩) ∈ Word 𝑋)
82		swrdcl 13706	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝑋 → (𝑆 substr ⟨𝐹, 𝑇⟩) ∈ Word 𝑋)
83	9, 82	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑆 substr ⟨𝐹, 𝑇⟩) ∈ Word 𝑋)
84		uztrn 11986	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ (ℤ≥‘𝑇) ∧ 𝑇 ∈ (ℤ≥‘𝐹)) → (♯‘𝑆) ∈ (ℤ≥‘𝐹))
85	51, 31, 84	syl2anc 581	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑆) ∈ (ℤ≥‘𝐹))
86		elfzuzb 12630	. . . . . . . . . 10 ⊢ (𝐹 ∈ (0...(♯‘𝑆)) ↔ (𝐹 ∈ (ℤ≥‘0) ∧ (♯‘𝑆) ∈ (ℤ≥‘𝐹)))
87	23, 85, 86	sylanbrc 580	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (0...(♯‘𝑆)))
88		swrd0lenOLD 13709	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝑋 ∧ 𝐹 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨0, 𝐹⟩)) = 𝐹)
89	9, 87, 88	syl2anc 581	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝑆 substr ⟨0, 𝐹⟩)) = 𝐹)
90	89, 10	eqtrd 2862	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑆 substr ⟨0, 𝐹⟩)) = (♯‘𝐴))
91		ccatopth 13805	. . . . . . 7 ⊢ ((((𝑆 substr ⟨0, 𝐹⟩) ∈ Word 𝑋 ∧ (𝑆 substr ⟨𝐹, 𝑇⟩) ∈ Word 𝑋) ∧ (𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (♯‘(𝑆 substr ⟨0, 𝐹⟩)) = (♯‘𝐴)) → (((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝐴 ++ 𝐵) ↔ ((𝑆 substr ⟨0, 𝐹⟩) = 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝑇⟩) = 𝐵)))
92	81, 83, 2, 3, 90, 91	syl221anc 1506	. . . . . 6 ⊢ (𝜑 → (((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝐴 ++ 𝐵) ↔ ((𝑆 substr ⟨0, 𝐹⟩) = 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝑇⟩) = 𝐵)))
93	79, 92	mpbid 224	. . . . 5 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) = 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝑇⟩) = 𝐵))
94	93	simpld 490	. . . 4 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝐹⟩) = 𝐴)
95	94	oveq1d 6921	. . 3 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) = (𝐴 ++ 𝑅))
96	77	simprd 491	. . 3 ⊢ (𝜑 → (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩) = 𝐶)
97	95, 96	oveq12d 6924	. 2 ⊢ (𝜑 → (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) = ((𝐴 ++ 𝑅) ++ 𝐶))
98	21, 97	eqtrd 2862	1 ⊢ (𝜑 → (𝑆 spliceOLD ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4404  ⟨cotp 4406  ‘cfv 6124  (class class class)co 6906  0cc0 10253   + caddc 10256  ℕ₀cn0 11619  ℤcz 11705  ℤ_≥cuz 11969  ...cfz 12620  ♯chash 13411  Word cword 13575   ++ cconcat 13631   substr csubstr 13701   spliceOLD cspliceold 13857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-ot 4407  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-card 9079  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-hash 13412  df-word 13576  df-concat 13632  df-substr 13702  df-pfx 13751  df-spliceOLD 13859

This theorem is referenced by: (None)