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Theorem splval2 13359

Description: Value of a splice, assuming the input word 𝑆 has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)

**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
splval2	⊢ (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))

**Proof of Theorem splval2**
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 13212	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (𝐴 ++ 𝐵) ∈ Word 𝑋)
5	2, 3, 4	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐴 ++ 𝐵) ∈ Word 𝑋)
6		splval2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Word 𝑋)
7		ccatcl 13212	. . . . 5 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) ++ 𝐶) ∈ Word 𝑋)
8	5, 6, 7	syl2anc 691	. . . 4 ⊢ (𝜑 → ((𝐴 ++ 𝐵) ++ 𝐶) ∈ Word 𝑋)
9	1, 8	eqeltrd 2688	. . 3 ⊢ (𝜑 → 𝑆 ∈ Word 𝑋)
10		splval2.f	. . . 4 ⊢ (𝜑 → 𝐹 = (#‘𝐴))
11		lencl 13179	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ₀)
12	2, 11	syl 17	. . . 4 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ₀)
13	10, 12	eqeltrd 2688	. . 3 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
14		splval2.t	. . . 4 ⊢ (𝜑 → 𝑇 = (𝐹 + (#‘𝐵)))
15		lencl 13179	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑋 → (#‘𝐵) ∈ ℕ₀)
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
17	13, 16	nn0addcld 11232	. . . 4 ⊢ (𝜑 → (𝐹 + (#‘𝐵)) ∈ ℕ₀)
18	14, 17	eqeltrd 2688	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℕ₀)
19		splval2.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Word 𝑋)
20		splval 13353	. . 3 ⊢ ((𝑆 ∈ Word 𝑋 ∧ (𝐹 ∈ ℕ₀ ∧ 𝑇 ∈ ℕ₀ ∧ 𝑅 ∈ Word 𝑋)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
21	9, 13, 18, 19, 20	syl13anc 1320	. 2 ⊢ (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
22		nn0uz 11598	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
23	13, 22	syl6eleq 2698	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℤ_≥‘0))
24		eluzfz1 12219	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝐹))
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝐹))
26	13	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ ℤ)
27		uzid 11578	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ℤ → 𝐹 ∈ (ℤ_≥‘𝐹))
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℤ_≥‘𝐹))
29		uzaddcl 11620	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (ℤ_≥‘𝐹) ∧ (#‘𝐵) ∈ ℕ₀) → (𝐹 + (#‘𝐵)) ∈ (ℤ_≥‘𝐹))
30	28, 16, 29	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 + (#‘𝐵)) ∈ (ℤ_≥‘𝐹))
31	14, 30	eqeltrd 2688	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (ℤ_≥‘𝐹))
32		elfzuzb 12207	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝑇) ↔ (𝐹 ∈ (ℤ_≥‘0) ∧ 𝑇 ∈ (ℤ_≥‘𝐹)))
33	23, 31, 32	sylanbrc 695	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))
34	18, 22	syl6eleq 2698	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (ℤ_≥‘0))
35		ccatlen 13213	. . . . . . . . . . . 12 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) → (#‘((𝐴 ++ 𝐵) ++ 𝐶)) = ((#‘(𝐴 ++ 𝐵)) + (#‘𝐶)))
36	5, 6, 35	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘((𝐴 ++ 𝐵) ++ 𝐶)) = ((#‘(𝐴 ++ 𝐵)) + (#‘𝐶)))
37	1	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑆) = (#‘((𝐴 ++ 𝐵) ++ 𝐶)))
38	10	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 + (#‘𝐵)) = ((#‘𝐴) + (#‘𝐵)))
39		ccatlen 13213	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
40	2, 3, 39	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
41	38, 14, 40	3eqtr4d 2654	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = (#‘(𝐴 ++ 𝐵)))
42	41	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 + (#‘𝐶)) = ((#‘(𝐴 ++ 𝐵)) + (#‘𝐶)))
43	36, 37, 42	3eqtr4d 2654	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑆) = (𝑇 + (#‘𝐶)))
44	18	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℤ)
45		uzid 11578	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ℤ → 𝑇 ∈ (ℤ_≥‘𝑇))
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ (ℤ_≥‘𝑇))
47		lencl 13179	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Word 𝑋 → (#‘𝐶) ∈ ℕ₀)
48	6, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐶) ∈ ℕ₀)
49		uzaddcl 11620	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (ℤ_≥‘𝑇) ∧ (#‘𝐶) ∈ ℕ₀) → (𝑇 + (#‘𝐶)) ∈ (ℤ_≥‘𝑇))
50	46, 48, 49	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 + (#‘𝐶)) ∈ (ℤ_≥‘𝑇))
51	43, 50	eqeltrd 2688	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘𝑇))
52		elfzuzb 12207	. . . . . . . . 9 ⊢ (𝑇 ∈ (0...(#‘𝑆)) ↔ (𝑇 ∈ (ℤ_≥‘0) ∧ (#‘𝑆) ∈ (ℤ_≥‘𝑇)))
53	34, 51, 52	sylanbrc 695	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (0...(#‘𝑆)))
54		ccatswrd 13308	. . . . . . . 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 1320	. . . . . . 7 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝑆 substr ⟨0, 𝑇⟩))
56		eluzfz1 12219	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑇))
57	34, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ (0...𝑇))
58		lencl 13179	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Word 𝑋 → (#‘𝑆) ∈ ℕ₀)
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ₀)
60	59, 22	syl6eleq 2698	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘0))
61		eluzfz2 12220	. . . . . . . . . . . 12 ⊢ ((#‘𝑆) ∈ (ℤ_≥‘0) → (#‘𝑆) ∈ (0...(#‘𝑆)))
62	60, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑆) ∈ (0...(#‘𝑆)))
63		ccatswrd 13308	. . . . . . . . . . 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 1320	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
65		swrdid 13280	. . . . . . . . . . 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 2648	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = ((𝐴 ++ 𝐵) ++ 𝐶))
68		swrdcl 13271	. . . . . . . . . . 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 13271	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝑋 → (𝑆 substr ⟨𝑇, (#‘𝑆)⟩) ∈ Word 𝑋)
71	9, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 substr ⟨𝑇, (#‘𝑆)⟩) ∈ Word 𝑋)
72		swrd0len 13274	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝑋 ∧ 𝑇 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨0, 𝑇⟩)) = 𝑇)
73	9, 53, 72	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝑇⟩)) = 𝑇)
74	73, 41	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝑇⟩)) = (#‘(𝐴 ++ 𝐵)))
75		ccatopth 13322	. . . . . . . . . 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 1329	. . . . . . . . 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 221	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵) ∧ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩) = 𝐶))
78	77	simpld 474	. . . . . . 7 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵))
79	55, 78	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝐴 ++ 𝐵))
80		swrdcl 13271	. . . . . . . 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 13271	. . . . . . . 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 11580	. . . . . . . . . . 11 ⊢ (((#‘𝑆) ∈ (ℤ_≥‘𝑇) ∧ 𝑇 ∈ (ℤ_≥‘𝐹)) → (#‘𝑆) ∈ (ℤ_≥‘𝐹))
85	51, 31, 84	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘𝐹))
86		elfzuzb 12207	. . . . . . . . . 10 ⊢ (𝐹 ∈ (0...(#‘𝑆)) ↔ (𝐹 ∈ (ℤ_≥‘0) ∧ (#‘𝑆) ∈ (ℤ_≥‘𝐹)))
87	23, 85, 86	sylanbrc 695	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (0...(#‘𝑆)))
88		swrd0len 13274	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝑋 ∧ 𝐹 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨0, 𝐹⟩)) = 𝐹)
89	9, 87, 88	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝐹⟩)) = 𝐹)
90	89, 10	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝐹⟩)) = (#‘𝐴))
91		ccatopth 13322	. . . . . . 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 1329	. . . . . 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 221	. . . . 5 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) = 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝑇⟩) = 𝐵))
94	93	simpld 474	. . . 4 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝐹⟩) = 𝐴)
95	94	oveq1d 6564	. . 3 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) = (𝐴 ++ 𝑅))
96	77	simprd 478	. . 3 ⊢ (𝜑 → (𝑆 substr ⟨𝑇, (#‘𝑆)⟩) = 𝐶)
97	95, 96	oveq12d 6567	. 2 ⊢ (𝜑 → (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = ((𝐴 ++ 𝑅) ++ 𝐶))
98	21, 97	eqtrd 2644	1 ⊢ (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⟨cop 4131 ⟨cotp 4133 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979 Word cword 13146 ++ cconcat 13148 substr csubstr 13150 splice csplice 13151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-splice 13159

This theorem is referenced by: efginvrel2 17963 efgredleme 17979 efgcpbllemb 17991 frgpnabllem1 18099

W3C validator