splval2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > splval2		Structured version Visualization version GIF version

Theorem splval2 13508

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 13359	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (𝐴 ++ 𝐵) ∈ Word 𝑋)
5	2, 3, 4	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐴 ++ 𝐵) ∈ Word 𝑋)
6		splval2.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Word 𝑋)
7		ccatcl 13359	. . . . 5 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) ++ 𝐶) ∈ Word 𝑋)
8	5, 6, 7	syl2anc 693	. . . 4 ⊢ (𝜑 → ((𝐴 ++ 𝐵) ++ 𝐶) ∈ Word 𝑋)
9	1, 8	eqeltrd 2701	. . 3 ⊢ (𝜑 → 𝑆 ∈ Word 𝑋)
10		splval2.f	. . . 4 ⊢ (𝜑 → 𝐹 = (#‘𝐴))
11		lencl 13324	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ₀)
12	2, 11	syl 17	. . . 4 ⊢ (𝜑 → (#‘𝐴) ∈ ℕ₀)
13	10, 12	eqeltrd 2701	. . 3 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
14		splval2.t	. . . 4 ⊢ (𝜑 → 𝑇 = (𝐹 + (#‘𝐵)))
15		lencl 13324	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑋 → (#‘𝐵) ∈ ℕ₀)
16	3, 15	syl 17	. . . . 5 ⊢ (𝜑 → (#‘𝐵) ∈ ℕ₀)
17	13, 16	nn0addcld 11355	. . . 4 ⊢ (𝜑 → (𝐹 + (#‘𝐵)) ∈ ℕ₀)
18	14, 17	eqeltrd 2701	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℕ₀)
19		splval2.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Word 𝑋)
20		splval 13502	. . 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 1328	. 2 ⊢ (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
22		nn0uz 11722	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
23	13, 22	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℤ_≥‘0))
24		eluzfz1 12348	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝐹))
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝐹))
26	13	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ ℤ)
27		uzid 11702	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ℤ → 𝐹 ∈ (ℤ_≥‘𝐹))
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℤ_≥‘𝐹))
29		uzaddcl 11744	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (ℤ_≥‘𝐹) ∧ (#‘𝐵) ∈ ℕ₀) → (𝐹 + (#‘𝐵)) ∈ (ℤ_≥‘𝐹))
30	28, 16, 29	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 + (#‘𝐵)) ∈ (ℤ_≥‘𝐹))
31	14, 30	eqeltrd 2701	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (ℤ_≥‘𝐹))
32		elfzuzb 12336	. . . . . . . . 9 ⊢ (𝐹 ∈ (0...𝑇) ↔ (𝐹 ∈ (ℤ_≥‘0) ∧ 𝑇 ∈ (ℤ_≥‘𝐹)))
33	23, 31, 32	sylanbrc 698	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))
34	18, 22	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (ℤ_≥‘0))
35		ccatlen 13360	. . . . . . . . . . . 12 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑋 ∧ 𝐶 ∈ Word 𝑋) → (#‘((𝐴 ++ 𝐵) ++ 𝐶)) = ((#‘(𝐴 ++ 𝐵)) + (#‘𝐶)))
36	5, 6, 35	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘((𝐴 ++ 𝐵) ++ 𝐶)) = ((#‘(𝐴 ++ 𝐵)) + (#‘𝐶)))
37	1	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑆) = (#‘((𝐴 ++ 𝐵) ++ 𝐶)))
38	10	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 + (#‘𝐵)) = ((#‘𝐴) + (#‘𝐵)))
39		ccatlen 13360	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
40	2, 3, 39	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
41	38, 14, 40	3eqtr4d 2666	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = (#‘(𝐴 ++ 𝐵)))
42	41	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 + (#‘𝐶)) = ((#‘(𝐴 ++ 𝐵)) + (#‘𝐶)))
43	36, 37, 42	3eqtr4d 2666	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑆) = (𝑇 + (#‘𝐶)))
44	18	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℤ)
45		uzid 11702	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ ℤ → 𝑇 ∈ (ℤ_≥‘𝑇))
46	44, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ (ℤ_≥‘𝑇))
47		lencl 13324	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ Word 𝑋 → (#‘𝐶) ∈ ℕ₀)
48	6, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝐶) ∈ ℕ₀)
49		uzaddcl 11744	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ (ℤ_≥‘𝑇) ∧ (#‘𝐶) ∈ ℕ₀) → (𝑇 + (#‘𝐶)) ∈ (ℤ_≥‘𝑇))
50	46, 48, 49	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 + (#‘𝐶)) ∈ (ℤ_≥‘𝑇))
51	43, 50	eqeltrd 2701	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘𝑇))
52		elfzuzb 12336	. . . . . . . . 9 ⊢ (𝑇 ∈ (0...(#‘𝑆)) ↔ (𝑇 ∈ (ℤ_≥‘0) ∧ (#‘𝑆) ∈ (ℤ_≥‘𝑇)))
53	34, 51, 52	sylanbrc 698	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (0...(#‘𝑆)))
54		ccatswrd 13456	. . . . . . . 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 1328	. . . . . . 7 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝑆 substr ⟨0, 𝑇⟩))
56		eluzfz1 12348	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑇))
57	34, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ (0...𝑇))
58		lencl 13324	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Word 𝑋 → (#‘𝑆) ∈ ℕ₀)
59	9, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ₀)
60	59, 22	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘0))
61		eluzfz2 12349	. . . . . . . . . . . 12 ⊢ ((#‘𝑆) ∈ (ℤ_≥‘0) → (#‘𝑆) ∈ (0...(#‘𝑆)))
62	60, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑆) ∈ (0...(#‘𝑆)))
63		ccatswrd 13456	. . . . . . . . . . 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 1328	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = (𝑆 substr ⟨0, (#‘𝑆)⟩))
65		swrdid 13428	. . . . . . . . . . 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 2660	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = ((𝐴 ++ 𝐵) ++ 𝐶))
68		swrdcl 13419	. . . . . . . . . . 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 13419	. . . . . . . . . . 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 13422	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝑋 ∧ 𝑇 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨0, 𝑇⟩)) = 𝑇)
73	9, 53, 72	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝑇⟩)) = 𝑇)
74	73, 41	eqtrd 2656	. . . . . . . . . 10 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝑇⟩)) = (#‘(𝐴 ++ 𝐵)))
75		ccatopth 13470	. . . . . . . . . 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 1337	. . . . . . . . 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 222	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵) ∧ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩) = 𝐶))
78	77	simpld 475	. . . . . . 7 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝑇⟩) = (𝐴 ++ 𝐵))
79	55, 78	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ (𝑆 substr ⟨𝐹, 𝑇⟩)) = (𝐴 ++ 𝐵))
80		swrdcl 13419	. . . . . . . 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 13419	. . . . . . . 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 11704	. . . . . . . . . . 11 ⊢ (((#‘𝑆) ∈ (ℤ_≥‘𝑇) ∧ 𝑇 ∈ (ℤ_≥‘𝐹)) → (#‘𝑆) ∈ (ℤ_≥‘𝐹))
85	51, 31, 84	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑆) ∈ (ℤ_≥‘𝐹))
86		elfzuzb 12336	. . . . . . . . . 10 ⊢ (𝐹 ∈ (0...(#‘𝑆)) ↔ (𝐹 ∈ (ℤ_≥‘0) ∧ (#‘𝑆) ∈ (ℤ_≥‘𝐹)))
87	23, 85, 86	sylanbrc 698	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (0...(#‘𝑆)))
88		swrd0len 13422	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝑋 ∧ 𝐹 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨0, 𝐹⟩)) = 𝐹)
89	9, 87, 88	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝐹⟩)) = 𝐹)
90	89, 10	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (#‘(𝑆 substr ⟨0, 𝐹⟩)) = (#‘𝐴))
91		ccatopth 13470	. . . . . . 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 1337	. . . . . 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 222	. . . . 5 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) = 𝐴 ∧ (𝑆 substr ⟨𝐹, 𝑇⟩) = 𝐵))
94	93	simpld 475	. . . 4 ⊢ (𝜑 → (𝑆 substr ⟨0, 𝐹⟩) = 𝐴)
95	94	oveq1d 6665	. . 3 ⊢ (𝜑 → ((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) = (𝐴 ++ 𝑅))
96	77	simprd 479	. . 3 ⊢ (𝜑 → (𝑆 substr ⟨𝑇, (#‘𝑆)⟩) = 𝐶)
97	95, 96	oveq12d 6668	. 2 ⊢ (𝜑 → (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = ((𝐴 ++ 𝑅) ++ 𝐶))
98	21, 97	eqtrd 2656	1 ⊢ (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⟨cop 4183 ⟨cotp 4185 ‘cfv 5888 (class class class)co 6650 0cc0 9936 + caddc 9939 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 #chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 splice csplice 13296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-substr 13303 df-splice 13304

This theorem is referenced by: efginvrel2 18140 efgredleme 18156 efgcpbllemb 18168 frgpnabllem1 18276

W3C validator