Theorem efgtf 18842

Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypotheses

Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
efgval.r	⊢ ∼ = ( ~FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))

Assertion

Ref	Expression
efgtf	⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊))

Distinct variable groups:   𝑎,𝑏,𝑦,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧,𝑎   𝑀,𝑎   𝑛,𝑏,𝑣,𝑤,𝑀   𝑇,𝑎,𝑏   𝑋,𝑎,𝑏   𝑊,𝑎,𝑏,𝑛,𝑣,𝑤,𝑦,𝑧   ∼ ,𝑎,𝑏,𝑦,𝑧   𝐼,𝑎,𝑏,𝑛,𝑣,𝑤,𝑦,𝑧

Allowed substitution hints:   ∼ (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)   𝑋(𝑦,𝑧,𝑤,𝑣,𝑛)

Proof of Theorem efgtf

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . . . 10 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		fviss 6735	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 4000	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4		simpl 485	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ 𝑊)
5	3, 4	sseldi 3964	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ Word (𝐼 × 2o))
6		simprr 771	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
7		efgval2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
8	7	efgmf 18833	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
9	8	ffvelrni 6844	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
10	9	ad2antll 727	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
11	6, 10	s2cld 14227	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
12		splcl 14108	. . . . . . . 8 ⊢ ((𝑋 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o)) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2o))
13	5, 11, 12	syl2anc 586	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2o))
14	1	efgrcl 18835	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
15	14	simprd 498	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
16	15	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
17	13, 16	eleqtrrd 2916	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
18	17	ralrimivva 3191	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2o)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
19		eqid 2821	. . . . . 6 ⊢ (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
20	19	fmpo 7760	. . . . 5 ⊢ (∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2o)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊 ↔ (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
21	18, 20	sylib 220	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
22		ovex 7183	. . . . 5 ⊢ (0...(♯‘𝑋)) ∈ V
23	14	simpld 497	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V)
24		2on 8105	. . . . . 6 ⊢ 2o ∈ On
25		xpexg 7467	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
26	23, 24, 25	sylancl 588	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V)
27		xpexg 7467	. . . . 5 ⊢ (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2o) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
28	22, 26, 27	sylancr 589	. . . 4 ⊢ (𝑋 ∈ 𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
29	1	fvexi 6678	. . . . 5 ⊢ 𝑊 ∈ V
30	29	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝑊 → 𝑊 ∈ V)
31		fex2 7632	. . . 4 ⊢ (((𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊 ∧ ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V ∧ 𝑊 ∈ V) → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
32	21, 28, 30, 31	syl3anc 1367	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
33		fveq2 6664	. . . . . 6 ⊢ (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
34	33	oveq2d 7166	. . . . 5 ⊢ (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
35		eqidd 2822	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝐼 × 2o) = (𝐼 × 2o))
36		oveq1 7157	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
37	34, 35, 36	mpoeq123dv 7223	. . . 4 ⊢ (𝑢 = 𝑋 → (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
38		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
39		oteq1 4805	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
40		oteq2 4806	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
41	39, 40	eqtrd 2856	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
42	41	oveq2d 7166	. . . . . . . 8 ⊢ (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))
43		id 22	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → 𝑤 = 𝑏)
44		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → (𝑀‘𝑤) = (𝑀‘𝑏))
45	43, 44	s2eqd 14219	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → ⟨“𝑤(𝑀‘𝑤)”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
46	45	oteq3d 4810	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
47	46	oveq2d 7166	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
48	42, 47	cbvmpov 7243	. . . . . . 7 ⊢ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑣)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
49		fveq2 6664	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
50	49	oveq2d 7166	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
51		eqidd 2822	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝐼 × 2o) = (𝐼 × 2o))
52		oveq1 7157	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
53	50, 51, 52	mpoeq123dv 7223	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑎 ∈ (0...(♯‘𝑣)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
54	48, 53	syl5eq 2868	. . . . . 6 ⊢ (𝑣 = 𝑢 → (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
55	54	cbvmptv 5161	. . . . 5 ⊢ (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
56	38, 55	eqtri 2844	. . . 4 ⊢ 𝑇 = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
57	37, 56	fvmptg 6760	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V) → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
58	32, 57	mpdan 685	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
59	58	feq1d 6493	. . 3 ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊 ↔ (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊))
60	21, 59	mpbird 259	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
61	58, 60	jca 514	1 ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ∖ cdif 3932  ⟨cop 4566  ⟨cotp 4568   ↦ cmpt 5138   I cid 5453   × cxp 5547  Oncon0 6185  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  1_oc1o 8089  2_oc2o 8090  0cc0 10531  ...cfz 12886  ♯chash 13684  Word cword 13855   splice csplice 14105  ⟨“cs2 14197   ~_FG cefg 18826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-ot 4569  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-fzo 13028  df-hash 13685  df-word 13856  df-concat 13917  df-s1 13944  df-substr 13997  df-pfx 14027  df-splice 14106  df-s2 14204

This theorem is referenced by:  efgtval  18843  efgval2  18844  efgtlen  18846  efginvrel2  18847  efgsp1  18857  efgredleme  18863  efgredlem  18867  efgrelexlemb  18870  efgcpbllemb  18875  frgpnabllem1  18987