Theorem efgtf 19601

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 6900	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3982	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4		simpl 482	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ 𝑊)
5	3, 4	sselid 3933	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ Word (𝐼 × 2o))
6		simprr 772	. . . . . . . . 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 19592	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
9	8	ffvelcdmi 7017	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
10	9	ad2antll 729	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
11	6, 10	s2cld 14778	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
12		splcl 14658	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2o))
14	1	efgrcl 19594	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
15	14	simprd 495	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
16	15	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
17	13, 16	eleqtrrd 2831	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
18	17	ralrimivva 3172	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2o)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
19		eqid 2729	. . . . . 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 8003	. . . . 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 218	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
22		ovex 7382	. . . . 5 ⊢ (0...(♯‘𝑋)) ∈ V
23	14	simpld 494	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V)
24		2on 8401	. . . . . 6 ⊢ 2o ∈ On
25		xpexg 7686	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
26	23, 24, 25	sylancl 586	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V)
27		xpexg 7686	. . . . 5 ⊢ (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2o) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
28	22, 26, 27	sylancr 587	. . . 4 ⊢ (𝑋 ∈ 𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
29	21, 28	fexd 7163	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
30		fveq2 6822	. . . . . 6 ⊢ (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
31	30	oveq2d 7365	. . . . 5 ⊢ (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
32		eqidd 2730	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝐼 × 2o) = (𝐼 × 2o))
33		oveq1 7356	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
34	31, 32, 33	mpoeq123dv 7424	. . . 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)”⟩⟩)))
35		efgval2.t	. . . . 5 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
36		oteq1 4833	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
37		oteq2 4834	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
38	36, 37	eqtrd 2764	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
39	38	oveq2d 7365	. . . . . . . 8 ⊢ (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))
40		id 22	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → 𝑤 = 𝑏)
41		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → (𝑀‘𝑤) = (𝑀‘𝑏))
42	40, 41	s2eqd 14770	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → ⟨“𝑤(𝑀‘𝑤)”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
43	42	oteq3d 4838	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
44	43	oveq2d 7365	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
45	39, 44	cbvmpov 7444	. . . . . . 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)”⟩⟩))
46		fveq2 6822	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
47	46	oveq2d 7365	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
48		eqidd 2730	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝐼 × 2o) = (𝐼 × 2o))
49		oveq1 7356	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
50	47, 48, 49	mpoeq123dv 7424	. . . . . . 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)”⟩⟩)))
51	45, 50	eqtrid 2776	. . . . . 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)”⟩⟩)))
52	51	cbvmptv 5196	. . . . 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)”⟩⟩)))
53	35, 52	eqtri 2752	. . . 4 ⊢ 𝑇 = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
54	34, 53	fvmptg 6928	. . 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)”⟩⟩)))
55	29, 54	mpdan 687	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
56	55	feq1d 6634	. . 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))
57	21, 56	mpbird 257	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
58	55, 57	jca 511	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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∖ cdif 3900  ⟨cop 4583  ⟨cotp 4585   ↦ cmpt 5173   I cid 5513   × cxp 5617  Oncon0 6307  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1_oc1o 8381  2_oc2o 8382  0cc0 11009  ...cfz 13410  ♯chash 14237  Word cword 14420   splice csplice 14655  ⟨“cs2 14748   ~_FG cefg 19585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14503  df-substr 14548  df-pfx 14578  df-splice 14656  df-s2 14755

This theorem is referenced by:  efgtval  19602  efgval2  19603  efgtlen  19605  efginvrel2  19606  efgsp1  19616  efgredleme  19622  efgredlem  19626  efgrelexlemb  19629  efgcpbllemb  19634  frgpnabllem1  19752