Theorem efgtf 19787

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 6958	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3983	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4		simpl 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ 𝑊)
5	3, 4	sselid 3935	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ Word (𝐼 × 2o))
6		simprr 784	. . . . . . . . 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 19778	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
9	8	ffvelcdmi 7078	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
10	9	ad2antll 741	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
11	6, 10	s2cld 14904	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
12		splcl 14785	. . . . . . . 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 595	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2o))
14	1	efgrcl 19780	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
15	14	simprd 500	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
16	15	adantr 485	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
17	13, 16	eleqtrrd 2866	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
18	17	ralrimivva 3208	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2o)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
19		eqid 2763	. . . . . 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 8061	. . . . 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 221	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
22		ovex 7443	. . . . 5 ⊢ (0...(♯‘𝑋)) ∈ V
23	14	simpld 499	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V)
24		2on 8463	. . . . . 6 ⊢ 2o ∈ On
25		xpexg 7745	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
26	23, 24, 25	sylancl 597	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V)
27		xpexg 7745	. . . . 5 ⊢ (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2o) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
28	22, 26, 27	sylancr 598	. . . 4 ⊢ (𝑋 ∈ 𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
29	21, 28	fexd 7225	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
30		fveq2 6881	. . . . . 6 ⊢ (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
31	30	oveq2d 7426	. . . . 5 ⊢ (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
32		eqidd 2764	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝐼 × 2o) = (𝐼 × 2o))
33		oveq1 7417	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
34	31, 32, 33	mpoeq123dv 7485	. . . 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 4847	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
37		oteq2 4848	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
38	36, 37	eqtrd 2798	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
39	38	oveq2d 7426	. . . . . . . 8 ⊢ (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))
40		id 23	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → 𝑤 = 𝑏)
41		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → (𝑀‘𝑤) = (𝑀‘𝑏))
42	40, 41	s2eqd 14896	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → ⟨“𝑤(𝑀‘𝑤)”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
43	42	oteq3d 4852	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
44	43	oveq2d 7426	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
45	39, 44	cbvmpov 7505	. . . . . . 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 6881	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
47	46	oveq2d 7426	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
48		eqidd 2764	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝐼 × 2o) = (𝐼 × 2o))
49		oveq1 7417	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
50	47, 48, 49	mpoeq123dv 7485	. . . . . . 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 2810	. . . . . 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 5215	. . . . 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 2786	. . . 4 ⊢ 𝑇 = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
54	34, 53	fvmptg 6987	. . 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 699	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
56	55	feq1d 6687	. . 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 260	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)
58	55, 57	jca 520	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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3902  ⟨cop 4595  ⟨cotp 4597   ↦ cmpt 5192   I cid 5555   × cxp 5659  Oncon0 6360  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1_oc1o 8442  2_oc2o 8443  0cc0 11095  ...cfz 13530  ♯chash 14362  Word cword 14546   splice csplice 14782  ⟨“cs2 14874   ~_FG cefg 19771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-hash 14363  df-word 14547  df-concat 14604  df-s1 14630  df-substr 14675  df-pfx 14705  df-splice 14783  df-s2 14881

This theorem is referenced by:  efgtval  19788  efgval2  19789  efgtlen  19791  efginvrel2  19792  efgsp1  19802  efgredleme  19808  efgredlem  19812  efgrelexlemb  19815  efgcpbllemb  19820  frgpnabllem1  19938