Theorem efgtf 19945

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 6967	. . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
3	1, 2	eqsstri 3955	. . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
4		simpl 488	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ 𝑊)
5	3, 4	sselid 3907	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ Word (𝐼 × 2o))
6		simprr 785	. . . . . . . . 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 19936	. . . . . . . . . . 11 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
9	8	ffvelcdmi 7088	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
10	9	ad2antll 742	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
11	6, 10	s2cld 15030	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
12		splcl 14909	. . . . . . . 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 596	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ Word (𝐼 × 2o))
14	1	efgrcl 19938	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
15	14	simprd 501	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o))
16	15	adantr 486	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
17	13, 16	eleqtrrd 2863	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
18	17	ralrimivva 3205	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2o)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ 𝑊)
19		eqid 2760	. . . . . 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 8083	. . . . 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 7457	. . . . 5 ⊢ (0...(♯‘𝑋)) ∈ V
23	14	simpld 500	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → 𝐼 ∈ V)
24		2on 8497	. . . . . 6 ⊢ 2o ∈ On
25		xpexg 7768	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V)
26	23, 24, 25	sylancl 598	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → (𝐼 × 2o) ∈ V)
27		xpexg 7768	. . . . 5 ⊢ (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2o) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
28	22, 26, 27	sylancr 599	. . . 4 ⊢ (𝑋 ∈ 𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2o)) ∈ V)
29	21, 28	fexd 7237	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∈ V)
30		fveq2 6890	. . . . . 6 ⊢ (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
31	30	oveq2d 7440	. . . . 5 ⊢ (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
32		eqidd 2761	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝐼 × 2o) = (𝐼 × 2o))
33		oveq1 7431	. . . . 5 ⊢ (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
34	31, 32, 33	mpoeq123dv 7499	. . . 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 4859	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
37		oteq2 4860	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
38	36, 37	eqtrd 2795	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)
39	38	oveq2d 7440	. . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑏 → (𝑀‘𝑤) = (𝑀‘𝑏))
42	40, 41	s2eqd 15022	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → ⟨“𝑤(𝑀‘𝑤)”⟩ = ⟨“𝑏(𝑀‘𝑏)”⟩)
43	42	oteq3d 4864	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
44	43	oveq2d 7440	. . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀‘𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
45	39, 44	cbvmpov 7519	. . . . . . 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 6890	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
47	46	oveq2d 7440	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
48		eqidd 2761	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝐼 × 2o) = (𝐼 × 2o))
49		oveq1 7431	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
50	47, 48, 49	mpoeq123dv 7499	. . . . . . 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 2807	. . . . . 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 5208	. . . . 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 2783	. . . 4 ⊢ 𝑇 = (𝑢 ∈ 𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
54	34, 53	fvmptg 6996	. . 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 700	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
56	55	feq1d 6696	. . 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 521	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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∖ cdif 4044  ⟨cop 4834  ⟨cotp 4857   ↦ cmpt 5185   I cid 5545   × cxp 5649  Oncon0 6365  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  1_oc1o 8476  2_oc2o 8477  0cc0 11207  ...cfz 13646  ♯chash 14481  Word cword 14665   splice csplice 14906  ⟨“cs2 15000   ~_FG cefg 19929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-er 8724  df-map 8856  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-n0 12614  df-z 12701  df-uz 12973  df-fz 13647  df-fzo 13796  df-hash 14482  df-word 14666  df-concat 14723  df-s1 14750  df-substr 14797  df-pfx 14829  df-splice 14907  df-s2 15007

This theorem is used by:  efgtval  19946  efgval2  19947  efgtlen  19949  efginvrel2  19950  efgsp1  19960  efgredleme  19966  efgredlem  19970  efgrelexlemb  19973  efgcpbllemb  19978  frgpnabllem1  20096