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Theorem efgtf 17958
 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 = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
Assertion
Ref Expression
efgtf (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((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.
StepHypRef Expression
1 efgval.w . . . . . . . . . 10 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 fviss 6166 . . . . . . . . . 10 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3598 . . . . . . . . 9 𝑊 ⊆ Word (𝐼 × 2𝑜)
4 simpl 472 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
53, 4sseldi 3566 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
6 simprr 792 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
7 efgval2.m . . . . . . . . . . . 12 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
87efgmf 17949 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
98ffvelrni 6266 . . . . . . . . . 10 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
109ad2antll 761 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
116, 10s2cld 13466 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
12 splcl 13354 . . . . . . . 8 ((𝑋 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
135, 11, 12syl2anc 691 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
141efgrcl 17951 . . . . . . . . 9 (𝑋𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
1514simprd 478 . . . . . . . 8 (𝑋𝑊𝑊 = Word (𝐼 × 2𝑜))
1615adantr 480 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
1713, 16eleqtrrd 2691 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
1817ralrimivva 2954 . . . . 5 (𝑋𝑊 → ∀𝑎 ∈ (0...(#‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
19 eqid 2610 . . . . . 6 (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
2019fmpt2 7126 . . . . 5 (∀𝑎 ∈ (0...(#‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊 ↔ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
2118, 20sylib 207 . . . 4 (𝑋𝑊 → (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
22 ovex 6577 . . . . 5 (0...(#‘𝑋)) ∈ V
2314simpld 474 . . . . . 6 (𝑋𝑊𝐼 ∈ V)
24 2on 7455 . . . . . 6 2𝑜 ∈ On
25 xpexg 6858 . . . . . 6 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
2623, 24, 25sylancl 693 . . . . 5 (𝑋𝑊 → (𝐼 × 2𝑜) ∈ V)
27 xpexg 6858 . . . . 5 (((0...(#‘𝑋)) ∈ V ∧ (𝐼 × 2𝑜) ∈ V) → ((0...(#‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
2822, 26, 27sylancr 694 . . . 4 (𝑋𝑊 → ((0...(#‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
29 fvex 6113 . . . . . 6 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
301, 29eqeltri 2684 . . . . 5 𝑊 ∈ V
3130a1i 11 . . . 4 (𝑋𝑊𝑊 ∈ V)
32 fex2 7014 . . . 4 (((𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊 ∧ ((0...(#‘𝑋)) × (𝐼 × 2𝑜)) ∈ V ∧ 𝑊 ∈ V) → (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∈ V)
3321, 28, 31, 32syl3anc 1318 . . 3 (𝑋𝑊 → (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∈ V)
34 fveq2 6103 . . . . . 6 (𝑢 = 𝑋 → (#‘𝑢) = (#‘𝑋))
3534oveq2d 6565 . . . . 5 (𝑢 = 𝑋 → (0...(#‘𝑢)) = (0...(#‘𝑋)))
36 eqidd 2611 . . . . 5 (𝑢 = 𝑋 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
37 oveq1 6556 . . . . 5 (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
3835, 36, 37mpt2eq123dv 6615 . . . 4 (𝑢 = 𝑋 → (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
39 efgval2.t . . . . 5 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
40 oteq1 4349 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)
41 oteq2 4350 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4240, 41eqtrd 2644 . . . . . . . . 9 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4342oveq2d 6565 . . . . . . . 8 (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩))
44 id 22 . . . . . . . . . . 11 (𝑤 = 𝑏𝑤 = 𝑏)
45 fveq2 6103 . . . . . . . . . . 11 (𝑤 = 𝑏 → (𝑀𝑤) = (𝑀𝑏))
4644, 45s2eqd 13459 . . . . . . . . . 10 (𝑤 = 𝑏 → ⟨“𝑤(𝑀𝑤)”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
4746oteq3d 4354 . . . . . . . . 9 (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)
4847oveq2d 6565 . . . . . . . 8 (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
4943, 48cbvmpt2v 6633 . . . . . . 7 (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑣)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
50 fveq2 6103 . . . . . . . . 9 (𝑣 = 𝑢 → (#‘𝑣) = (#‘𝑢))
5150oveq2d 6565 . . . . . . . 8 (𝑣 = 𝑢 → (0...(#‘𝑣)) = (0...(#‘𝑢)))
52 eqidd 2611 . . . . . . . 8 (𝑣 = 𝑢 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
53 oveq1 6556 . . . . . . . 8 (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
5451, 52, 53mpt2eq123dv 6615 . . . . . . 7 (𝑣 = 𝑢 → (𝑎 ∈ (0...(#‘𝑣)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5549, 54syl5eq 2656 . . . . . 6 (𝑣 = 𝑢 → (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5655cbvmptv 4678 . . . . 5 (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩))) = (𝑢𝑊 ↦ (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5739, 56eqtri 2632 . . . 4 𝑇 = (𝑢𝑊 ↦ (𝑎 ∈ (0...(#‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5838, 57fvmptg 6189 . . 3 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∈ V) → (𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5933, 58mpdan 699 . 2 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
6059feq1d 5943 . . 3 (𝑋𝑊 → ((𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊 ↔ (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
6121, 60mpbird 246 . 2 (𝑋𝑊 → (𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
6259, 61jca 553 1 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ⟨cop 4131  ⟨cotp 4133   ↦ cmpt 4643   I cid 4948   × cxp 5036  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441  0cc0 9815  ...cfz 12197  #chash 12979  Word cword 13146   splice csplice 13151  ⟨“cs2 13437   ~FG cefg 17942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444 This theorem is referenced by:  efgtval  17959  efgval2  17960  efgtlen  17962  efginvrel2  17963  efgsp1  17973  efgredleme  17979  efgredlem  17983  efgrelexlemb  17986  efgcpbllemb  17991  frgpnabllem1  18099
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