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Theorem efgtf 18401
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 6445 . . . . . . . . . 10 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3795 . . . . . . . . 9 𝑊 ⊆ Word (𝐼 × 2𝑜)
4 simpl 474 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
53, 4sseldi 3759 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
6 simprr 789 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
7 efgval2.m . . . . . . . . . . . 12 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
87efgmf 18392 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
98ffvelrni 6548 . . . . . . . . . 10 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
109ad2antll 720 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
116, 10s2cld 13902 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
12 splcl 13770 . . . . . . . 8 ((𝑋 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
135, 11, 12syl2anc 579 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
141efgrcl 18394 . . . . . . . . 9 (𝑋𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
1514simprd 489 . . . . . . . 8 (𝑋𝑊𝑊 = Word (𝐼 × 2𝑜))
1615adantr 472 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
1713, 16eleqtrrd 2847 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
1817ralrimivva 3118 . . . . 5 (𝑋𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
19 eqid 2765 . . . . . 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 7438 . . . . 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 209 . . . 4 (𝑋𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
22 ovex 6874 . . . . 5 (0...(♯‘𝑋)) ∈ V
2314simpld 488 . . . . . 6 (𝑋𝑊𝐼 ∈ V)
24 2on 7773 . . . . . 6 2𝑜 ∈ On
25 xpexg 7158 . . . . . 6 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
2623, 24, 25sylancl 580 . . . . 5 (𝑋𝑊 → (𝐼 × 2𝑜) ∈ V)
27 xpexg 7158 . . . . 5 (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2𝑜) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
2822, 26, 27sylancr 581 . . . 4 (𝑋𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
291fvexi 6389 . . . . 5 𝑊 ∈ V
3029a1i 11 . . . 4 (𝑋𝑊𝑊 ∈ V)
31 fex2 7319 . . . 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)
3221, 28, 30, 31syl3anc 1490 . . 3 (𝑋𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∈ V)
33 fveq2 6375 . . . . . 6 (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
3433oveq2d 6858 . . . . 5 (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
35 eqidd 2766 . . . . 5 (𝑢 = 𝑋 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
36 oveq1 6849 . . . . 5 (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
3734, 35, 36mpt2eq123dv 6915 . . . 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)”⟩⟩)))
38 efgval2.t . . . . 5 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
39 oteq1 4568 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)
40 oteq2 4569 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4139, 40eqtrd 2799 . . . . . . . . 9 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4241oveq2d 6858 . . . . . . . 8 (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩))
43 id 22 . . . . . . . . . . 11 (𝑤 = 𝑏𝑤 = 𝑏)
44 fveq2 6375 . . . . . . . . . . 11 (𝑤 = 𝑏 → (𝑀𝑤) = (𝑀𝑏))
4543, 44s2eqd 13894 . . . . . . . . . 10 (𝑤 = 𝑏 → ⟨“𝑤(𝑀𝑤)”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
4645oteq3d 4573 . . . . . . . . 9 (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)
4746oveq2d 6858 . . . . . . . 8 (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
4842, 47cbvmpt2v 6933 . . . . . . 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)”⟩⟩))
49 fveq2 6375 . . . . . . . . 9 (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
5049oveq2d 6858 . . . . . . . 8 (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
51 eqidd 2766 . . . . . . . 8 (𝑣 = 𝑢 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
52 oveq1 6849 . . . . . . . 8 (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
5350, 51, 52mpt2eq123dv 6915 . . . . . . 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)”⟩⟩)))
5448, 53syl5eq 2811 . . . . . 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)”⟩⟩)))
5554cbvmptv 4909 . . . . 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)”⟩⟩)))
5638, 55eqtri 2787 . . . 4 𝑇 = (𝑢𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5737, 56fvmptg 6469 . . 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)”⟩⟩)))
5832, 57mpdan 678 . 2 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5958feq1d 6208 . . 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))
6021, 59mpbird 248 . 2 (𝑋𝑊 → (𝑇𝑋):((0...(♯‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
6158, 60jca 507 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 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3729  cop 4340  cotp 4342  cmpt 4888   I cid 5184   × cxp 5275  Oncon0 5908  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  1𝑜c1o 7757  2𝑜c2o 7758  0cc0 10189  ...cfz 12533  chash 13321  Word cword 13486   splice csplice 13763  ⟨“cs2 13872   ~FG cefg 18385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-concat 13542  df-s1 13567  df-substr 13617  df-pfx 13662  df-splice 13765  df-s2 13879
This theorem is referenced by:  efgtval  18402  efgval2  18403  efgtlen  18405  efginvrel2  18406  efgsp1  18416  efgredleme  18422  efgredlem  18426  efgrelexlemb  18429  efgcpbllemb  18434  frgpnabllem1  18542
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