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Theorem efgtf 18335
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 6418 . . . . . . . . . 10 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3776 . . . . . . . . 9 𝑊 ⊆ Word (𝐼 × 2𝑜)
4 simpl 474 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
53, 4sseldi 3742 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
6 simprr 813 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
7 efgval2.m . . . . . . . . . . . 12 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
87efgmf 18326 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
98ffvelrni 6521 . . . . . . . . . 10 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
109ad2antll 767 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
116, 10s2cld 13816 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
12 splcl 13703 . . . . . . . 8 ((𝑋 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
135, 11, 12syl2anc 696 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
141efgrcl 18328 . . . . . . . . 9 (𝑋𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
1514simprd 482 . . . . . . . 8 (𝑋𝑊𝑊 = Word (𝐼 × 2𝑜))
1615adantr 472 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
1713, 16eleqtrrd 2842 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
1817ralrimivva 3109 . . . . 5 (𝑋𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
19 eqid 2760 . . . . . 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 7405 . . . . 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 208 . . . 4 (𝑋𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)):((0...(♯‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
22 ovex 6841 . . . . 5 (0...(♯‘𝑋)) ∈ V
2314simpld 477 . . . . . 6 (𝑋𝑊𝐼 ∈ V)
24 2on 7737 . . . . . 6 2𝑜 ∈ On
25 xpexg 7125 . . . . . 6 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
2623, 24, 25sylancl 697 . . . . 5 (𝑋𝑊 → (𝐼 × 2𝑜) ∈ V)
27 xpexg 7125 . . . . 5 (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2𝑜) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
2822, 26, 27sylancr 698 . . . 4 (𝑋𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
29 fvex 6362 . . . . . 6 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
301, 29eqeltri 2835 . . . . 5 𝑊 ∈ V
3130a1i 11 . . . 4 (𝑋𝑊𝑊 ∈ V)
32 fex2 7286 . . . 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 1477 . . 3 (𝑋𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∈ V)
34 fveq2 6352 . . . . . 6 (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
3534oveq2d 6829 . . . . 5 (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
36 eqidd 2761 . . . . 5 (𝑢 = 𝑋 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
37 oveq1 6820 . . . . 5 (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
3835, 36, 37mpt2eq123dv 6882 . . . 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 4562 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)
41 oteq2 4563 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4240, 41eqtrd 2794 . . . . . . . . 9 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4342oveq2d 6829 . . . . . . . 8 (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩))
44 id 22 . . . . . . . . . . 11 (𝑤 = 𝑏𝑤 = 𝑏)
45 fveq2 6352 . . . . . . . . . . 11 (𝑤 = 𝑏 → (𝑀𝑤) = (𝑀𝑏))
4644, 45s2eqd 13808 . . . . . . . . . 10 (𝑤 = 𝑏 → ⟨“𝑤(𝑀𝑤)”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
4746oteq3d 4567 . . . . . . . . 9 (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)
4847oveq2d 6829 . . . . . . . 8 (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
4943, 48cbvmpt2v 6900 . . . . . . 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 6352 . . . . . . . . 9 (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
5150oveq2d 6829 . . . . . . . 8 (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
52 eqidd 2761 . . . . . . . 8 (𝑣 = 𝑢 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
53 oveq1 6820 . . . . . . . 8 (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
5451, 52, 53mpt2eq123dv 6882 . . . . . . 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 2806 . . . . . 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 4902 . . . . 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 2782 . . . 4 𝑇 = (𝑢𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5838, 57fvmptg 6442 . . 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 705 . 2 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
6059feq1d 6191 . . 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 247 . 2 (𝑋𝑊 → (𝑇𝑋):((0...(♯‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊)
6259, 61jca 555 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 1632  wcel 2139  wral 3050  Vcvv 3340  cdif 3712  cop 4327  cotp 4329  cmpt 4881   I cid 5173   × cxp 5264  Oncon0 5884  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  1𝑜c1o 7722  2𝑜c2o 7723  0cc0 10128  ...cfz 12519  chash 13311  Word cword 13477   splice csplice 13482  ⟨“cs2 13786   ~FG cefg 18319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-concat 13487  df-s1 13488  df-substr 13489  df-splice 13490  df-s2 13793
This theorem is referenced by:  efgtval  18336  efgval2  18337  efgtlen  18339  efginvrel2  18340  efgsp1  18350  efgredleme  18356  efgredlem  18360  efgrelexlemb  18363  efgcpbllemb  18368  frgpnabllem1  18476
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