MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgtf Structured version   Visualization version   GIF version

Theorem efgtf 18342
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 6398 . . . . . . . . . 10 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3784 . . . . . . . . 9 𝑊 ⊆ Word (𝐼 × 2𝑜)
4 simpl 468 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
53, 4sseldi 3750 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
6 simprr 756 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
7 efgval2.m . . . . . . . . . . . 12 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
87efgmf 18333 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
98ffvelrni 6501 . . . . . . . . . 10 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
109ad2antll 708 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
116, 10s2cld 13825 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
12 splcl 13712 . . . . . . . 8 ((𝑋 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
135, 11, 12syl2anc 573 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ Word (𝐼 × 2𝑜))
141efgrcl 18335 . . . . . . . . 9 (𝑋𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
1514simprd 483 . . . . . . . 8 (𝑋𝑊𝑊 = Word (𝐼 × 2𝑜))
1615adantr 466 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
1713, 16eleqtrrd 2853 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
1817ralrimivva 3120 . . . . 5 (𝑋𝑊 → ∀𝑎 ∈ (0...(♯‘𝑋))∀𝑏 ∈ (𝐼 × 2𝑜)(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ 𝑊)
19 eqid 2771 . . . . . 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 7387 . . . . 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 6823 . . . . 5 (0...(♯‘𝑋)) ∈ V
2314simpld 482 . . . . . 6 (𝑋𝑊𝐼 ∈ V)
24 2on 7722 . . . . . 6 2𝑜 ∈ On
25 xpexg 7107 . . . . . 6 ((𝐼 ∈ V ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
2623, 24, 25sylancl 574 . . . . 5 (𝑋𝑊 → (𝐼 × 2𝑜) ∈ V)
27 xpexg 7107 . . . . 5 (((0...(♯‘𝑋)) ∈ V ∧ (𝐼 × 2𝑜) ∈ V) → ((0...(♯‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
2822, 26, 27sylancr 575 . . . 4 (𝑋𝑊 → ((0...(♯‘𝑋)) × (𝐼 × 2𝑜)) ∈ V)
29 fvex 6342 . . . . . 6 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
301, 29eqeltri 2846 . . . . 5 𝑊 ∈ V
3130a1i 11 . . . 4 (𝑋𝑊𝑊 ∈ V)
32 fex2 7268 . . . 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 1476 . . 3 (𝑋𝑊 → (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∈ V)
34 fveq2 6332 . . . . . 6 (𝑢 = 𝑋 → (♯‘𝑢) = (♯‘𝑋))
3534oveq2d 6809 . . . . 5 (𝑢 = 𝑋 → (0...(♯‘𝑢)) = (0...(♯‘𝑋)))
36 eqidd 2772 . . . . 5 (𝑢 = 𝑋 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
37 oveq1 6800 . . . . 5 (𝑢 = 𝑋 → (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
3835, 36, 37mpt2eq123dv 6864 . . . 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 4548 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)
41 oteq2 4549 . . . . . . . . . 10 (𝑛 = 𝑎 → ⟨𝑎, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4240, 41eqtrd 2805 . . . . . . . . 9 (𝑛 = 𝑎 → ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩)
4342oveq2d 6809 . . . . . . . 8 (𝑛 = 𝑎 → (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩))
44 id 22 . . . . . . . . . . 11 (𝑤 = 𝑏𝑤 = 𝑏)
45 fveq2 6332 . . . . . . . . . . 11 (𝑤 = 𝑏 → (𝑀𝑤) = (𝑀𝑏))
4644, 45s2eqd 13817 . . . . . . . . . 10 (𝑤 = 𝑏 → ⟨“𝑤(𝑀𝑤)”⟩ = ⟨“𝑏(𝑀𝑏)”⟩)
4746oteq3d 4553 . . . . . . . . 9 (𝑤 = 𝑏 → ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩ = ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)
4847oveq2d 6809 . . . . . . . 8 (𝑤 = 𝑏 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑤(𝑀𝑤)”⟩⟩) = (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
4943, 48cbvmpt2v 6882 . . . . . . 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 6332 . . . . . . . . 9 (𝑣 = 𝑢 → (♯‘𝑣) = (♯‘𝑢))
5150oveq2d 6809 . . . . . . . 8 (𝑣 = 𝑢 → (0...(♯‘𝑣)) = (0...(♯‘𝑢)))
52 eqidd 2772 . . . . . . . 8 (𝑣 = 𝑢 → (𝐼 × 2𝑜) = (𝐼 × 2𝑜))
53 oveq1 6800 . . . . . . . 8 (𝑣 = 𝑢 → (𝑣 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
5451, 52, 53mpt2eq123dv 6864 . . . . . . 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 2817 . . . . . 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 4884 . . . . 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 2793 . . . 4 𝑇 = (𝑢𝑊 ↦ (𝑎 ∈ (0...(♯‘𝑢)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑢 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
5838, 57fvmptg 6422 . . 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 667 . 2 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
6059feq1d 6170 . . 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 501 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 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cdif 3720  cop 4322  cotp 4324  cmpt 4863   I cid 5156   × cxp 5247  Oncon0 5866  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  1𝑜c1o 7706  2𝑜c2o 7707  0cc0 10138  ...cfz 12533  chash 13321  Word cword 13487   splice csplice 13492  ⟨“cs2 13795   ~FG cefg 18326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-ot 4325  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-substr 13499  df-splice 13500  df-s2 13802
This theorem is referenced by:  efgtval  18343  efgval2  18344  efgtlen  18346  efginvrel2  18347  efgsp1  18357  efgredleme  18363  efgredlem  18367  efgrelexlemb  18370  efgcpbllemb  18375  frgpnabllem1  18483
  Copyright terms: Public domain W3C validator