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Theorem efgval2 19757
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
efgval2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Distinct variable groups:   𝑦,𝑟,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧,𝑟,𝑥   𝑛,𝑀   𝑣,𝑟,𝑤,𝑥,𝑀   𝑇,𝑟,𝑥   𝑛,𝑊,𝑟,𝑣,𝑤   𝑥,𝑦,𝑧,𝑊   ,𝑟,𝑥,𝑦,𝑧   𝑛,𝐼,𝑟,𝑣,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efgval2
Dummy variables 𝑎 𝑏 𝑢 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . 3 𝑊 = ( I ‘Word (𝐼 × 2o))
2 efgval.r . . 3 = ( ~FG𝐼)
31, 2efgval 19750 . 2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))}
4 efgval2.m . . . . . . . . . . 11 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
5 efgval2.t . . . . . . . . . . 11 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
61, 2, 4, 5efgtf 19755 . . . . . . . . . 10 (𝑥𝑊 → ((𝑇𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ∧ (𝑇𝑥):((0...(♯‘𝑥)) × (𝐼 × 2o))⟶𝑊))
76simpld 494 . . . . . . . . 9 (𝑥𝑊 → (𝑇𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
87rneqd 5952 . . . . . . . 8 (𝑥𝑊 → ran (𝑇𝑥) = ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
98sseq1d 4027 . . . . . . 7 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟))
10 dfss3 3984 . . . . . . . 8 (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟)
11 ovex 7464 . . . . . . . . . . 11 (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
1211rgen2w 3064 . . . . . . . . . 10 𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
13 eqid 2735 . . . . . . . . . . 11 (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))
14 vex 3482 . . . . . . . . . . . . 13 𝑎 ∈ V
15 vex 3482 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15elec 8790 . . . . . . . . . . . 12 (𝑎 ∈ [𝑥]𝑟𝑥𝑟𝑎)
17 breq2 5152 . . . . . . . . . . . 12 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑥𝑟𝑎𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1816, 17bitrid 283 . . . . . . . . . . 11 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑎 ∈ [𝑥]𝑟𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1913, 18ralrnmpo 7572 . . . . . . . . . 10 (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V → (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
2012, 19ax-mp 5 . . . . . . . . 9 (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))
21 id 22 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → 𝑢 = ⟨𝑎, 𝑏⟩)
22 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 7434 . . . . . . . . . . . . . . . . 17 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23eqtr4di 2793 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑎𝑀𝑏))
2521, 24s2eqd 14899 . . . . . . . . . . . . . . 15 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
2625oteq3d 4892 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
2726oveq2d 7447 . . . . . . . . . . . . 13 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
2827breq2d 5160 . . . . . . . . . . . 12 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
2928ralxp 5855 . . . . . . . . . . 11 (∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
30 eqidd 2736 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
314efgmval 19745 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎𝑀𝑏) = ⟨𝑎, (1o𝑏)⟩)
3230, 31s2eqd 14899 . . . . . . . . . . . . . . . 16 ((𝑎𝐼𝑏 ∈ 2o) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)
3332oteq3d 4892 . . . . . . . . . . . . . . 15 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)
3433oveq2d 7447 . . . . . . . . . . . . . 14 ((𝑎𝐼𝑏 ∈ 2o) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
3534breq2d 5160 . . . . . . . . . . . . 13 ((𝑎𝐼𝑏 ∈ 2o) → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
3635ralbidva 3174 . . . . . . . . . . . 12 (𝑎𝐼 → (∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
3736ralbiia 3089 . . . . . . . . . . 11 (∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
3829, 37bitri 275 . . . . . . . . . 10 (∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
3938ralbii 3091 . . . . . . . . 9 (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2o)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
4020, 39bitri 275 . . . . . . . 8 (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
4110, 40bitri 275 . . . . . . 7 (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2o) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
429, 41bitrdi 287 . . . . . 6 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
4342ralbiia 3089 . . . . 5 (∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
4443anbi2i 623 . . . 4 ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
4544abbii 2807 . . 3 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))}
4645inteqi 4955 . 2 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))}
473, 46eqtr4i 2766 1 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  cdif 3960  wss 3963  cop 4637  cotp 4639   cint 4951   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1oc1o 8498  2oc2o 8499   Er wer 8741  [cec 8742  0cc0 11153  ...cfz 13544  chash 14366  Word cword 14549   splice csplice 14784  ⟨“cs2 14877   ~FG cefg 19739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-s2 14884  df-efg 19742
This theorem is referenced by:  efgi2  19758  efgrelexlemb  19783  efgcpbllemb  19788
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