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Theorem efgi 18837
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2o))
efgval.r = ( ~FG𝐼)
Assertion
Ref Expression
efgi (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → 𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩))

Proof of Theorem efgi
Dummy variables 𝑎 𝑏 𝑖 𝑟 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . . . . . . . . . 11 (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
21oveq2d 7164 . . . . . . . . . 10 (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3 id 22 . . . . . . . . . . . 12 (𝑢 = 𝐴𝑢 = 𝐴)
4 oveq1 7155 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
53, 4breq12d 5070 . . . . . . . . . . 11 (𝑢 = 𝐴 → (𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
652ralbidv 3197 . . . . . . . . . 10 (𝑢 = 𝐴 → (∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
72, 6raleqbidv 3400 . . . . . . . . 9 (𝑢 = 𝐴 → (∀𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ ∀𝑖 ∈ (0...(♯‘𝐴))∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
87rspcv 3616 . . . . . . . 8 (𝐴𝑊 → (∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → ∀𝑖 ∈ (0...(♯‘𝐴))∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
9 oteq1 4804 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)
10 oteq2 4805 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)
119, 10eqtrd 2854 . . . . . . . . . . . 12 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)
1211oveq2d 7164 . . . . . . . . . . 11 (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
1312breq2d 5069 . . . . . . . . . 10 (𝑖 = 𝑁 → (𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
14132ralbidv 3197 . . . . . . . . 9 (𝑖 = 𝑁 → (∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
1514rspcv 3616 . . . . . . . 8 (𝑁 ∈ (0...(♯‘𝐴)) → (∀𝑖 ∈ (0...(♯‘𝐴))∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → ∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
168, 15sylan9 510 . . . . . . 7 ((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) → (∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → ∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
17 opeq1 4795 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18 opeq1 4795 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, (1o𝑏)⟩ = ⟨𝐽, (1o𝑏)⟩)
1917, 18s2eqd 14217 . . . . . . . . . . 11 (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩)
2019oteq3d 4809 . . . . . . . . . 10 (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩)
2120oveq2d 7164 . . . . . . . . 9 (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩))
2221breq2d 5069 . . . . . . . 8 (𝑎 = 𝐽 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩)))
23 opeq2 4796 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24 difeq2 4091 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (1o𝑏) = (1o𝐾))
2524opeq2d 4802 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, (1o𝑏)⟩ = ⟨𝐽, (1o𝐾)⟩)
2623, 25s2eqd 14217 . . . . . . . . . . . 12 (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩)
2726oteq3d 4809 . . . . . . . . . . 11 (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)
2827oveq2d 7164 . . . . . . . . . 10 (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩))
2928breq2d 5069 . . . . . . . . 9 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)))
30 df-br 5058 . . . . . . . . 9 (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟)
3129, 30syl6bb 289 . . . . . . . 8 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1o𝑏)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
3222, 31rspc2v 3631 . . . . . . 7 ((𝐽𝐼𝐾 ∈ 2o) → (∀𝑎𝐼𝑏 ∈ 2o 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
3316, 32sylan9 510 . . . . . 6 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → (∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
3433adantld 493 . . . . 5 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → ((𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
3534alrimiv 1921 . . . 4 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
36 opex 5347 . . . . 5 𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ V
3736elintab 4878 . . . 4 (⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))} ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
3835, 37sylibr 236 . . 3 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))})
39 efgval.w . . . 4 𝑊 = ( I ‘Word (𝐼 × 2o))
40 efgval.r . . . 4 = ( ~FG𝐼)
4139, 40efgval 18835 . . 3 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2o 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))}
4238, 41eleqtrrdi 2922 . 2 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ )
43 df-br 5058 . 2 (𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩)⟩ ∈ )
4442, 43sylibr 236 1 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2o)) → 𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o𝐾)⟩”⟩⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1528   = wceq 1530  wcel 2107  {cab 2797  wral 3136  cdif 3931  cop 4565  cotp 4567   cint 4867   class class class wbr 5057   I cid 5452   × cxp 5546  cfv 6348  (class class class)co 7148  1oc1o 8087  2oc2o 8088   Er wer 8278  0cc0 10529  ...cfz 12884  chash 13682  Word cword 13853   splice csplice 14103  ⟨“cs2 14195   ~FG cefg 18824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-hash 13683  df-word 13854  df-concat 13915  df-s1 13942  df-substr 13995  df-pfx 14025  df-splice 14104  df-s2 14202  df-efg 18827
This theorem is referenced by:  efgi0  18838  efgi1  18839
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