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Theorem efgi 18339
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 6332 . . . . . . . . . . 11 (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
21oveq2d 6809 . . . . . . . . . 10 (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3 id 22 . . . . . . . . . . . 12 (𝑢 = 𝐴𝑢 = 𝐴)
4 oveq1 6800 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
53, 4breq12d 4799 . . . . . . . . . . 11 (𝑢 = 𝐴 → (𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
652ralbidv 3138 . . . . . . . . . 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 3301 . . . . . . . . 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 3456 . . . . . . . 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 4548 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
10 oteq2 4549 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
119, 10eqtrd 2805 . . . . . . . . . . . 12 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
1211oveq2d 6809 . . . . . . . . . . 11 (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
1312breq2d 4798 . . . . . . . . . 10 (𝑖 = 𝑁 → (𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
14132ralbidv 3138 . . . . . . . . 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 3456 . . . . . . . 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 497 . . . . . . 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 4539 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18 opeq1 4539 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝑏)⟩)
1917, 18s2eqd 13817 . . . . . . . . . . 11 (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩)
2019oteq3d 4553 . . . . . . . . . 10 (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)
2120oveq2d 6809 . . . . . . . . 9 (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩))
2221breq2d 4798 . . . . . . . 8 (𝑎 = 𝐽 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)))
23 opeq2 4540 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24 difeq2 3873 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (1𝑜𝑏) = (1𝑜𝐾))
2524opeq2d 4546 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝐾)⟩)
2623, 25s2eqd 13817 . . . . . . . . . . . 12 (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩)
2726oteq3d 4553 . . . . . . . . . . 11 (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)
2827oveq2d 6809 . . . . . . . . . 10 (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩))
2928breq2d 4798 . . . . . . . . 9 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)))
30 df-br 4787 . . . . . . . . 9 (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ 𝑟)
3129, 30syl6bb 276 . . . . . . . 8 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ 𝑟))
3222, 31rspc2v 3472 . . . . . . 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 497 . . . . . 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 478 . . . . 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 2007 . . . 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 5060 . . . . 5 𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ V
3736elintab 4622 . . . 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 224 . . 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 18337 . . 3 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4238, 41syl6eleqr 2861 . 2 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2𝑜)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ )
43 df-br 4787 . 2 (𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩) ↔ ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ )
4442, 43sylibr 224 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 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  cdif 3720  cop 4322  cotp 4324   cint 4611   class class class wbr 4786   I cid 5156   × cxp 5247  cfv 6031  (class class class)co 6793  1𝑜c1o 7706  2𝑜c2o 7707   Er wer 7893  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  df-efg 18329
This theorem is referenced by:  efgi0  18340  efgi1  18341
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