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Theorem efgi 18332
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 6352 . . . . . . . . . . 11 (𝑢 = 𝐴 → (♯‘𝑢) = (♯‘𝐴))
21oveq2d 6829 . . . . . . . . . 10 (𝑢 = 𝐴 → (0...(♯‘𝑢)) = (0...(♯‘𝐴)))
3 id 22 . . . . . . . . . . . 12 (𝑢 = 𝐴𝑢 = 𝐴)
4 oveq1 6820 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
53, 4breq12d 4817 . . . . . . . . . . 11 (𝑢 = 𝐴 → (𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
652ralbidv 3127 . . . . . . . . . 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 3291 . . . . . . . . 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 3445 . . . . . . . 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 4562 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
10 oteq2 4563 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
119, 10eqtrd 2794 . . . . . . . . . . . 12 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
1211oveq2d 6829 . . . . . . . . . . 11 (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
1312breq2d 4816 . . . . . . . . . 10 (𝑖 = 𝑁 → (𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
14132ralbidv 3127 . . . . . . . . 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 3445 . . . . . . . 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 692 . . . . . . 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 4553 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18 opeq1 4553 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝑏)⟩)
1917, 18s2eqd 13808 . . . . . . . . . . 11 (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩)
2019oteq3d 4567 . . . . . . . . . 10 (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)
2120oveq2d 6829 . . . . . . . . 9 (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩))
2221breq2d 4816 . . . . . . . 8 (𝑎 = 𝐽 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)))
23 opeq2 4554 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24 difeq2 3865 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (1𝑜𝑏) = (1𝑜𝐾))
2524opeq2d 4560 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝐾)⟩)
2623, 25s2eqd 13808 . . . . . . . . . . . 12 (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩)
2726oteq3d 4567 . . . . . . . . . . 11 (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)
2827oveq2d 6829 . . . . . . . . . 10 (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩))
2928breq2d 4816 . . . . . . . . 9 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)))
30 df-br 4805 . . . . . . . . 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 3461 . . . . . . 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 692 . . . . . 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 484 . . . . 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 2004 . . . 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 5081 . . . . 5 𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ V
3736elintab 4639 . . . 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 18330 . . 3 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(♯‘𝑢))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4238, 41syl6eleqr 2850 . 2 (((𝐴𝑊𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2𝑜)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ )
43 df-br 4805 . 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 383  wal 1630   = wceq 1632  wcel 2139  {cab 2746  wral 3050  cdif 3712  cop 4327  cotp 4329   cint 4627   class class class wbr 4804   I cid 5173   × cxp 5264  cfv 6049  (class class class)co 6813  1𝑜c1o 7722  2𝑜c2o 7723   Er wer 7908  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  df-efg 18322
This theorem is referenced by:  efgi0  18333  efgi1  18334
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