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Theorem efgi 18053
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 6148 . . . . . . . . . . 11 (𝑢 = 𝐴 → (#‘𝑢) = (#‘𝐴))
21oveq2d 6620 . . . . . . . . . 10 (𝑢 = 𝐴 → (0...(#‘𝑢)) = (0...(#‘𝐴)))
3 id 22 . . . . . . . . . . . 12 (𝑢 = 𝐴𝑢 = 𝐴)
4 oveq1 6611 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
53, 4breq12d 4626 . . . . . . . . . . 11 (𝑢 = 𝐴 → (𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
652ralbidv 2983 . . . . . . . . . 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 3141 . . . . . . . . 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 3291 . . . . . . . 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 4379 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
10 oteq2 4380 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
119, 10eqtrd 2655 . . . . . . . . . . . 12 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
1211oveq2d 6620 . . . . . . . . . . 11 (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
1312breq2d 4625 . . . . . . . . . 10 (𝑖 = 𝑁 → (𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
14132ralbidv 2983 . . . . . . . . 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 3291 . . . . . . . 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 688 . . . . . . 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 4370 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18 opeq1 4370 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝑏)⟩)
1917, 18s2eqd 13545 . . . . . . . . . . 11 (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩)
2019oteq3d 4384 . . . . . . . . . 10 (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)
2120oveq2d 6620 . . . . . . . . 9 (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩))
2221breq2d 4625 . . . . . . . 8 (𝑎 = 𝐽 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)))
23 opeq2 4371 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24 difeq2 3700 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (1𝑜𝑏) = (1𝑜𝐾))
2524opeq2d 4377 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝐾)⟩)
2623, 25s2eqd 13545 . . . . . . . . . . . 12 (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩)
2726oteq3d 4384 . . . . . . . . . . 11 (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)
2827oveq2d 6620 . . . . . . . . . 10 (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩))
2928breq2d 4625 . . . . . . . . 9 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)))
30 df-br 4614 . . . . . . . . 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 3306 . . . . . . 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 688 . . . . . 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 483 . . . . 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 1852 . . . 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 4893 . . . . 5 𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ V
3736elintab 4452 . . . 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 18051 . . 3 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑢𝑊𝑖 ∈ (0...(#‘𝑢))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4238, 41syl6eleqr 2709 . 2 (((𝐴𝑊𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2𝑜)) → ⟨𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ )
43 df-br 4614 . 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 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  cdif 3552  cop 4154  cotp 4156   cint 4440   class class class wbr 4613   I cid 4984   × cxp 5072  cfv 5847  (class class class)co 6604  1𝑜c1o 7498  2𝑜c2o 7499   Er wer 7684  0cc0 9880  ...cfz 12268  #chash 13057  Word cword 13230   splice csplice 13235  ⟨“cs2 13523   ~FG cefg 18040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-s2 13530  df-efg 18043
This theorem is referenced by:  efgi0  18054  efgi1  18055
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