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Theorem efgi 18072
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 6158 . . . . . . . . . . 11 (𝑢 = 𝐴 → (#‘𝑢) = (#‘𝐴))
21oveq2d 6631 . . . . . . . . . 10 (𝑢 = 𝐴 → (0...(#‘𝑢)) = (0...(#‘𝐴)))
3 id 22 . . . . . . . . . . . 12 (𝑢 = 𝐴𝑢 = 𝐴)
4 oveq1 6622 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
53, 4breq12d 4636 . . . . . . . . . . 11 (𝑢 = 𝐴 → (𝑢𝑟(𝑢 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
652ralbidv 2985 . . . . . . . . . 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 3145 . . . . . . . . 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 3295 . . . . . . . 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 4386 . . . . . . . . . . . . 13 (𝑖 = 𝑁 → ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
10 oteq2 4387 . . . . . . . . . . . . 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 6631 . . . . . . . . . . 11 (𝑖 = 𝑁 → (𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
1312breq2d 4635 . . . . . . . . . 10 (𝑖 = 𝑁 → (𝐴𝑟(𝐴 splice ⟨𝑖, 𝑖, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
14132ralbidv 2985 . . . . . . . . 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 3295 . . . . . . . 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 4377 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, 𝑏⟩ = ⟨𝐽, 𝑏⟩)
18 opeq1 4377 . . . . . . . . . . . 12 (𝑎 = 𝐽 → ⟨𝑎, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝑏)⟩)
1917, 18s2eqd 13561 . . . . . . . . . . 11 (𝑎 = 𝐽 → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩)
2019oteq3d 4391 . . . . . . . . . 10 (𝑎 = 𝐽 → ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)
2120oveq2d 6631 . . . . . . . . 9 (𝑎 = 𝐽 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩))
2221breq2d 4635 . . . . . . . 8 (𝑎 = 𝐽 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩)))
23 opeq2 4378 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, 𝑏⟩ = ⟨𝐽, 𝐾⟩)
24 difeq2 3706 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (1𝑜𝑏) = (1𝑜𝐾))
2524opeq2d 4384 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ⟨𝐽, (1𝑜𝑏)⟩ = ⟨𝐽, (1𝑜𝐾)⟩)
2623, 25s2eqd 13561 . . . . . . . . . . . 12 (𝑏 = 𝐾 → ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩ = ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩)
2726oteq3d 4391 . . . . . . . . . . 11 (𝑏 = 𝐾 → ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩ = ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)
2827oveq2d 6631 . . . . . . . . . 10 (𝑏 = 𝐾 → (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) = (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩))
2928breq2d 4635 . . . . . . . . 9 (𝑏 = 𝐾 → (𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝑏⟩⟨𝐽, (1𝑜𝑏)⟩”⟩⟩) ↔ 𝐴𝑟(𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)))
30 df-br 4624 . . . . . . . . 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 3311 . . . . . . 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 4903 . . . . 5 𝐴, (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩)⟩ ∈ V
3736elintab 4459 . . . 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 18070 . . 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 4624 . 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 2908  cdif 3557  cop 4161  cotp 4163   cint 4447   class class class wbr 4623   I cid 4994   × cxp 5082  cfv 5857  (class class class)co 6615  1𝑜c1o 7513  2𝑜c2o 7514   Er wer 7699  0cc0 9896  ...cfz 12284  #chash 13073  Word cword 13246   splice csplice 13251  ⟨“cs2 13539   ~FG cefg 18059
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-ot 4164  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-concat 13256  df-s1 13257  df-substr 13258  df-splice 13259  df-s2 13546  df-efg 18062
This theorem is referenced by:  efgi0  18073  efgi1  18074
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