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Theorem efgval2 18450
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
Assertion
Ref Expression
efgval2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Distinct variable groups:   𝑦,𝑟,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧,𝑟,𝑥   𝑛,𝑀   𝑣,𝑟,𝑤,𝑥,𝑀   𝑇,𝑟,𝑥   𝑛,𝑊,𝑟,𝑣,𝑤   𝑥,𝑦,𝑧,𝑊   ,𝑟,𝑥,𝑦,𝑧   𝑛,𝐼,𝑟,𝑣,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efgval2
Dummy variables 𝑎 𝑏 𝑢 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . 3 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 efgval.r . . 3 = ( ~FG𝐼)
31, 2efgval 18443 . 2 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4 efgval2.m . . . . . . . . . . 11 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
5 efgval2.t . . . . . . . . . . 11 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
61, 2, 4, 5efgtf 18448 . . . . . . . . . 10 (𝑥𝑊 → ((𝑇𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ∧ (𝑇𝑥):((0...(♯‘𝑥)) × (𝐼 × 2𝑜))⟶𝑊))
76simpld 489 . . . . . . . . 9 (𝑥𝑊 → (𝑇𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
87rneqd 5556 . . . . . . . 8 (𝑥𝑊 → ran (𝑇𝑥) = ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
98sseq1d 3828 . . . . . . 7 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟))
10 dfss3 3787 . . . . . . . 8 (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟)
11 ovex 6910 . . . . . . . . . . 11 (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
1211rgen2w 3106 . . . . . . . . . 10 𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
13 eqid 2799 . . . . . . . . . . 11 (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))
14 vex 3388 . . . . . . . . . . . . 13 𝑎 ∈ V
15 vex 3388 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15elec 8024 . . . . . . . . . . . 12 (𝑎 ∈ [𝑥]𝑟𝑥𝑟𝑎)
17 breq2 4847 . . . . . . . . . . . 12 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑥𝑟𝑎𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1816, 17syl5bb 275 . . . . . . . . . . 11 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑎 ∈ [𝑥]𝑟𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1913, 18ralrnmpt2 7009 . . . . . . . . . 10 (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V → (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
2012, 19ax-mp 5 . . . . . . . . 9 (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))
21 id 22 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → 𝑢 = ⟨𝑎, 𝑏⟩)
22 fveq2 6411 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 6881 . . . . . . . . . . . . . . . . 17 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23syl6eqr 2851 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑎𝑀𝑏))
2521, 24s2eqd 13948 . . . . . . . . . . . . . . 15 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
2625oteq3d 4607 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
2726oveq2d 6894 . . . . . . . . . . . . 13 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
2827breq2d 4855 . . . . . . . . . . . 12 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
2928ralxp 5467 . . . . . . . . . . 11 (∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
30 eqidd 2800 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
314efgmval 18438 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
3230, 31s2eqd 13948 . . . . . . . . . . . . . . . 16 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)
3332oteq3d 4607 . . . . . . . . . . . . . . 15 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
3433oveq2d 6894 . . . . . . . . . . . . . 14 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3534breq2d 4855 . . . . . . . . . . . . 13 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
3635ralbidva 3166 . . . . . . . . . . . 12 (𝑎𝐼 → (∀𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
3736ralbiia 3160 . . . . . . . . . . 11 (∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3829, 37bitri 267 . . . . . . . . . 10 (∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3938ralbii 3161 . . . . . . . . 9 (∀𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4020, 39bitri 267 . . . . . . . 8 (∀𝑎 ∈ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩))𝑎 ∈ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4110, 40bitri 267 . . . . . . 7 (ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
429, 41syl6bb 279 . . . . . 6 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
4342ralbiia 3160 . . . . 5 (∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4443anbi2i 617 . . . 4 ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟) ↔ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
4544abbii 2916 . . 3 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4645inteqi 4671 . 2 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)} = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
473, 46eqtr4i 2824 1 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157  {cab 2785  wral 3089  Vcvv 3385  cdif 3766  wss 3769  cop 4374  cotp 4376   cint 4667   class class class wbr 4843  cmpt 4922   I cid 5219   × cxp 5310  ran crn 5313  wf 6097  cfv 6101  (class class class)co 6878  cmpt2 6880  1𝑜c1o 7792  2𝑜c2o 7793   Er wer 7979  [cec 7980  0cc0 10224  ...cfz 12580  chash 13370  Word cword 13534   splice csplice 13819  ⟨“cs2 13926   ~FG cefg 18432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-ot 4377  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-ec 7984  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-concat 13591  df-s1 13616  df-substr 13665  df-pfx 13714  df-splice 13821  df-s2 13933  df-efg 18435
This theorem is referenced by:  efgi2  18451  efgrelexlemb  18478  efgcpbllemb  18483
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