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Theorem efgval2 18344
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 18337 . 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 18342 . . . . . . . . . 10 (𝑥𝑊 → ((𝑇𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ∧ (𝑇𝑥):((0...(♯‘𝑥)) × (𝐼 × 2𝑜))⟶𝑊))
76simpld 476 . . . . . . . . 9 (𝑥𝑊 → (𝑇𝑥) = (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
87rneqd 5491 . . . . . . . 8 (𝑥𝑊 → ran (𝑇𝑥) = ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
98sseq1d 3781 . . . . . . 7 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ran (𝑚 ∈ (0...(♯‘𝑥)), 𝑢 ∈ (𝐼 × 2𝑜) ↦ (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ⊆ [𝑥]𝑟))
10 dfss3 3741 . . . . . . . 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 6823 . . . . . . . . . . 11 (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
1211rgen2w 3074 . . . . . . . . . 10 𝑚 ∈ (0...(♯‘𝑥))∀𝑢 ∈ (𝐼 × 2𝑜)(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ∈ V
13 eqid 2771 . . . . . . . . . . 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 3354 . . . . . . . . . . . . 13 𝑎 ∈ V
15 vex 3354 . . . . . . . . . . . . 13 𝑥 ∈ V
1614, 15elec 7938 . . . . . . . . . . . 12 (𝑎 ∈ [𝑥]𝑟𝑥𝑟𝑎)
17 breq2 4790 . . . . . . . . . . . 12 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑥𝑟𝑎𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1816, 17syl5bb 272 . . . . . . . . . . 11 (𝑎 = (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) → (𝑎 ∈ [𝑥]𝑟𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)))
1913, 18ralrnmpt2 6922 . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . . 17 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑀‘⟨𝑎, 𝑏⟩))
23 df-ov 6796 . . . . . . . . . . . . . . . . 17 (𝑎𝑀𝑏) = (𝑀‘⟨𝑎, 𝑏⟩)
2422, 23syl6eqr 2823 . . . . . . . . . . . . . . . 16 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑀𝑢) = (𝑎𝑀𝑏))
2521, 24s2eqd 13817 . . . . . . . . . . . . . . 15 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨“𝑢(𝑀𝑢)”⟩ = ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩)
2625oteq3d 4553 . . . . . . . . . . . . . 14 (𝑢 = ⟨𝑎, 𝑏⟩ → ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)
2726oveq2d 6809 . . . . . . . . . . . . 13 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
2827breq2d 4798 . . . . . . . . . . . 12 (𝑢 = ⟨𝑎, 𝑏⟩ → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩)))
2928ralxp 5402 . . . . . . . . . . 11 (∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩))
30 eqidd 2772 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, 𝑏⟩)
314efgmval 18332 . . . . . . . . . . . . . . . . 17 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎𝑀𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
3230, 31s2eqd 13817 . . . . . . . . . . . . . . . 16 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩ = ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)
3332oteq3d 4553 . . . . . . . . . . . . . . 15 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩ = ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)
3433oveq2d 6809 . . . . . . . . . . . . . 14 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) = (𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3534breq2d 4798 . . . . . . . . . . . . 13 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
3635ralbidva 3134 . . . . . . . . . . . 12 (𝑎𝐼 → (∀𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
3736ralbiia 3128 . . . . . . . . . . 11 (∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩(𝑎𝑀𝑏)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3829, 37bitri 264 . . . . . . . . . 10 (∀𝑢 ∈ (𝐼 × 2𝑜)𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
3938ralbii 3129 . . . . . . . . 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 264 . . . . . . . 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 264 . . . . . . 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 276 . . . . . 6 (𝑥𝑊 → (ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
4342ralbiia 3128 . . . . 5 (∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟 ↔ ∀𝑥𝑊𝑚 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑚, 𝑚, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))
4443anbi2i 601 . . . 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 2888 . . 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 4615 . 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 2796 1 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wral 3061  Vcvv 3351  cdif 3720  wss 3723  cop 4322  cotp 4324   cint 4611   class class class wbr 4786  cmpt 4863   I cid 5156   × cxp 5247  ran crn 5250  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  1𝑜c1o 7706  2𝑜c2o 7707   Er wer 7893  [cec 7894  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 827  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-ec 7898  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:  efgi2  18345  efgrelexlemb  18370  efgcpbllemb  18375
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