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Theorem efginvrel2 18406
Description: The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-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
efginvrel2 (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)
Distinct variable groups:   𝑦,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧   𝑛,𝑀,𝑣,𝑤   𝑛,𝑊,𝑣,𝑤,𝑦,𝑧   𝑦, ,𝑧   𝑛,𝐼,𝑣,𝑤,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑤,𝑣,𝑛)   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efginvrel2
Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . 4 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 fviss 6445 . . . 4 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3795 . . 3 𝑊 ⊆ Word (𝐼 × 2𝑜)
43sseli 3757 . 2 (𝐴𝑊𝐴 ∈ Word (𝐼 × 2𝑜))
5 id 22 . . . . . 6 (𝑐 = ∅ → 𝑐 = ∅)
6 fveq2 6375 . . . . . . . . 9 (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7 rev0 13788 . . . . . . . . 9 (reverse‘∅) = ∅
86, 7syl6eq 2815 . . . . . . . 8 (𝑐 = ∅ → (reverse‘𝑐) = ∅)
98coeq2d 5453 . . . . . . 7 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10 co02 5835 . . . . . . 7 (𝑀 ∘ ∅) = ∅
119, 10syl6eq 2815 . . . . . 6 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
125, 11oveq12d 6860 . . . . 5 (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
1312breq1d 4819 . . . 4 (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (∅ ++ ∅) ∅))
1413imbi2d 331 . . 3 (𝑐 = ∅ → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (∅ ++ ∅) ∅)))
15 id 22 . . . . . 6 (𝑐 = 𝑎𝑐 = 𝑎)
16 fveq2 6375 . . . . . . 7 (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
1716coeq2d 5453 . . . . . 6 (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
1815, 17oveq12d 6860 . . . . 5 (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
1918breq1d 4819 . . . 4 (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅))
2019imbi2d 331 . . 3 (𝑐 = 𝑎 → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅)))
21 id 22 . . . . . 6 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → 𝑐 = (𝑎 ++ ⟨“𝑏”⟩))
22 fveq2 6375 . . . . . . 7 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
2322coeq2d 5453 . . . . . 6 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
2421, 23oveq12d 6860 . . . . 5 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
2524breq1d 4819 . . . 4 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
2625imbi2d 331 . . 3 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅)))
27 id 22 . . . . . 6 (𝑐 = 𝐴𝑐 = 𝐴)
28 fveq2 6375 . . . . . . 7 (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
2928coeq2d 5453 . . . . . 6 (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
3027, 29oveq12d 6860 . . . . 5 (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
3130breq1d 4819 . . . 4 (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
3231imbi2d 331 . . 3 (𝑐 = 𝐴 → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)))
33 wrd0 13511 . . . . 5 ∅ ∈ Word (𝐼 × 2𝑜)
34 ccatlid 13557 . . . . 5 (∅ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ∅) = ∅)
3533, 34ax-mp 5 . . . 4 (∅ ++ ∅) = ∅
36 efgval.r . . . . . . 7 = ( ~FG𝐼)
371, 36efger 18397 . . . . . 6 Er 𝑊
3837a1i 11 . . . . 5 (𝐴𝑊 Er 𝑊)
391efgrcl 18394 . . . . . . 7 (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
4039simprd 489 . . . . . 6 (𝐴𝑊𝑊 = Word (𝐼 × 2𝑜))
4133, 40syl5eleqr 2851 . . . . 5 (𝐴𝑊 → ∅ ∈ 𝑊)
4238, 41erref 7967 . . . 4 (𝐴𝑊 → ∅ ∅)
4335, 42syl5eqbr 4844 . . 3 (𝐴𝑊 → (∅ ++ ∅) ∅)
4437a1i 11 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → Er 𝑊)
45 simprl 787 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ Word (𝐼 × 2𝑜))
46 revcl 13785 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝑎) ∈ Word (𝐼 × 2𝑜))
4746ad2antrl 719 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘𝑎) ∈ Word (𝐼 × 2𝑜))
48 efgval2.m . . . . . . . . . . . 12 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
4948efgmf 18392 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
50 wrdco 13862 . . . . . . . . . . 11 (((reverse‘𝑎) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜))
5147, 49, 50sylancl 580 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜))
52 ccatcl 13545 . . . . . . . . . 10 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2𝑜))
5345, 51, 52syl2anc 579 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2𝑜))
5440adantr 472 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
5553, 54eleqtrrd 2847 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
56 lencl 13505 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2𝑜) → (♯‘𝑎) ∈ ℕ0)
5756ad2antrl 719 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) ∈ ℕ0)
58 nn0uz 11922 . . . . . . . . . . . . 13 0 = (ℤ‘0)
5957, 58syl6eleq 2854 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) ∈ (ℤ‘0))
60 ccatlen 13546 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
6145, 51, 60syl2anc 579 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
6257nn0zd 11727 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) ∈ ℤ)
63 uzid 11901 . . . . . . . . . . . . . . 15 ((♯‘𝑎) ∈ ℤ → (♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)))
6462, 63syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)))
65 lencl 13505 . . . . . . . . . . . . . . 15 ((𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
6651, 65syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
67 uzaddcl 11944 . . . . . . . . . . . . . 14 (((♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)) ∧ (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
6864, 66, 67syl2anc 579 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
6961, 68eqeltrd 2844 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
70 elfzuzb 12543 . . . . . . . . . . . 12 ((♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((♯‘𝑎) ∈ (ℤ‘0) ∧ (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎))))
7159, 69, 70sylanbrc 578 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
72 simprr 789 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
73 efgval2.t . . . . . . . . . . . 12 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
741, 36, 48, 73efgtval 18402 . . . . . . . . . . 11 (((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 ∧ (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ∧ 𝑏 ∈ (𝐼 × 2𝑜)) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀𝑏)”⟩⟩))
7555, 71, 72, 74syl3anc 1490 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀𝑏)”⟩⟩))
7633a1i 11 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ∅ ∈ Word (𝐼 × 2𝑜))
7749ffvelrni 6548 . . . . . . . . . . . . 13 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
7872, 77syl 17 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
7972, 78s2cld 13902 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
80 ccatrid 13558 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2𝑜) → (𝑎 ++ ∅) = 𝑎)
8180ad2antrl 719 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ ∅) = 𝑎)
8281eqcomd 2771 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (𝑎 ++ ∅))
8382oveq1d 6857 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
84 eqidd 2766 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) = (♯‘𝑎))
85 hash0 13360 . . . . . . . . . . . . 13 (♯‘∅) = 0
8685oveq2i 6853 . . . . . . . . . . . 12 ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
8757nn0cnd 11600 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) ∈ ℂ)
8887addid1d 10490 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
8986, 88syl5req 2812 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
9045, 76, 51, 79, 83, 84, 89splval2 13780 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
9172s1cld 13574 . . . . . . . . . . . . . . . 16 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜))
92 revccat 13790 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
9345, 91, 92syl2anc 579 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
94 revs1 13789 . . . . . . . . . . . . . . . 16 (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
9594oveq1i 6852 . . . . . . . . . . . . . . 15 ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
9693, 95syl6eq 2815 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
9796coeq2d 5453 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))))
9849a1i 11 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜))
99 ccatco 13866 . . . . . . . . . . . . . 14 ((⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜) ∧ (reverse‘𝑎) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
10091, 47, 98, 99syl3anc 1490 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
101 s1co 13864 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
10272, 49, 101sylancl 580 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
103102oveq1d 6857 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
10497, 100, 1033eqtrd 2803 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
105104oveq2d 6858 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
106 ccatcl 13545 . . . . . . . . . . . . 13 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜))
10745, 91, 106syl2anc 579 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜))
10878s1cld 13574 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
109 ccatass 13559 . . . . . . . . . . . 12 (((𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜) ∧ ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
110107, 108, 51, 109syl3anc 1490 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
111 ccatass 13559 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜) ∧ ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
11245, 91, 108, 111syl3anc 1490 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
113 df-s2 13879 . . . . . . . . . . . . . 14 ⟨“𝑏(𝑀𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)
114113oveq2i 6853 . . . . . . . . . . . . 13 (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩))
115112, 114syl6eqr 2817 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩))
116115oveq1d 6857 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
117105, 110, 1163eqtr2rd 2806 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
11875, 90, 1173eqtrd 2803 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
1191, 36, 48, 73efgtf 18401 . . . . . . . . . . . 12 ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → ((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = (𝑚 ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))), 𝑢 ∈ (𝐼 × 2𝑜) ↦ ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨𝑚, 𝑚, ⟨“𝑢(𝑀𝑢)”⟩⟩)) ∧ (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜))⟶𝑊))
120119simprd 489 . . . . . . . . . . 11 ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜))⟶𝑊)
121 ffn 6223 . . . . . . . . . . 11 ((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜))⟶𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜)))
12255, 120, 1213syl 18 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜)))
123 fnovrn 7007 . . . . . . . . . 10 (((𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜)) ∧ (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ∧ 𝑏 ∈ (𝐼 × 2𝑜)) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
124122, 71, 72, 123syl3anc 1490 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
125118, 124eqeltrrd 2845 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
1261, 36, 48, 73efgi2 18404 . . . . . . . 8 (((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 ∧ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
12755, 125, 126syl2anc 579 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
12844, 127ersym 7959 . . . . . 6 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
12944ertr 7962 . . . . . 6 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∧ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
130128, 129mpand 686 . . . . 5 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
131130expcom 402 . . . 4 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜)) → (𝐴𝑊 → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅)))
132131a2d 29 . . 3 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜)) → ((𝐴𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅) → (𝐴𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅)))
13314, 20, 26, 32, 43, 132wrdind 13720 . 2 (𝐴 ∈ Word (𝐼 × 2𝑜) → (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
1344, 133mpcom 38 1 (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  c0 4079  cop 4340  cotp 4342   class class class wbr 4809  cmpt 4888   I cid 5184   × cxp 5275  ran crn 5278  ccom 5281   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  1𝑜c1o 7757  2𝑜c2o 7758   Er wer 7944  0cc0 10189   + caddc 10192  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  chash 13321  Word cword 13486   ++ cconcat 13541  ⟨“cs1 13566   splice csplice 13763  reversecreverse 13782  ⟨“cs2 13872   ~FG cefg 18385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-ec 7949  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-lsw 13534  df-concat 13542  df-s1 13567  df-substr 13617  df-pfx 13662  df-splice 13765  df-reverse 13783  df-s2 13879  df-efg 18388
This theorem is referenced by:  efginvrel1  18407  frgpinv  18443
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