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Theorem efginvrel2 18080
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 6223 . . . 4 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3620 . . 3 𝑊 ⊆ Word (𝐼 × 2𝑜)
43sseli 3584 . 2 (𝐴𝑊𝐴 ∈ Word (𝐼 × 2𝑜))
5 id 22 . . . . . 6 (𝑐 = ∅ → 𝑐 = ∅)
6 fveq2 6158 . . . . . . . . 9 (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7 rev0 13466 . . . . . . . . 9 (reverse‘∅) = ∅
86, 7syl6eq 2671 . . . . . . . 8 (𝑐 = ∅ → (reverse‘𝑐) = ∅)
98coeq2d 5254 . . . . . . 7 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10 co02 5618 . . . . . . 7 (𝑀 ∘ ∅) = ∅
119, 10syl6eq 2671 . . . . . 6 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
125, 11oveq12d 6633 . . . . 5 (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
1312breq1d 4633 . . . 4 (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (∅ ++ ∅) ∅))
1413imbi2d 330 . . 3 (𝑐 = ∅ → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (∅ ++ ∅) ∅)))
15 id 22 . . . . . 6 (𝑐 = 𝑎𝑐 = 𝑎)
16 fveq2 6158 . . . . . . 7 (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
1716coeq2d 5254 . . . . . 6 (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
1815, 17oveq12d 6633 . . . . 5 (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
1918breq1d 4633 . . . 4 (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅))
2019imbi2d 330 . . 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 6158 . . . . . . 7 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
2322coeq2d 5254 . . . . . 6 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
2421, 23oveq12d 6633 . . . . 5 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
2524breq1d 4633 . . . 4 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
2625imbi2d 330 . . 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 6158 . . . . . . 7 (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
2928coeq2d 5254 . . . . . 6 (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
3027, 29oveq12d 6633 . . . . 5 (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
3130breq1d 4633 . . . 4 (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
3231imbi2d 330 . . 3 (𝑐 = 𝐴 → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)))
33 wrd0 13285 . . . . 5 ∅ ∈ Word (𝐼 × 2𝑜)
34 ccatlid 13324 . . . . 5 (∅ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ∅) = ∅)
3533, 34ax-mp 5 . . . 4 (∅ ++ ∅) = ∅
36 efgval.r . . . . . . 7 = ( ~FG𝐼)
371, 36efger 18071 . . . . . 6 Er 𝑊
3837a1i 11 . . . . 5 (𝐴𝑊 Er 𝑊)
391efgrcl 18068 . . . . . . 7 (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
4039simprd 479 . . . . . 6 (𝐴𝑊𝑊 = Word (𝐼 × 2𝑜))
4133, 40syl5eleqr 2705 . . . . 5 (𝐴𝑊 → ∅ ∈ 𝑊)
4238, 41erref 7722 . . . 4 (𝐴𝑊 → ∅ ∅)
4335, 42syl5eqbr 4658 . . 3 (𝐴𝑊 → (∅ ++ ∅) ∅)
4437a1i 11 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → Er 𝑊)
45 simprl 793 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ Word (𝐼 × 2𝑜))
46 revcl 13463 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝑎) ∈ Word (𝐼 × 2𝑜))
4746ad2antrl 763 . . . . . . . . . . 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 18066 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
50 wrdco 13530 . . . . . . . . . . 11 (((reverse‘𝑎) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜))
5147, 49, 50sylancl 693 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜))
52 ccatcl 13314 . . . . . . . . . 10 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2𝑜))
5345, 51, 52syl2anc 692 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2𝑜))
5440adantr 481 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
5553, 54eleqtrrd 2701 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
56 lencl 13279 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2𝑜) → (#‘𝑎) ∈ ℕ0)
5756ad2antrl 763 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ ℕ0)
58 nn0uz 11682 . . . . . . . . . . . . 13 0 = (ℤ‘0)
5957, 58syl6eleq 2708 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ (ℤ‘0))
60 ccatlen 13315 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))))
6145, 51, 60syl2anc 692 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))))
6257nn0zd 11440 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ ℤ)
63 uzid 11662 . . . . . . . . . . . . . . 15 ((#‘𝑎) ∈ ℤ → (#‘𝑎) ∈ (ℤ‘(#‘𝑎)))
6462, 63syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ (ℤ‘(#‘𝑎)))
65 lencl 13279 . . . . . . . . . . . . . . 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 11704 . . . . . . . . . . . . . 14 (((#‘𝑎) ∈ (ℤ‘(#‘𝑎)) ∧ (#‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎)))
6864, 66, 67syl2anc 692 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎)))
6961, 68eqeltrd 2698 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎)))
70 elfzuzb 12294 . . . . . . . . . . . 12 ((#‘𝑎) ∈ (0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((#‘𝑎) ∈ (ℤ‘0) ∧ (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎))))
7159, 69, 70sylanbrc 697 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ (0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
72 simprr 795 . . . . . . . . . . 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 18076 . . . . . . . . . . 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 1323 . . . . . . . . . 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 6324 . . . . . . . . . . . . 13 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
7872, 77syl 17 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
7972, 78s2cld 13568 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
80 ccatrid 13325 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2𝑜) → (𝑎 ++ ∅) = 𝑎)
8180ad2antrl 763 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ ∅) = 𝑎)
8281eqcomd 2627 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (𝑎 ++ ∅))
8382oveq1d 6630 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
84 eqidd 2622 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) = (#‘𝑎))
85 hash0 13114 . . . . . . . . . . . . 13 (#‘∅) = 0
8685oveq2i 6626 . . . . . . . . . . . 12 ((#‘𝑎) + (#‘∅)) = ((#‘𝑎) + 0)
8757nn0cnd 11313 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ ℂ)
8887addid1d 10196 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑎) + 0) = (#‘𝑎))
8986, 88syl5req 2668 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) = ((#‘𝑎) + (#‘∅)))
9045, 76, 51, 79, 83, 84, 89splval2 13461 . . . . . . . . . 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 13338 . . . . . . . . . . . . . . . 16 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜))
92 revccat 13468 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
9345, 91, 92syl2anc 692 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
94 revs1 13467 . . . . . . . . . . . . . . . 16 (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
9594oveq1i 6625 . . . . . . . . . . . . . . 15 ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
9693, 95syl6eq 2671 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
9796coeq2d 5254 . . . . . . . . . . . . 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 13534 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
101 s1co 13532 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
10272, 49, 101sylancl 693 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
103102oveq1d 6630 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
10497, 100, 1033eqtrd 2659 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
105104oveq2d 6631 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
106 ccatcl 13314 . . . . . . . . . . . . 13 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜))
10745, 91, 106syl2anc 692 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜))
10878s1cld 13338 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
109 ccatass 13326 . . . . . . . . . . . 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 1323 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
111 ccatass 13326 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜) ∧ ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
11245, 91, 108, 111syl3anc 1323 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
113 df-s2 13546 . . . . . . . . . . . . . 14 ⟨“𝑏(𝑀𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)
114113oveq2i 6626 . . . . . . . . . . . . 13 (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩))
115112, 114syl6eqr 2673 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩))
116115oveq1d 6630 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
117105, 110, 1163eqtr2rd 2662 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
11875, 90, 1173eqtrd 2659 . . . . . . . . 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 18075 . . . . . . . . . . . 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 479 . . . . . . . . . . 11 ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜))⟶𝑊)
121 ffn 6012 . . . . . . . . . . 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 6774 . . . . . . . . . 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 1323 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
125118, 124eqeltrrd 2699 . . . . . . . 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 18078 . . . . . . . 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 692 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
12844, 127ersym 7714 . . . . . 6 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
12944ertr 7717 . . . . . 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 710 . . . . 5 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
131130expcom 451 . . . 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 13430 . 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 1480  wcel 1987  Vcvv 3190  cdif 3557  c0 3897  cop 4161  cotp 4163   class class class wbr 4623  cmpt 4683   I cid 4994   × cxp 5082  ran crn 5085  ccom 5088   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617  1𝑜c1o 7513  2𝑜c2o 7514   Er wer 7699  0cc0 9896   + caddc 9899  0cn0 11252  cz 11337  cuz 11647  ...cfz 12284  #chash 13073  Word cword 13246   ++ cconcat 13248  ⟨“cs1 13249   splice csplice 13251  reversecreverse 13252  ⟨“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-iin 4495  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-ec 7704  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-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-lsw 13255  df-concat 13256  df-s1 13257  df-substr 13258  df-splice 13259  df-reverse 13260  df-s2 13546  df-efg 18062
This theorem is referenced by:  efginvrel1  18081  frgpinv  18117
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