MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efginvrel2 Structured version   Visualization version   GIF version

Theorem efginvrel2 17905
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 6147 . . . 4 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3593 . . 3 𝑊 ⊆ Word (𝐼 × 2𝑜)
43sseli 3559 . 2 (𝐴𝑊𝐴 ∈ Word (𝐼 × 2𝑜))
5 id 22 . . . . . 6 (𝑐 = ∅ → 𝑐 = ∅)
6 fveq2 6084 . . . . . . . . 9 (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7 rev0 13306 . . . . . . . . 9 (reverse‘∅) = ∅
86, 7syl6eq 2655 . . . . . . . 8 (𝑐 = ∅ → (reverse‘𝑐) = ∅)
98coeq2d 5190 . . . . . . 7 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10 co02 5548 . . . . . . 7 (𝑀 ∘ ∅) = ∅
119, 10syl6eq 2655 . . . . . 6 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
125, 11oveq12d 6541 . . . . 5 (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
1312breq1d 4583 . . . 4 (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (∅ ++ ∅) ∅))
1413imbi2d 328 . . 3 (𝑐 = ∅ → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (∅ ++ ∅) ∅)))
15 id 22 . . . . . 6 (𝑐 = 𝑎𝑐 = 𝑎)
16 fveq2 6084 . . . . . . 7 (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
1716coeq2d 5190 . . . . . 6 (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
1815, 17oveq12d 6541 . . . . 5 (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
1918breq1d 4583 . . . 4 (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅))
2019imbi2d 328 . . 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 6084 . . . . . . 7 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
2322coeq2d 5190 . . . . . 6 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
2421, 23oveq12d 6541 . . . . 5 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
2524breq1d 4583 . . . 4 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
2625imbi2d 328 . . 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 6084 . . . . . . 7 (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
2928coeq2d 5190 . . . . . 6 (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
3027, 29oveq12d 6541 . . . . 5 (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
3130breq1d 4583 . . . 4 (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
3231imbi2d 328 . . 3 (𝑐 = 𝐴 → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)))
33 wrd0 13127 . . . . 5 ∅ ∈ Word (𝐼 × 2𝑜)
34 ccatlid 13164 . . . . 5 (∅ ∈ Word (𝐼 × 2𝑜) → (∅ ++ ∅) = ∅)
3533, 34ax-mp 5 . . . 4 (∅ ++ ∅) = ∅
36 efgval.r . . . . . . 7 = ( ~FG𝐼)
371, 36efger 17896 . . . . . 6 Er 𝑊
3837a1i 11 . . . . 5 (𝐴𝑊 Er 𝑊)
391efgrcl 17893 . . . . . . 7 (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
4039simprd 477 . . . . . 6 (𝐴𝑊𝑊 = Word (𝐼 × 2𝑜))
4133, 40syl5eleqr 2690 . . . . 5 (𝐴𝑊 → ∅ ∈ 𝑊)
4238, 41erref 7622 . . . 4 (𝐴𝑊 → ∅ ∅)
4335, 42syl5eqbr 4608 . . 3 (𝐴𝑊 → (∅ ++ ∅) ∅)
4437a1i 11 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → Er 𝑊)
45 simprl 789 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ Word (𝐼 × 2𝑜))
46 revcl 13303 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝑎) ∈ Word (𝐼 × 2𝑜))
4746ad2antrl 759 . . . . . . . . . . 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 17891 . . . . . . . . . . 11 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
50 wrdco 13370 . . . . . . . . . . 11 (((reverse‘𝑎) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜))
5147, 49, 50sylancl 692 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜))
52 ccatcl 13154 . . . . . . . . . 10 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2𝑜))
5345, 51, 52syl2anc 690 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2𝑜))
5440adantr 479 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑊 = Word (𝐼 × 2𝑜))
5553, 54eleqtrrd 2686 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
56 lencl 13121 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2𝑜) → (#‘𝑎) ∈ ℕ0)
5756ad2antrl 759 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ ℕ0)
58 nn0uz 11550 . . . . . . . . . . . . 13 0 = (ℤ‘0)
5957, 58syl6eleq 2693 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ (ℤ‘0))
60 ccatlen 13155 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2𝑜)) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))))
6145, 51, 60syl2anc 690 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))))
6257nn0zd 11308 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ ℤ)
63 uzid 11530 . . . . . . . . . . . . . . 15 ((#‘𝑎) ∈ ℤ → (#‘𝑎) ∈ (ℤ‘(#‘𝑎)))
6462, 63syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ (ℤ‘(#‘𝑎)))
65 lencl 13121 . . . . . . . . . . . . . . 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 11572 . . . . . . . . . . . . . 14 (((#‘𝑎) ∈ (ℤ‘(#‘𝑎)) ∧ (#‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎)))
6864, 66, 67syl2anc 690 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑎) + (#‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎)))
6961, 68eqeltrd 2683 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎)))
70 elfzuzb 12158 . . . . . . . . . . . 12 ((#‘𝑎) ∈ (0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((#‘𝑎) ∈ (ℤ‘0) ∧ (#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(#‘𝑎))))
7159, 69, 70sylanbrc 694 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ (0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
72 simprr 791 . . . . . . . . . . 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 17901 . . . . . . . . . . 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 1317 . . . . . . . . . 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 6247 . . . . . . . . . . . . 13 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
7872, 77syl 17 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
7972, 78s2cld 13408 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
80 ccatrid 13165 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2𝑜) → (𝑎 ++ ∅) = 𝑎)
8180ad2antrl 759 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ ∅) = 𝑎)
8281eqcomd 2611 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 = (𝑎 ++ ∅))
8382oveq1d 6538 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
84 eqidd 2606 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) = (#‘𝑎))
85 hash0 12967 . . . . . . . . . . . . 13 (#‘∅) = 0
8685oveq2i 6534 . . . . . . . . . . . 12 ((#‘𝑎) + (#‘∅)) = ((#‘𝑎) + 0)
8757nn0cnd 11196 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) ∈ ℂ)
8887addid1d 10083 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑎) + 0) = (#‘𝑎))
8986, 88syl5req 2652 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘𝑎) = ((#‘𝑎) + (#‘∅)))
9045, 76, 51, 79, 83, 84, 89splval2 13301 . . . . . . . . . 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 13178 . . . . . . . . . . . . . . . 16 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜))
92 revccat 13308 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
9345, 91, 92syl2anc 690 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
94 revs1 13307 . . . . . . . . . . . . . . . 16 (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
9594oveq1i 6533 . . . . . . . . . . . . . . 15 ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
9693, 95syl6eq 2655 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
9796coeq2d 5190 . . . . . . . . . . . . 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 13374 . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
101 s1co 13372 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
10272, 49, 101sylancl 692 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
103102oveq1d 6538 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
10497, 100, 1033eqtrd 2643 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
105104oveq2d 6539 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
106 ccatcl 13154 . . . . . . . . . . . . 13 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜))
10745, 91, 106syl2anc 690 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2𝑜))
10878s1cld 13178 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
109 ccatass 13166 . . . . . . . . . . . 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 1317 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
111 ccatass 13166 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2𝑜) ∧ ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
11245, 91, 108, 111syl3anc 1317 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
113 df-s2 13386 . . . . . . . . . . . . . 14 ⟨“𝑏(𝑀𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)
114113oveq2i 6534 . . . . . . . . . . . . 13 (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩))
115112, 114syl6eqr 2657 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩))
116115oveq1d 6538 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
117105, 110, 1163eqtr2rd 2646 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
11875, 90, 1173eqtrd 2643 . . . . . . . . 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 17900 . . . . . . . . . . . 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 477 . . . . . . . . . . 11 ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(#‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2𝑜))⟶𝑊)
121 ffn 5940 . . . . . . . . . . 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 6680 . . . . . . . . . 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 1317 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
125118, 124eqeltrrd 2684 . . . . . . . 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 17903 . . . . . . . 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 690 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
12844, 127ersym 7614 . . . . . 6 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
12944ertr 7617 . . . . . 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 706 . . . . 5 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2𝑜) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
131130expcom 449 . . . 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 13270 . 2 (𝐴 ∈ Word (𝐼 × 2𝑜) → (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
1344, 133mpcom 37 1 (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cdif 3532  c0 3869  cop 4126  cotp 4128   class class class wbr 4573  cmpt 4633   I cid 4934   × cxp 5022  ran crn 5025  ccom 5028   Fn wfn 5781  wf 5782  cfv 5786  (class class class)co 6523  cmpt2 6525  1𝑜c1o 7413  2𝑜c2o 7414   Er wer 7599  0cc0 9788   + caddc 9791  0cn0 11135  cz 11206  cuz 11515  ...cfz 12148  #chash 12930  Word cword 13088   ++ cconcat 13090  ⟨“cs1 13091   splice csplice 13093  reversecreverse 13094  ⟨“cs2 13379   ~FG cefg 17884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-ot 4129  df-uni 4363  df-int 4401  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-ec 7604  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-hash 12931  df-word 13096  df-lsw 13097  df-concat 13098  df-s1 13099  df-substr 13100  df-splice 13101  df-reverse 13102  df-s2 13386  df-efg 17887
This theorem is referenced by:  efginvrel1  17906  frgpinv  17942
  Copyright terms: Public domain W3C validator