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Theorem efginvrel2 18782
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 6734 . . . 4 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
31, 2eqsstri 3998 . . 3 𝑊 ⊆ Word (𝐼 × 2o)
43sseli 3960 . 2 (𝐴𝑊𝐴 ∈ Word (𝐼 × 2o))
5 id 22 . . . . . 6 (𝑐 = ∅ → 𝑐 = ∅)
6 fveq2 6663 . . . . . . . . 9 (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7 rev0 14114 . . . . . . . . 9 (reverse‘∅) = ∅
86, 7syl6eq 2869 . . . . . . . 8 (𝑐 = ∅ → (reverse‘𝑐) = ∅)
98coeq2d 5726 . . . . . . 7 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10 co02 6106 . . . . . . 7 (𝑀 ∘ ∅) = ∅
119, 10syl6eq 2869 . . . . . 6 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
125, 11oveq12d 7163 . . . . 5 (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
1312breq1d 5067 . . . 4 (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (∅ ++ ∅) ∅))
1413imbi2d 342 . . 3 (𝑐 = ∅ → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (∅ ++ ∅) ∅)))
15 id 22 . . . . . 6 (𝑐 = 𝑎𝑐 = 𝑎)
16 fveq2 6663 . . . . . . 7 (𝑐 = 𝑎 → (reverse‘𝑐) = (reverse‘𝑎))
1716coeq2d 5726 . . . . . 6 (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
1815, 17oveq12d 7163 . . . . 5 (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
1918breq1d 5067 . . . 4 (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅))
2019imbi2d 342 . . 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 6663 . . . . . . 7 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (reverse‘𝑐) = (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))
2322coeq2d 5726 . . . . . 6 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
2421, 23oveq12d 7163 . . . . 5 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
2524breq1d 5067 . . . 4 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
2625imbi2d 342 . . 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 6663 . . . . . . 7 (𝑐 = 𝐴 → (reverse‘𝑐) = (reverse‘𝐴))
2928coeq2d 5726 . . . . . 6 (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
3027, 29oveq12d 7163 . . . . 5 (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
3130breq1d 5067 . . . 4 (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
3231imbi2d 342 . . 3 (𝑐 = 𝐴 → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)))
33 ccatidid 13932 . . . 4 (∅ ++ ∅) = ∅
34 efgval.r . . . . . . 7 = ( ~FG𝐼)
351, 34efger 18773 . . . . . 6 Er 𝑊
3635a1i 11 . . . . 5 (𝐴𝑊 Er 𝑊)
37 wrd0 13877 . . . . . 6 ∅ ∈ Word (𝐼 × 2o)
381efgrcl 18770 . . . . . . 7 (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3938simprd 496 . . . . . 6 (𝐴𝑊𝑊 = Word (𝐼 × 2o))
4037, 39eleqtrrid 2917 . . . . 5 (𝐴𝑊 → ∅ ∈ 𝑊)
4136, 40erref 8298 . . . 4 (𝐴𝑊 → ∅ ∅)
4233, 41eqbrtrid 5092 . . 3 (𝐴𝑊 → (∅ ++ ∅) ∅)
4335a1i 11 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → Er 𝑊)
44 simprl 767 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ Word (𝐼 × 2o))
45 revcl 14111 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2o) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
4645ad2antrl 724 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
47 efgval2.m . . . . . . . . . . . 12 𝑀 = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
4847efgmf 18768 . . . . . . . . . . 11 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
49 wrdco 14181 . . . . . . . . . . 11 (((reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
5046, 48, 49sylancl 586 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
51 ccatcl 13914 . . . . . . . . . 10 ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
5244, 50, 51syl2anc 584 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
5339adantr 481 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
5452, 53eleqtrrd 2913 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
55 lencl 13871 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
5655ad2antrl 724 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℕ0)
57 nn0uz 12268 . . . . . . . . . . . . 13 0 = (ℤ‘0)
5856, 57eleqtrdi 2920 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ‘0))
59 ccatlen 13915 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
6044, 50, 59syl2anc 584 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
6156nn0zd 12073 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℤ)
6261uzidd 12247 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)))
63 lencl 13871 . . . . . . . . . . . . . . 15 ((𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
6450, 63syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0)
65 uzaddcl 12292 . . . . . . . . . . . . . 14 (((♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)) ∧ (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
6662, 64, 65syl2anc 584 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
6760, 66eqeltrd 2910 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
68 elfzuzb 12890 . . . . . . . . . . . 12 ((♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((♯‘𝑎) ∈ (ℤ‘0) ∧ (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎))))
6958, 67, 68sylanbrc 583 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
70 simprr 769 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
71 efgval2.t . . . . . . . . . . . 12 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
721, 34, 47, 71efgtval 18778 . . . . . . . . . . 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)”⟩⟩))
7354, 69, 70, 72syl3anc 1363 . . . . . . . . . 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)”⟩⟩))
7437a1i 11 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ∅ ∈ Word (𝐼 × 2o))
7548ffvelrni 6842 . . . . . . . . . . . . 13 (𝑏 ∈ (𝐼 × 2o) → (𝑀𝑏) ∈ (𝐼 × 2o))
7675ad2antll 725 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀𝑏) ∈ (𝐼 × 2o))
7770, 76s2cld 14221 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
78 ccatrid 13929 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2o) → (𝑎 ++ ∅) = 𝑎)
7978ad2antrl 724 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ∅) = 𝑎)
8079eqcomd 2824 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 ++ ∅))
8180oveq1d 7160 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
82 eqidd 2819 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = (♯‘𝑎))
83 hash0 13716 . . . . . . . . . . . . 13 (♯‘∅) = 0
8483oveq2i 7156 . . . . . . . . . . . 12 ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
8556nn0cnd 11945 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℂ)
8685addid1d 10828 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
8784, 86syl5req 2866 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
8844, 74, 50, 77, 81, 82, 87splval2 14107 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) splice ⟨(♯‘𝑎), (♯‘𝑎), ⟨“𝑏(𝑀𝑏)”⟩⟩) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
8970s1cld 13945 . . . . . . . . . . . . . . . 16 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o))
90 revccat 14116 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
9144, 89, 90syl2anc 584 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
92 revs1 14115 . . . . . . . . . . . . . . . 16 (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
9392oveq1i 7155 . . . . . . . . . . . . . . 15 ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
9491, 93syl6eq 2869 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
9594coeq2d 5726 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))))
9648a1i 11 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o))
97 ccatco 14185 . . . . . . . . . . . . . 14 ((⟨“𝑏”⟩ ∈ Word (𝐼 × 2o) ∧ (reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
9889, 46, 96, 97syl3anc 1363 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
99 s1co 14183 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
10070, 48, 99sylancl 586 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
101100oveq1d 7160 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
10295, 98, 1013eqtrd 2857 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
103102oveq2d 7161 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
104 ccatcl 13914 . . . . . . . . . . . . 13 ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
10544, 89, 104syl2anc 584 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
10676s1cld 13945 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
107 ccatass 13930 . . . . . . . . . . . 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)))))
108105, 106, 50, 107syl3anc 1363 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
109 ccatass 13930 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o) ∧ ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
11044, 89, 106, 109syl3anc 1363 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
111 df-s2 14198 . . . . . . . . . . . . . 14 ⟨“𝑏(𝑀𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)
112111oveq2i 7156 . . . . . . . . . . . . 13 (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩))
113110, 112syl6eqr 2871 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩))
114113oveq1d 7160 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
115103, 108, 1143eqtr2rd 2860 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
11673, 88, 1153eqtrd 2857 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
1171, 34, 47, 71efgtf 18777 . . . . . . . . . . . . 13 ((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))
118117simprd 496 . . . . . . . . . . . 12 ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊 → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o))⟶𝑊)
11954, 118syl 17 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))):((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o))⟶𝑊)
120119ffnd 6508 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o)))
121 fnovrn 7312 . . . . . . . . . 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)))))
122120, 69, 70, 121syl3anc 1363 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
123116, 122eqeltrrd 2911 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
1241, 34, 47, 71efgi2 18780 . . . . . . . 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”⟩)))))
12554, 123, 124syl2anc 584 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
12643, 125ersym 8290 . . . . . 6 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
12743ertr 8293 . . . . . 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”⟩)))) ∅))
128126, 127mpand 691 . . . . 5 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
129128expcom 414 . . . 4 ((𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o)) → (𝐴𝑊 → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅)))
130129a2d 29 . . 3 ((𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o)) → ((𝐴𝑊 → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅) → (𝐴𝑊 → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅)))
13114, 20, 26, 32, 42, 130wrdind 14072 . 2 (𝐴 ∈ Word (𝐼 × 2o) → (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
1324, 131mpcom 38 1 (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  c0 4288  cop 4563  cotp 4565   class class class wbr 5057  cmpt 5137   I cid 5452   × cxp 5546  ran crn 5549  ccom 5552   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  1oc1o 8084  2oc2o 8085   Er wer 8275  0cc0 10525   + caddc 10528  0cn0 11885  cuz 12231  ...cfz 12880  chash 13678  Word cword 13849   ++ cconcat 13910  ⟨“cs1 13937   splice csplice 14099  reversecreverse 14108  ⟨“cs2 14191   ~FG cefg 18761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-ec 8280  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-splice 14100  df-reverse 14109  df-s2 14198  df-efg 18764
This theorem is referenced by:  efginvrel1  18783  frgpinv  18819
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