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Theorem efginvrel2 18933
 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 3928 . . 3 𝑊 ⊆ Word (𝐼 × 2o)
43sseli 3890 . 2 (𝐴𝑊𝐴 ∈ Word (𝐼 × 2o))
5 id 22 . . . . . 6 (𝑐 = ∅ → 𝑐 = ∅)
6 fveq2 6663 . . . . . . . . 9 (𝑐 = ∅ → (reverse‘𝑐) = (reverse‘∅))
7 rev0 14186 . . . . . . . . 9 (reverse‘∅) = ∅
86, 7eqtrdi 2809 . . . . . . . 8 (𝑐 = ∅ → (reverse‘𝑐) = ∅)
98coeq2d 5708 . . . . . . 7 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ ∅))
10 co02 6095 . . . . . . 7 (𝑀 ∘ ∅) = ∅
119, 10eqtrdi 2809 . . . . . 6 (𝑐 = ∅ → (𝑀 ∘ (reverse‘𝑐)) = ∅)
125, 11oveq12d 7174 . . . . 5 (𝑐 = ∅ → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (∅ ++ ∅))
1312breq1d 5046 . . . 4 (𝑐 = ∅ → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (∅ ++ ∅) ∅))
1413imbi2d 344 . . 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 5708 . . . . . 6 (𝑐 = 𝑎 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝑎)))
1815, 17oveq12d 7174 . . . . 5 (𝑐 = 𝑎 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
1918breq1d 5046 . . . 4 (𝑐 = 𝑎 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅))
2019imbi2d 344 . . 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 5708 . . . . . 6 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))))
2421, 23oveq12d 7174 . . . . 5 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
2524breq1d 5046 . . . 4 (𝑐 = (𝑎 ++ ⟨“𝑏”⟩) → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
2625imbi2d 344 . . 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 5708 . . . . . 6 (𝑐 = 𝐴 → (𝑀 ∘ (reverse‘𝑐)) = (𝑀 ∘ (reverse‘𝐴)))
3027, 29oveq12d 7174 . . . . 5 (𝑐 = 𝐴 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) = (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))))
3130breq1d 5046 . . . 4 (𝑐 = 𝐴 → ((𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅ ↔ (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅))
3231imbi2d 344 . . 3 (𝑐 = 𝐴 → ((𝐴𝑊 → (𝑐 ++ (𝑀 ∘ (reverse‘𝑐))) ∅) ↔ (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)))
33 ccatidid 14004 . . . 4 (∅ ++ ∅) = ∅
34 efgval.r . . . . . . 7 = ( ~FG𝐼)
351, 34efger 18924 . . . . . 6 Er 𝑊
3635a1i 11 . . . . 5 (𝐴𝑊 Er 𝑊)
37 wrd0 13951 . . . . . 6 ∅ ∈ Word (𝐼 × 2o)
381efgrcl 18921 . . . . . . 7 (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
3938simprd 499 . . . . . 6 (𝐴𝑊𝑊 = Word (𝐼 × 2o))
4037, 39eleqtrrid 2859 . . . . 5 (𝐴𝑊 → ∅ ∈ 𝑊)
4136, 40erref 8325 . . . 4 (𝐴𝑊 → ∅ ∅)
4233, 41eqbrtrid 5071 . . 3 (𝐴𝑊 → (∅ ++ ∅) ∅)
4335a1i 11 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → Er 𝑊)
44 simprl 770 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ Word (𝐼 × 2o))
45 revcl 14183 . . . . . . . . . . . 12 (𝑎 ∈ Word (𝐼 × 2o) → (reverse‘𝑎) ∈ Word (𝐼 × 2o))
4645ad2antrl 727 . . . . . . . . . . 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 18919 . . . . . . . . . . 11 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
49 wrdco 14253 . . . . . . . . . . 11 (((reverse‘𝑎) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
5046, 48, 49sylancl 589 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o))
51 ccatcl 13986 . . . . . . . . . 10 ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
5244, 50, 51syl2anc 587 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ Word (𝐼 × 2o))
5339adantr 484 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑊 = Word (𝐼 × 2o))
5452, 53eleqtrrd 2855 . . . . . . . 8 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∈ 𝑊)
55 lencl 13945 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2o) → (♯‘𝑎) ∈ ℕ0)
5655ad2antrl 727 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℕ0)
57 nn0uz 12333 . . . . . . . . . . . . 13 0 = (ℤ‘0)
5856, 57eleqtrdi 2862 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ‘0))
59 ccatlen 13987 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2o) ∧ (𝑀 ∘ (reverse‘𝑎)) ∈ Word (𝐼 × 2o)) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
6044, 50, 59syl2anc 587 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) = ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))))
6156nn0zd 12137 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℤ)
6261uzidd 12311 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)))
63 lencl 13945 . . . . . . . . . . . . . . 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 12357 . . . . . . . . . . . . . 14 (((♯‘𝑎) ∈ (ℤ‘(♯‘𝑎)) ∧ (♯‘(𝑀 ∘ (reverse‘𝑎))) ∈ ℕ0) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
6662, 64, 65syl2anc 587 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + (♯‘(𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
6760, 66eqeltrd 2852 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎)))
68 elfzuzb 12963 . . . . . . . . . . . 12 ((♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) ↔ ((♯‘𝑎) ∈ (ℤ‘0) ∧ (♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) ∈ (ℤ‘(♯‘𝑎))))
6958, 67, 68sylanbrc 586 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ (0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))))
70 simprr 772 . . . . . . . . . . 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 18929 . . . . . . . . . . 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 1368 . . . . . . . . . 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 6847 . . . . . . . . . . . . 13 (𝑏 ∈ (𝐼 × 2o) → (𝑀𝑏) ∈ (𝐼 × 2o))
7675ad2antll 728 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀𝑏) ∈ (𝐼 × 2o))
7770, 76s2cld 14293 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
78 ccatrid 14001 . . . . . . . . . . . . . 14 (𝑎 ∈ Word (𝐼 × 2o) → (𝑎 ++ ∅) = 𝑎)
7978ad2antrl 727 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ∅) = 𝑎)
8079eqcomd 2764 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 = (𝑎 ++ ∅))
8180oveq1d 7171 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ∅) ++ (𝑀 ∘ (reverse‘𝑎))))
82 eqidd 2759 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = (♯‘𝑎))
83 hash0 13791 . . . . . . . . . . . . 13 (♯‘∅) = 0
8483oveq2i 7167 . . . . . . . . . . . 12 ((♯‘𝑎) + (♯‘∅)) = ((♯‘𝑎) + 0)
8556nn0cnd 12009 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) ∈ ℂ)
8685addid1d 10891 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎) + 0) = (♯‘𝑎))
8784, 86syl5req 2806 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘𝑎) = ((♯‘𝑎) + (♯‘∅)))
8844, 74, 50, 77, 81, 82, 87splval2 14179 . . . . . . . . . 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 14017 . . . . . . . . . . . . . . . 16 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o))
90 revccat 14188 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
9144, 89, 90syl2anc 587 . . . . . . . . . . . . . . 15 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)))
92 revs1 14187 . . . . . . . . . . . . . . . 16 (reverse‘⟨“𝑏”⟩) = ⟨“𝑏”⟩
9392oveq1i 7166 . . . . . . . . . . . . . . 15 ((reverse‘⟨“𝑏”⟩) ++ (reverse‘𝑎)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎))
9491, 93eqtrdi 2809 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (reverse‘(𝑎 ++ ⟨“𝑏”⟩)) = (⟨“𝑏”⟩ ++ (reverse‘𝑎)))
9594coeq2d 5708 . . . . . . . . . . . . 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 14257 . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (⟨“𝑏”⟩ ++ (reverse‘𝑎))) = ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
99 s1co 14255 . . . . . . . . . . . . . . 15 ((𝑏 ∈ (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
10070, 48, 99sylancl 589 . . . . . . . . . . . . . 14 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ ⟨“𝑏”⟩) = ⟨“(𝑀𝑏)”⟩)
101100oveq1d 7171 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑀 ∘ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
10295, 98, 1013eqtrd 2797 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩))) = (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎))))
103102oveq2d 7172 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
104 ccatcl 13986 . . . . . . . . . . . . 13 ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o)) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
10544, 89, 104syl2anc 587 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ ⟨“𝑏”⟩) ∈ Word (𝐼 × 2o))
10676s1cld 14017 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
107 ccatass 14002 . . . . . . . . . . . 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 1368 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (⟨“(𝑀𝑏)”⟩ ++ (𝑀 ∘ (reverse‘𝑎)))))
109 ccatass 14002 . . . . . . . . . . . . . 14 ((𝑎 ∈ Word (𝐼 × 2o) ∧ ⟨“𝑏”⟩ ∈ Word (𝐼 × 2o) ∧ ⟨“(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o)) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
11044, 89, 106, 109syl3anc 1368 . . . . . . . . . . . . 13 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)))
111 df-s2 14270 . . . . . . . . . . . . . 14 ⟨“𝑏(𝑀𝑏)”⟩ = (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩)
112111oveq2i 7167 . . . . . . . . . . . . 13 (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) = (𝑎 ++ (⟨“𝑏”⟩ ++ ⟨“(𝑀𝑏)”⟩))
113110, 112eqtr4di 2811 . . . . . . . . . . . 12 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) = (𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩))
114113oveq1d 7171 . . . . . . . . . . 11 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (((𝑎 ++ ⟨“𝑏”⟩) ++ ⟨“(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))))
115103, 108, 1143eqtr2rd 2800 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏(𝑀𝑏)”⟩) ++ (𝑀 ∘ (reverse‘𝑎))) = ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
11673, 88, 1153eqtrd 2797 . . . . . . . . 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 18928 . . . . . . . . . . . . 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 499 . . . . . . . . . . . 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 6504 . . . . . . . . . 10 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))) Fn ((0...(♯‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))) × (𝐼 × 2o)))
121 fnovrn 7325 . . . . . . . . . 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 1368 . . . . . . . . 9 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑎)(𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))𝑏) ∈ ran (𝑇‘(𝑎 ++ (𝑀 ∘ (reverse‘𝑎)))))
123116, 122eqeltrrd 2853 . . . . . . . 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 18931 . . . . . . . 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 587 . . . . . . 7 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))))
12643, 125ersym 8317 . . . . . 6 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) (𝑎 ++ (𝑀 ∘ (reverse‘𝑎))))
12743ertr 8320 . . . . . 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 694 . . . . 5 ((𝐴𝑊 ∧ (𝑎 ∈ Word (𝐼 × 2o) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((𝑎 ++ (𝑀 ∘ (reverse‘𝑎))) ∅ → ((𝑎 ++ ⟨“𝑏”⟩) ++ (𝑀 ∘ (reverse‘(𝑎 ++ ⟨“𝑏”⟩)))) ∅))
129128expcom 417 . . . 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 14144 . 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 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3857  ∅c0 4227  ⟨cop 4531  ⟨cotp 4533   class class class wbr 5036   ↦ cmpt 5116   I cid 5433   × cxp 5526  ran crn 5529   ∘ ccom 5532   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1oc1o 8111  2oc2o 8112   Er wer 8302  0cc0 10588   + caddc 10591  ℕ0cn0 11947  ℤ≥cuz 12295  ...cfz 12952  ♯chash 13753  Word cword 13926   ++ cconcat 13982  ⟨“cs1 14009   splice csplice 14171  reversecreverse 14180  ⟨“cs2 14263   ~FG cefg 18912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-2o 8119  df-er 8305  df-ec 8307  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-n0 11948  df-xnn0 12020  df-z 12034  df-uz 12296  df-fz 12953  df-fzo 13096  df-hash 13754  df-word 13927  df-lsw 13975  df-concat 13983  df-s1 14010  df-substr 14063  df-pfx 14093  df-splice 14172  df-reverse 14181  df-s2 14270  df-efg 18915 This theorem is referenced by:  efginvrel1  18934  frgpinv  18970
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