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Theorem efginvrel1 17964
 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
efginvrel1 (𝐴𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 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 efginvrel1
Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . . . 10 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 fviss 6166 . . . . . . . . . 10 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
31, 2eqsstri 3598 . . . . . . . . 9 𝑊 ⊆ Word (𝐼 × 2𝑜)
43sseli 3564 . . . . . . . 8 (𝐴𝑊𝐴 ∈ Word (𝐼 × 2𝑜))
5 revcl 13361 . . . . . . . 8 (𝐴 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝐴) ∈ Word (𝐼 × 2𝑜))
64, 5syl 17 . . . . . . 7 (𝐴𝑊 → (reverse‘𝐴) ∈ Word (𝐼 × 2𝑜))
7 efgval2.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
87efgmf 17949 . . . . . . 7 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
9 revco 13431 . . . . . . 7 (((reverse‘𝐴) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (reverse‘(𝑀 ∘ (reverse‘𝐴))))
106, 8, 9sylancl 693 . . . . . 6 (𝐴𝑊 → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (reverse‘(𝑀 ∘ (reverse‘𝐴))))
11 revrev 13367 . . . . . . . 8 (𝐴 ∈ Word (𝐼 × 2𝑜) → (reverse‘(reverse‘𝐴)) = 𝐴)
124, 11syl 17 . . . . . . 7 (𝐴𝑊 → (reverse‘(reverse‘𝐴)) = 𝐴)
1312coeq2d 5206 . . . . . 6 (𝐴𝑊 → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (𝑀𝐴))
1410, 13eqtr3d 2646 . . . . 5 (𝐴𝑊 → (reverse‘(𝑀 ∘ (reverse‘𝐴))) = (𝑀𝐴))
1514coeq2d 5206 . . . 4 (𝐴𝑊 → (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴)))) = (𝑀 ∘ (𝑀𝐴)))
16 wrdf 13165 . . . . . . . . 9 (𝐴 ∈ Word (𝐼 × 2𝑜) → 𝐴:(0..^(#‘𝐴))⟶(𝐼 × 2𝑜))
174, 16syl 17 . . . . . . . 8 (𝐴𝑊𝐴:(0..^(#‘𝐴))⟶(𝐼 × 2𝑜))
1817ffvelrnda 6267 . . . . . . 7 ((𝐴𝑊𝑐 ∈ (0..^(#‘𝐴))) → (𝐴𝑐) ∈ (𝐼 × 2𝑜))
197efgmnvl 17950 . . . . . . 7 ((𝐴𝑐) ∈ (𝐼 × 2𝑜) → (𝑀‘(𝑀‘(𝐴𝑐))) = (𝐴𝑐))
2018, 19syl 17 . . . . . 6 ((𝐴𝑊𝑐 ∈ (0..^(#‘𝐴))) → (𝑀‘(𝑀‘(𝐴𝑐))) = (𝐴𝑐))
2120mpteq2dva 4672 . . . . 5 (𝐴𝑊 → (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝑀‘(𝐴𝑐)))) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝐴𝑐)))
228ffvelrni 6266 . . . . . . 7 ((𝐴𝑐) ∈ (𝐼 × 2𝑜) → (𝑀‘(𝐴𝑐)) ∈ (𝐼 × 2𝑜))
2318, 22syl 17 . . . . . 6 ((𝐴𝑊𝑐 ∈ (0..^(#‘𝐴))) → (𝑀‘(𝐴𝑐)) ∈ (𝐼 × 2𝑜))
24 fcompt 6306 . . . . . . 7 ((𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) ∧ 𝐴:(0..^(#‘𝐴))⟶(𝐼 × 2𝑜)) → (𝑀𝐴) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝐴𝑐))))
258, 17, 24sylancr 694 . . . . . 6 (𝐴𝑊 → (𝑀𝐴) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝐴𝑐))))
268a1i 11 . . . . . . 7 (𝐴𝑊𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜))
2726feqmptd 6159 . . . . . 6 (𝐴𝑊𝑀 = (𝑎 ∈ (𝐼 × 2𝑜) ↦ (𝑀𝑎)))
28 fveq2 6103 . . . . . 6 (𝑎 = (𝑀‘(𝐴𝑐)) → (𝑀𝑎) = (𝑀‘(𝑀‘(𝐴𝑐))))
2923, 25, 27, 28fmptco 6303 . . . . 5 (𝐴𝑊 → (𝑀 ∘ (𝑀𝐴)) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝑀‘(𝐴𝑐)))))
3017feqmptd 6159 . . . . 5 (𝐴𝑊𝐴 = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝐴𝑐)))
3121, 29, 303eqtr4d 2654 . . . 4 (𝐴𝑊 → (𝑀 ∘ (𝑀𝐴)) = 𝐴)
3215, 31eqtrd 2644 . . 3 (𝐴𝑊 → (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴)))) = 𝐴)
3332oveq2d 6565 . 2 (𝐴𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) = ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴))
34 wrdco 13428 . . . . 5 (((reverse‘𝐴) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2𝑜))
356, 8, 34sylancl 693 . . . 4 (𝐴𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2𝑜))
361efgrcl 17951 . . . . 5 (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
3736simprd 478 . . . 4 (𝐴𝑊𝑊 = Word (𝐼 × 2𝑜))
3835, 37eleqtrrd 2691 . . 3 (𝐴𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊)
39 efgval.r . . . 4 = ( ~FG𝐼)
40 efgval2.t . . . 4 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
411, 39, 7, 40efginvrel2 17963 . . 3 ((𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) ∅)
4238, 41syl 17 . 2 (𝐴𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) ∅)
4333, 42eqbrtrrd 4607 1 (𝐴𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ⟨cop 4131  ⟨cotp 4133   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441  0cc0 9815  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   splice csplice 13151  reversecreverse 13152  ⟨“cs2 13437   ~FG cefg 17942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-efg 17945 This theorem is referenced by:  frgp0  17996
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