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Mirrors > Home > MPE Home > Th. List > efginvrel1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) |
efgval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
efgval2.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) |
efgval2.t | ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) |
Ref | Expression |
---|---|
efginvrel1 | ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∼ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . . . . . . . 10 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
2 | fviss 6256 | . . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜) | |
3 | 1, 2 | eqsstri 3635 | . . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2𝑜) |
4 | 3 | sseli 3599 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2𝑜)) |
5 | revcl 13510 | . . . . . . . 8 ⊢ (𝐴 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝐴) ∈ Word (𝐼 × 2𝑜)) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (reverse‘𝐴) ∈ Word (𝐼 × 2𝑜)) |
7 | efgval2.m | . . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) | |
8 | 7 | efgmf 18126 | . . . . . . 7 ⊢ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) |
9 | revco 13580 | . . . . . . 7 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (reverse‘(𝑀 ∘ (reverse‘𝐴)))) | |
10 | 6, 8, 9 | sylancl 694 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (reverse‘(𝑀 ∘ (reverse‘𝐴)))) |
11 | revrev 13516 | . . . . . . . 8 ⊢ (𝐴 ∈ Word (𝐼 × 2𝑜) → (reverse‘(reverse‘𝐴)) = 𝐴) | |
12 | 4, 11 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (reverse‘(reverse‘𝐴)) = 𝐴) |
13 | 12 | coeq2d 5284 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (𝑀 ∘ 𝐴)) |
14 | 10, 13 | eqtr3d 2658 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (reverse‘(𝑀 ∘ (reverse‘𝐴))) = (𝑀 ∘ 𝐴)) |
15 | 14 | coeq2d 5284 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴)))) = (𝑀 ∘ (𝑀 ∘ 𝐴))) |
16 | wrdf 13310 | . . . . . . . . 9 ⊢ (𝐴 ∈ Word (𝐼 × 2𝑜) → 𝐴:(0..^(#‘𝐴))⟶(𝐼 × 2𝑜)) | |
17 | 4, 16 | syl 17 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → 𝐴:(0..^(#‘𝐴))⟶(𝐼 × 2𝑜)) |
18 | 17 | ffvelrnda 6359 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑐 ∈ (0..^(#‘𝐴))) → (𝐴‘𝑐) ∈ (𝐼 × 2𝑜)) |
19 | 7 | efgmnvl 18127 | . . . . . . 7 ⊢ ((𝐴‘𝑐) ∈ (𝐼 × 2𝑜) → (𝑀‘(𝑀‘(𝐴‘𝑐))) = (𝐴‘𝑐)) |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑐 ∈ (0..^(#‘𝐴))) → (𝑀‘(𝑀‘(𝐴‘𝑐))) = (𝐴‘𝑐)) |
21 | 20 | mpteq2dva 4744 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝑀‘(𝐴‘𝑐)))) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝐴‘𝑐))) |
22 | 8 | ffvelrni 6358 | . . . . . . 7 ⊢ ((𝐴‘𝑐) ∈ (𝐼 × 2𝑜) → (𝑀‘(𝐴‘𝑐)) ∈ (𝐼 × 2𝑜)) |
23 | 18, 22 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑐 ∈ (0..^(#‘𝐴))) → (𝑀‘(𝐴‘𝑐)) ∈ (𝐼 × 2𝑜)) |
24 | fcompt 6400 | . . . . . . 7 ⊢ ((𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) ∧ 𝐴:(0..^(#‘𝐴))⟶(𝐼 × 2𝑜)) → (𝑀 ∘ 𝐴) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝐴‘𝑐)))) | |
25 | 8, 17, 24 | sylancr 695 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ 𝐴) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝐴‘𝑐)))) |
26 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) |
27 | 26 | feqmptd 6249 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑀 = (𝑎 ∈ (𝐼 × 2𝑜) ↦ (𝑀‘𝑎))) |
28 | fveq2 6191 | . . . . . 6 ⊢ (𝑎 = (𝑀‘(𝐴‘𝑐)) → (𝑀‘𝑎) = (𝑀‘(𝑀‘(𝐴‘𝑐)))) | |
29 | 23, 25, 27, 28 | fmptco 6396 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (𝑀 ∘ 𝐴)) = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝑀‘(𝑀‘(𝐴‘𝑐))))) |
30 | 17 | feqmptd 6249 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝐴 = (𝑐 ∈ (0..^(#‘𝐴)) ↦ (𝐴‘𝑐))) |
31 | 21, 29, 30 | 3eqtr4d 2666 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (𝑀 ∘ 𝐴)) = 𝐴) |
32 | 15, 31 | eqtrd 2656 | . . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴)))) = 𝐴) |
33 | 32 | oveq2d 6666 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) = ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴)) |
34 | wrdco 13577 | . . . . 5 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2𝑜)) | |
35 | 6, 8, 34 | sylancl 694 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2𝑜)) |
36 | 1 | efgrcl 18128 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜))) |
37 | 36 | simprd 479 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜)) |
38 | 35, 37 | eleqtrrd 2704 | . . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) |
39 | efgval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
40 | efgval2.t | . . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
41 | 1, 39, 7, 40 | efginvrel2 18140 | . . 3 ⊢ ((𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) ∼ ∅) |
42 | 38, 41 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) ∼ ∅) |
43 | 33, 42 | eqbrtrrd 4677 | 1 ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∼ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∅c0 3915 〈cop 4183 〈cotp 4185 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1𝑜c1o 7553 2𝑜c2o 7554 0cc0 9936 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 splice csplice 13296 reversecreverse 13297 〈“cs2 13586 ~FG cefg 18119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-efg 18122 |
This theorem is referenced by: frgp0 18173 |
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