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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 6986 | . . . . . . . . . 10 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
| 3 | 1, 2 | eqsstri 4030 | . . . . . . . . 9 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) | 
| 4 | 3 | sseli 3979 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o)) | 
| 5 | revcl 14799 | . . . . . . . 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 19731 | . . . . . . 7 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) | 
| 9 | revco 14873 | . . . . . . 7 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (reverse‘(𝑀 ∘ (reverse‘𝐴)))) | |
| 10 | 6, 8, 9 | sylancl 586 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (reverse‘(𝑀 ∘ (reverse‘𝐴)))) | 
| 11 | revrev 14805 | . . . . . . . 8 ⊢ (𝐴 ∈ Word (𝐼 × 2o) → (reverse‘(reverse‘𝐴)) = 𝐴) | |
| 12 | 4, 11 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → (reverse‘(reverse‘𝐴)) = 𝐴) | 
| 13 | 12 | coeq2d 5873 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(reverse‘𝐴))) = (𝑀 ∘ 𝐴)) | 
| 14 | 10, 13 | eqtr3d 2779 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (reverse‘(𝑀 ∘ (reverse‘𝐴))) = (𝑀 ∘ 𝐴)) | 
| 15 | 14 | coeq2d 5873 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴)))) = (𝑀 ∘ (𝑀 ∘ 𝐴))) | 
| 16 | wrdf 14557 | . . . . . . . . 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 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑐 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝑐) ∈ (𝐼 × 2o)) | 
| 19 | 7 | efgmnvl 19732 | . . . . . . 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 5242 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑐 ∈ (0..^(♯‘𝐴)) ↦ (𝑀‘(𝑀‘(𝐴‘𝑐)))) = (𝑐 ∈ (0..^(♯‘𝐴)) ↦ (𝐴‘𝑐))) | 
| 22 | 8 | ffvelcdmi 7103 | . . . . . . 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 7153 | . . . . . . 7 ⊢ ((𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) ∧ 𝐴:(0..^(♯‘𝐴))⟶(𝐼 × 2o)) → (𝑀 ∘ 𝐴) = (𝑐 ∈ (0..^(♯‘𝐴)) ↦ (𝑀‘(𝐴‘𝑐)))) | |
| 25 | 8, 17, 24 | sylancr 587 | . . . . . 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 6977 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → 𝑀 = (𝑎 ∈ (𝐼 × 2o) ↦ (𝑀‘𝑎))) | 
| 28 | fveq2 6906 | . . . . . 6 ⊢ (𝑎 = (𝑀‘(𝐴‘𝑐)) → (𝑀‘𝑎) = (𝑀‘(𝑀‘(𝐴‘𝑐)))) | |
| 29 | 23, 25, 27, 28 | fmptco 7149 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (𝑀 ∘ 𝐴)) = (𝑐 ∈ (0..^(♯‘𝐴)) ↦ (𝑀‘(𝑀‘(𝐴‘𝑐))))) | 
| 30 | 17 | feqmptd 6977 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝐴 = (𝑐 ∈ (0..^(♯‘𝐴)) ↦ (𝐴‘𝑐))) | 
| 31 | 21, 29, 30 | 3eqtr4d 2787 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (𝑀 ∘ 𝐴)) = 𝐴) | 
| 32 | 15, 31 | eqtrd 2777 | . . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴)))) = 𝐴) | 
| 33 | 32 | oveq2d 7447 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ (𝑀 ∘ (reverse‘(𝑀 ∘ (reverse‘𝐴))))) = ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴)) | 
| 34 | wrdco 14870 | . . . . 5 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) | |
| 35 | 6, 8, 34 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) | 
| 36 | 1 | efgrcl 19733 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) | 
| 37 | 36 | simprd 495 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) | 
| 38 | 35, 37 | eleqtrrd 2844 | . . 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 19745 | . . 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 5167 | 1 ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∼ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 〈cop 4632 〈cotp 4634 class class class wbr 5143 ↦ cmpt 5225 I cid 5577 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1oc1o 8499 2oc2o 8500 0cc0 11155 ...cfz 13547 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608 splice csplice 14787 reversecreverse 14796 〈“cs2 14880 ~FG cefg 19724 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-efg 19727 | 
| This theorem is referenced by: frgp0 19778 | 
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