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| Mirrors > Home > MPE Home > Th. List > frgpinv | Structured version Visualization version GIF version | ||
| Description: The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| frgpadd.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
| frgpadd.g | ⊢ 𝐺 = (freeGrp‘𝐼) |
| frgpadd.r | ⊢ ∼ = ( ~FG ‘𝐼) |
| frgpinv.n | ⊢ 𝑁 = (invg‘𝐺) |
| frgpinv.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
| Ref | Expression |
|---|---|
| frgpinv | ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frgpadd.w | . . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
| 2 | fviss 6986 | . . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
| 3 | 1, 2 | eqsstri 4030 | . . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
| 4 | 3 | sseli 3979 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o)) |
| 5 | revcl 14799 | . . . . . . 7 ⊢ (𝐴 ∈ Word (𝐼 × 2o) → (reverse‘𝐴) ∈ Word (𝐼 × 2o)) | |
| 6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (reverse‘𝐴) ∈ Word (𝐼 × 2o)) |
| 7 | frgpinv.m | . . . . . . 7 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
| 8 | 7 | efgmf 19731 | . . . . . 6 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
| 9 | wrdco 14870 | . . . . . 6 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) | |
| 10 | 6, 8, 9 | sylancl 586 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) |
| 11 | 1 | efgrcl 19733 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
| 12 | 11 | simprd 495 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
| 13 | 10, 12 | eleqtrrd 2844 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) |
| 14 | frgpadd.g | . . . . 5 ⊢ 𝐺 = (freeGrp‘𝐼) | |
| 15 | frgpadd.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
| 16 | eqid 2737 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 17 | 1, 14, 15, 16 | frgpadd 19781 | . . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
| 18 | 13, 17 | mpdan 687 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
| 19 | 1, 15 | efger 19736 | . . . . 5 ⊢ ∼ Er 𝑊 |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊) |
| 21 | eqid 2737 | . . . . 5 ⊢ (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
| 22 | 1, 15, 7, 21 | efginvrel2 19745 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅) |
| 23 | 20, 22 | erthi 8798 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ = [∅] ∼ ) |
| 24 | 14, 15 | frgp0 19778 | . . . . . 6 ⊢ (𝐼 ∈ V → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
| 25 | 24 | adantr 480 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)) → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
| 26 | 11, 25 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
| 27 | 26 | simprd 495 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [∅] ∼ = (0g‘𝐺)) |
| 28 | 18, 23, 27 | 3eqtrd 2781 | . 2 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺)) |
| 29 | 26 | simpld 494 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐺 ∈ Grp) |
| 30 | eqid 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 31 | 14, 15, 1, 30 | frgpeccl 19779 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [𝐴] ∼ ∈ (Base‘𝐺)) |
| 32 | 14, 15, 1, 30 | frgpeccl 19779 | . . . 4 ⊢ ((𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊 → [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) |
| 33 | 13, 32 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) |
| 34 | eqid 2737 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 35 | frgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
| 36 | 30, 16, 34, 35 | grpinvid1 19009 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ [𝐴] ∼ ∈ (Base‘𝐺) ∧ [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺))) |
| 37 | 29, 31, 33, 36 | syl3anc 1373 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺))) |
| 38 | 28, 37 | mpbird 257 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 〈cop 4632 〈cotp 4634 ↦ cmpt 5225 I cid 5577 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1oc1o 8499 2oc2o 8500 Er wer 8742 [cec 8743 0cc0 11155 ...cfz 13547 ♯chash 14369 Word cword 14552 ++ cconcat 14608 splice csplice 14787 reversecreverse 14796 〈“cs2 14880 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 invgcminusg 18952 ~FG cefg 19724 freeGrpcfrgp 19725 |
| 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-rmo 3380 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-tp 4631 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-qs 8751 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 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-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 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-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-frmd 18862 df-grp 18954 df-minusg 18955 df-efg 19727 df-frgp 19728 |
| This theorem is referenced by: vrgpinv 19787 |
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