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| Mirrors > Home > MPE Home > Th. List > vrgpinv | Structured version Visualization version GIF version | ||
| Description: The inverse of a generating element is represented by 〈𝐴, 1〉 instead of 〈𝐴, 0〉. (Contributed by Mario Carneiro, 2-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| vrgpfval.r | ⊢ ∼ = ( ~FG ‘𝐼) | 
| vrgpfval.u | ⊢ 𝑈 = (varFGrp‘𝐼) | 
| vrgpf.m | ⊢ 𝐺 = (freeGrp‘𝐼) | 
| vrgpinv.n | ⊢ 𝑁 = (invg‘𝐺) | 
| Ref | Expression | 
|---|---|
| vrgpinv | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vrgpfval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
| 2 | vrgpfval.u | . . . 4 ⊢ 𝑈 = (varFGrp‘𝐼) | |
| 3 | 1, 2 | vrgpval 19786 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) | 
| 4 | 3 | fveq2d 6909 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ )) | 
| 5 | simpr 484 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼) | |
| 6 | 0ex 5306 | . . . . . . . 8 ⊢ ∅ ∈ V | |
| 7 | 6 | prid1 4761 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} | 
| 8 | df2o3 8515 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
| 9 | 7, 8 | eleqtrri 2839 | . . . . . 6 ⊢ ∅ ∈ 2o | 
| 10 | opelxpi 5721 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) | |
| 11 | 5, 9, 10 | sylancl 586 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) | 
| 12 | 11 | s1cld 14642 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2o)) | 
| 13 | simpl 482 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉) | |
| 14 | 2on 8521 | . . . . . 6 ⊢ 2o ∈ On | |
| 15 | xpexg 7771 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
| 16 | 13, 14, 15 | sylancl 586 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2o) ∈ V) | 
| 17 | wrdexg 14563 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
| 18 | fvi 6984 | . . . . 5 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) | |
| 19 | 16, 17, 18 | 3syl 18 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) | 
| 20 | 12, 19 | eleqtrrd 2843 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o))) | 
| 21 | eqid 2736 | . . . 4 ⊢ ( I ‘Word (𝐼 × 2o)) = ( I ‘Word (𝐼 × 2o)) | |
| 22 | vrgpf.m | . . . 4 ⊢ 𝐺 = (freeGrp‘𝐼) | |
| 23 | vrgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
| 24 | eqid 2736 | . . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) = (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) | |
| 25 | 21, 22, 1, 23, 24 | frgpinv 19783 | . . 3 ⊢ (〈“〈𝐴, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2o)) → (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ ) = [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ ) | 
| 26 | 20, 25 | syl 17 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ ) = [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ ) | 
| 27 | revs1 14804 | . . . . . 6 ⊢ (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉 | |
| 28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉) | 
| 29 | 28 | coeq2d 5872 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉)) | 
| 30 | 24 | efgmf 19732 | . . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉):(𝐼 × 2o)⟶(𝐼 × 2o) | 
| 31 | s1co 14873 | . . . . 5 ⊢ ((〈𝐴, ∅〉 ∈ (𝐼 × 2o) ∧ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉):(𝐼 × 2o)⟶(𝐼 × 2o)) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) | |
| 32 | 11, 30, 31 | sylancl 586 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) | 
| 33 | 24 | efgmval 19731 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) | 
| 34 | 5, 9, 33 | sylancl 586 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) | 
| 35 | df-ov 7435 | . . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) | |
| 36 | dif0 4377 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
| 37 | 36 | opeq2i 4876 | . . . . . 6 ⊢ 〈𝐴, (1o ∖ ∅)〉 = 〈𝐴, 1o〉 | 
| 38 | 34, 35, 37 | 3eqtr3g 2799 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) = 〈𝐴, 1o〉) | 
| 39 | 38 | s1eqd 14640 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉 = 〈“〈𝐴, 1o〉”〉) | 
| 40 | 29, 32, 39 | 3eqtrd 2780 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = 〈“〈𝐴, 1o〉”〉) | 
| 41 | 40 | eceq1d 8786 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ = [〈“〈𝐴, 1o〉”〉] ∼ ) | 
| 42 | 4, 26, 41 | 3eqtrd 2780 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ∅c0 4332 {cpr 4627 〈cop 4631 I cid 5576 × cxp 5682 ∘ ccom 5688 Oncon0 6383 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1oc1o 8500 2oc2o 8501 [cec 8744 Word cword 14553 〈“cs1 14634 reversecreverse 14797 invgcminusg 18953 ~FG cefg 19725 freeGrpcfrgp 19726 varFGrpcvrgp 19727 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-lsw 14602 df-concat 14610 df-s1 14635 df-substr 14680 df-pfx 14710 df-splice 14789 df-reverse 14798 df-s2 14888 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17487 df-imas 17554 df-qus 17555 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-frmd 18863 df-grp 18955 df-minusg 18956 df-efg 19728 df-frgp 19729 df-vrgp 19730 | 
| This theorem is referenced by: frgpup3lem 19796 | 
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