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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 19111 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
4 | 3 | fveq2d 6699 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ )) |
5 | simpr 488 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼) | |
6 | 0ex 5185 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4664 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
8 | df2o3 8195 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
9 | 7, 8 | eleqtrri 2830 | . . . . . 6 ⊢ ∅ ∈ 2o |
10 | opelxpi 5573 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) | |
11 | 5, 9, 10 | sylancl 589 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) |
12 | 11 | s1cld 14125 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2o)) |
13 | simpl 486 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉) | |
14 | 2on 8188 | . . . . . 6 ⊢ 2o ∈ On | |
15 | xpexg 7513 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
16 | 13, 14, 15 | sylancl 589 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2o) ∈ V) |
17 | wrdexg 14044 | . . . . 5 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
18 | fvi 6765 | . . . . 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 2834 | . . 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 19108 | . . 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 14295 | . . . . . 6 ⊢ (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉 | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉) |
29 | 28 | coeq2d 5716 | . . . 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 19057 | . . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉):(𝐼 × 2o)⟶(𝐼 × 2o) |
31 | s1co 14363 | . . . . 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 589 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) |
33 | 24 | efgmval 19056 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) |
34 | 5, 9, 33 | sylancl 589 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = 〈𝐴, (1o ∖ ∅)〉) |
35 | df-ov 7194 | . . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) | |
36 | dif0 4273 | . . . . . . 7 ⊢ (1o ∖ ∅) = 1o | |
37 | 36 | opeq2i 4774 | . . . . . 6 ⊢ 〈𝐴, (1o ∖ ∅)〉 = 〈𝐴, 1o〉 |
38 | 34, 35, 37 | 3eqtr3g 2794 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉) = 〈𝐴, 1o〉) |
39 | 38 | s1eqd 14123 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉 = 〈“〈𝐴, 1o〉”〉) |
40 | 29, 32, 39 | 3eqtrd 2775 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = 〈“〈𝐴, 1o〉”〉) |
41 | 40 | eceq1d 8408 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2o ↦ 〈𝑥, (1o ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ = [〈“〈𝐴, 1o〉”〉] ∼ ) |
42 | 4, 26, 41 | 3eqtrd 2775 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 ∅c0 4223 {cpr 4529 〈cop 4533 I cid 5439 × cxp 5534 ∘ ccom 5540 Oncon0 6191 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 1oc1o 8173 2oc2o 8174 [cec 8367 Word cword 14034 〈“cs1 14117 reversecreverse 14288 invgcminusg 18320 ~FG cefg 19050 freeGrpcfrgp 19051 varFGrpcvrgp 19052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-xnn0 12128 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-lsw 14083 df-concat 14091 df-s1 14118 df-substr 14171 df-pfx 14201 df-splice 14280 df-reverse 14289 df-s2 14378 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-0g 16900 df-imas 16967 df-qus 16968 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-frmd 18230 df-grp 18322 df-minusg 18323 df-efg 19053 df-frgp 19054 df-vrgp 19055 |
This theorem is referenced by: frgpup3lem 19121 |
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