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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 18380 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
4 | 3 | fveq2d 6356 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[〈“〈𝐴, ∅〉”〉] ∼ )) |
5 | simpr 479 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼) | |
6 | 0ex 4942 | . . . . . . . 8 ⊢ ∅ ∈ V | |
7 | 6 | prid1 4441 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1𝑜} |
8 | df2o3 7742 | . . . . . . 7 ⊢ 2𝑜 = {∅, 1𝑜} | |
9 | 7, 8 | eleqtrri 2838 | . . . . . 6 ⊢ ∅ ∈ 2𝑜 |
10 | opelxpi 5305 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → 〈𝐴, ∅〉 ∈ (𝐼 × 2𝑜)) | |
11 | 5, 9, 10 | sylancl 697 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈𝐴, ∅〉 ∈ (𝐼 × 2𝑜)) |
12 | 11 | s1cld 13573 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2𝑜)) |
13 | simpl 474 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉) | |
14 | 2on 7737 | . . . . . 6 ⊢ 2𝑜 ∈ On | |
15 | xpexg 7125 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V) | |
16 | 13, 14, 15 | sylancl 697 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2𝑜) ∈ V) |
17 | wrdexg 13501 | . . . . 5 ⊢ ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V) | |
18 | fvi 6417 | . . . . 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 2842 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“〈𝐴, ∅〉”〉 ∈ ( I ‘Word (𝐼 × 2𝑜))) |
21 | eqid 2760 | . . . 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 2760 | . . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉) = (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉) | |
25 | 21, 22, 1, 23, 24 | frgpinv 18377 | . . 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 13714 | . . . . . 6 ⊢ (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉 | |
28 | 27 | a1i 11 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘〈“〈𝐴, ∅〉”〉) = 〈“〈𝐴, ∅〉”〉) |
29 | 28 | coeq2d 5440 | . . . 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 18326 | . . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) |
31 | s1co 13779 | . . . . 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 697 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉) ∘ 〈“〈𝐴, ∅〉”〉) = 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉) |
33 | 24 | efgmval 18325 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)∅) = 〈𝐴, (1𝑜 ∖ ∅)〉) |
34 | 5, 9, 33 | sylancl 697 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)∅) = 〈𝐴, (1𝑜 ∖ ∅)〉) |
35 | df-ov 6816 | . . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)‘〈𝐴, ∅〉) | |
36 | dif0 4093 | . . . . . . 7 ⊢ (1𝑜 ∖ ∅) = 1𝑜 | |
37 | 36 | opeq2i 4557 | . . . . . 6 ⊢ 〈𝐴, (1𝑜 ∖ ∅)〉 = 〈𝐴, 1𝑜〉 |
38 | 34, 35, 37 | 3eqtr3g 2817 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)‘〈𝐴, ∅〉) = 〈𝐴, 1𝑜〉) |
39 | 38 | s1eqd 13571 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 〈“((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉)‘〈𝐴, ∅〉)”〉 = 〈“〈𝐴, 1𝑜〉”〉) |
40 | 29, 32, 39 | 3eqtrd 2798 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉)) = 〈“〈𝐴, 1𝑜〉”〉) |
41 | 40 | eceq1d 7950 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ 〈𝑥, (1𝑜 ∖ 𝑦)〉) ∘ (reverse‘〈“〈𝐴, ∅〉”〉))] ∼ = [〈“〈𝐴, 1𝑜〉”〉] ∼ ) |
42 | 4, 26, 41 | 3eqtrd 2798 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1𝑜〉”〉] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∖ cdif 3712 ∅c0 4058 {cpr 4323 〈cop 4327 I cid 5173 × cxp 5264 ∘ ccom 5270 Oncon0 5884 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↦ cmpt2 6815 1𝑜c1o 7722 2𝑜c2o 7723 [cec 7909 Word cword 13477 〈“cs1 13480 reversecreverse 13483 invgcminusg 17624 ~FG cefg 18319 freeGrpcfrgp 18320 varFGrpcvrgp 18321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-xnn0 11556 df-z 11570 df-dec 11686 df-uz 11880 df-fz 12520 df-fzo 12660 df-hash 13312 df-word 13485 df-lsw 13486 df-concat 13487 df-s1 13488 df-substr 13489 df-splice 13490 df-reverse 13491 df-s2 13793 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-0g 16304 df-imas 16370 df-qus 16371 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-frmd 17587 df-grp 17626 df-minusg 17627 df-efg 18322 df-frgp 18323 df-vrgp 18324 |
This theorem is referenced by: frgpup3lem 18390 |
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