![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > idrespermg | Structured version Visualization version GIF version |
Description: The structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation (constructed by (structure) restricting the symmetric group to that singleton) is a permutation group (group consisting of permutations). (Contributed by AV, 17-Mar-2019.) |
Ref | Expression |
---|---|
idressubgsymg.g | ⊢ 𝐺 = (SymGrp‘𝐴) |
idrespermg.e | ⊢ 𝐸 = (𝐺 ↾s {( I ↾ 𝐴)}) |
Ref | Expression |
---|---|
idrespermg | ⊢ (𝐴 ∈ 𝑉 → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idressubgsymg.g | . . 3 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | 1 | idressubgsymg 19200 | . 2 ⊢ (𝐴 ∈ 𝑉 → {( I ↾ 𝐴)} ∈ (SubGrp‘𝐺)) |
3 | eqid 2733 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 1, 3 | pgrpsubgsymgbi 19198 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({( I ↾ 𝐴)} ∈ (SubGrp‘𝐺) ↔ ({( I ↾ 𝐴)} ⊆ (Base‘𝐺) ∧ (𝐺 ↾s {( I ↾ 𝐴)}) ∈ Grp))) |
5 | snex 5392 | . . . . . . 7 ⊢ {( I ↾ 𝐴)} ∈ V | |
6 | idrespermg.e | . . . . . . . 8 ⊢ 𝐸 = (𝐺 ↾s {( I ↾ 𝐴)}) | |
7 | 6, 3 | ressbas 17126 | . . . . . . 7 ⊢ ({( I ↾ 𝐴)} ∈ V → ({( I ↾ 𝐴)} ∩ (Base‘𝐺)) = (Base‘𝐸)) |
8 | 5, 7 | mp1i 13 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({( I ↾ 𝐴)} ∩ (Base‘𝐺)) = (Base‘𝐸)) |
9 | inss2 4193 | . . . . . 6 ⊢ ({( I ↾ 𝐴)} ∩ (Base‘𝐺)) ⊆ (Base‘𝐺) | |
10 | 8, 9 | eqsstrrdi 4003 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐸) ⊆ (Base‘𝐺)) |
11 | 6 | eqcomi 2742 | . . . . . . . 8 ⊢ (𝐺 ↾s {( I ↾ 𝐴)}) = 𝐸 |
12 | 11 | eleq1i 2825 | . . . . . . 7 ⊢ ((𝐺 ↾s {( I ↾ 𝐴)}) ∈ Grp ↔ 𝐸 ∈ Grp) |
13 | 12 | biimpi 215 | . . . . . 6 ⊢ ((𝐺 ↾s {( I ↾ 𝐴)}) ∈ Grp → 𝐸 ∈ Grp) |
14 | 13 | adantl 483 | . . . . 5 ⊢ (({( I ↾ 𝐴)} ⊆ (Base‘𝐺) ∧ (𝐺 ↾s {( I ↾ 𝐴)}) ∈ Grp) → 𝐸 ∈ Grp) |
15 | 10, 14 | anim12ci 615 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ({( I ↾ 𝐴)} ⊆ (Base‘𝐺) ∧ (𝐺 ↾s {( I ↾ 𝐴)}) ∈ Grp)) → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺))) |
16 | 15 | ex 414 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (({( I ↾ 𝐴)} ⊆ (Base‘𝐺) ∧ (𝐺 ↾s {( I ↾ 𝐴)}) ∈ Grp) → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))) |
17 | 4, 16 | sylbid 239 | . 2 ⊢ (𝐴 ∈ 𝑉 → ({( I ↾ 𝐴)} ∈ (SubGrp‘𝐺) → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))) |
18 | 2, 17 | mpd 15 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 {csn 4590 I cid 5534 ↾ cres 5639 ‘cfv 6500 (class class class)co 7361 Basecbs 17091 ↾s cress 17120 Grpcgrp 18756 SubGrpcsubg 18930 SymGrpcsymg 19156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-tset 17160 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-efmnd 18687 df-grp 18759 df-minusg 18760 df-subg 18933 df-symg 19157 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |