![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > conjsubg | Structured version Visualization version GIF version |
Description: A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
conjghm.x | ⊢ 𝑋 = (Base‘𝐺) |
conjghm.p | ⊢ + = (+g‘𝐺) |
conjghm.m | ⊢ − = (-g‘𝐺) |
conjsubg.f | ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) |
Ref | Expression |
---|---|
conjsubg | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | conjghm.x | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
2 | 1 | subgss 17946 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋) |
3 | 2 | adantr 474 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋) |
4 | df-ima 5355 | . . . 4 ⊢ ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) “ 𝑆) = ran ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) | |
5 | resmpt 5686 | . . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))) | |
6 | conjsubg.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
7 | 5, 6 | syl6eqr 2879 | . . . . 5 ⊢ (𝑆 ⊆ 𝑋 → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹) |
8 | 7 | rneqd 5585 | . . . 4 ⊢ (𝑆 ⊆ 𝑋 → ran ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = ran 𝐹) |
9 | 4, 8 | syl5eq 2873 | . . 3 ⊢ (𝑆 ⊆ 𝑋 → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) “ 𝑆) = ran 𝐹) |
10 | 3, 9 | syl 17 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) “ 𝑆) = ran 𝐹) |
11 | subgrcl 17950 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
12 | conjghm.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
13 | conjghm.m | . . . . . 6 ⊢ − = (-g‘𝐺) | |
14 | eqid 2825 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
15 | 1, 12, 13, 14 | conjghm 18042 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋)) |
16 | 11, 15 | sylan 577 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋)) |
17 | 16 | simpld 490 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺)) |
18 | simpl 476 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ∈ (SubGrp‘𝐺)) | |
19 | ghmima 18032 | . . 3 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) “ 𝑆) ∈ (SubGrp‘𝐺)) | |
20 | 17, 18, 19 | syl2anc 581 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) “ 𝑆) ∈ (SubGrp‘𝐺)) |
21 | 10, 20 | eqeltrrd 2907 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 ↦ cmpt 4952 ran crn 5343 ↾ cres 5344 “ cima 5345 –1-1-onto→wf1o 6122 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 Grpcgrp 17776 -gcsg 17778 SubGrpcsubg 17939 GrpHom cghm 18008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-ghm 18009 |
This theorem is referenced by: slwhash 18390 sylow2 18392 sylow3lem1 18393 |
Copyright terms: Public domain | W3C validator |