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Mirrors > Home > MPE Home > Th. List > conjsubgen | Structured version Visualization version GIF version |
Description: A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-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 |
---|---|
conjsubgen | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 18276 | . . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | conjghm.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺) | |
3 | conjghm.p | . . . . . . . 8 ⊢ + = (+g‘𝐺) | |
4 | conjghm.m | . . . . . . . 8 ⊢ − = (-g‘𝐺) | |
5 | eqid 2798 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
6 | 2, 3, 4, 5 | conjghm 18381 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋)) |
7 | 1, 6 | sylan 583 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋)) |
8 | f1of1 6589 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋 → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋) | |
9 | 7, 8 | simpl2im 507 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋) |
10 | 2 | subgss 18272 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋) |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋) |
12 | f1ssres 6557 | . . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋 ∧ 𝑆 ⊆ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋) | |
13 | 9, 11, 12 | syl2anc 587 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋) |
14 | 11 | resmptd 5875 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))) |
15 | conjsubg.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
16 | 14, 15 | eqtr4di 2851 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹) |
17 | f1eq1 6544 | . . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹 → (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋)) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋)) |
19 | 13, 18 | mpbid 235 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1→𝑋) |
20 | f1f1orn 6601 | . . 3 ⊢ (𝐹:𝑆–1-1→𝑋 → 𝐹:𝑆–1-1-onto→ran 𝐹) | |
21 | 19, 20 | syl 17 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1-onto→ran 𝐹) |
22 | f1oeng 8511 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆–1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹) | |
23 | 21, 22 | syldan 594 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ≈ cen 8489 Basecbs 16475 +gcplusg 16557 Grpcgrp 18095 -gcsg 18097 SubGrpcsubg 18265 GrpHom cghm 18347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-en 8493 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 |
This theorem is referenced by: slwhash 18741 sylow2 18743 sylow3lem1 18744 |
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