Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17599 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
2 | conjghm.x | . . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺) | |
3 | conjghm.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
4 | conjghm.m | . . . . . . . . 9 ⊢ − = (-g‘𝐺) | |
5 | eqid 2622 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
6 | 2, 3, 4, 5 | conjghm 17691 | . . . . . . . 8 ⊢ ((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 488 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋)) |
8 | 7 | simprd 479 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋) |
9 | f1of1 6136 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋 → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋) |
11 | 2 | subgss 17595 | . . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋) |
12 | 11 | adantr 481 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋) |
13 | f1ssres 6108 | . . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋 ∧ 𝑆 ⊆ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋) | |
14 | 10, 12, 13 | syl2anc 693 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋) |
15 | 12 | resmptd 5452 | . . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))) |
16 | conjsubg.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
17 | 15, 16 | syl6eqr 2674 | . . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹) |
18 | f1eq1 6096 | . . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹 → (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋)) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋)) |
20 | 14, 19 | mpbid 222 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1→𝑋) |
21 | f1f1orn 6148 | . . 3 ⊢ (𝐹:𝑆–1-1→𝑋 → 𝐹:𝑆–1-1-onto→ran 𝐹) | |
22 | 20, 21 | syl 17 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1-onto→ran 𝐹) |
23 | f1oeng 7974 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆–1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹) | |
24 | 22, 23 | syldan 487 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ↾ cres 5116 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ≈ cen 7952 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 -gcsg 17424 SubGrpcsubg 17588 GrpHom cghm 17657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-en 7956 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 |
This theorem is referenced by: slwhash 18039 sylow2 18041 sylow3lem1 18042 |
Copyright terms: Public domain | W3C validator |