![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nsgconj | Structured version Visualization version GIF version |
Description: The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
isnsg3.1 | ⊢ 𝑋 = (Base‘𝐺) |
isnsg3.2 | ⊢ + = (+g‘𝐺) |
isnsg3.3 | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
nsgconj | ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐵) − 𝐴) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsgsubg 19189 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | 1 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺)) |
3 | subgrcl 19162 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝐺 ∈ Grp) |
5 | simp2 1136 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑋) | |
6 | isnsg3.1 | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
7 | 6 | subgss 19158 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝑆 ⊆ 𝑋) |
9 | simp3 1137 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆) | |
10 | 8, 9 | sseldd 3996 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑋) |
11 | isnsg3.2 | . . . 4 ⊢ + = (+g‘𝐺) | |
12 | isnsg3.3 | . . . 4 ⊢ − = (-g‘𝐺) | |
13 | 6, 11, 12 | grpaddsubass 19061 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴 + 𝐵) − 𝐴) = (𝐴 + (𝐵 − 𝐴))) |
14 | 4, 5, 10, 5, 13 | syl13anc 1371 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐵) − 𝐴) = (𝐴 + (𝐵 − 𝐴))) |
15 | 6, 11, 12 | grpnpcan 19063 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐵 − 𝐴) + 𝐴) = 𝐵) |
16 | 4, 10, 5, 15 | syl3anc 1370 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐵 − 𝐴) + 𝐴) = 𝐵) |
17 | 16, 9 | eqeltrd 2839 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐵 − 𝐴) + 𝐴) ∈ 𝑆) |
18 | simp1 1135 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺)) | |
19 | 6, 12 | grpsubcl 19051 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 − 𝐴) ∈ 𝑋) |
20 | 4, 10, 5, 19 | syl3anc 1370 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → (𝐵 − 𝐴) ∈ 𝑋) |
21 | 6, 11 | nsgbi 19188 | . . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐵 − 𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (((𝐵 − 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 − 𝐴)) ∈ 𝑆)) |
22 | 18, 20, 5, 21 | syl3anc 1370 | . . 3 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → (((𝐵 − 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 − 𝐴)) ∈ 𝑆)) |
23 | 17, 22 | mpbid 232 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → (𝐴 + (𝐵 − 𝐴)) ∈ 𝑆) |
24 | 14, 23 | eqeltrd 2839 | 1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐵) − 𝐴) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Grpcgrp 18964 -gcsg 18966 SubGrpcsubg 19151 NrmSGrpcnsg 19152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 |
This theorem is referenced by: isnsg3 19191 ghmnsgima 19271 ghmnsgpreima 19272 clsnsg 24134 |
Copyright terms: Public domain | W3C validator |