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Mirrors > Home > ILE Home > Th. List > isnsg4 | GIF version |
Description: A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
elnmz.1 | ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} |
nmzsubg.2 | ⊢ 𝑋 = (Base‘𝐺) |
nmzsubg.3 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
isnsg4 | ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmzsubg.2 | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | nmzsubg.3 | . . 3 ⊢ + = (+g‘𝐺) | |
3 | 1, 2 | isnsg 13258 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))) |
4 | eqcom 2195 | . . . 4 ⊢ (𝑁 = 𝑋 ↔ 𝑋 = 𝑁) | |
5 | elnmz.1 | . . . . 5 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} | |
6 | 5 | eqeq2i 2204 | . . . 4 ⊢ (𝑋 = 𝑁 ↔ 𝑋 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}) |
7 | rabid2 2671 | . . . 4 ⊢ (𝑋 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) | |
8 | 4, 6, 7 | 3bitri 206 | . . 3 ⊢ (𝑁 = 𝑋 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) |
9 | 8 | anbi2i 457 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 = 𝑋) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))) |
10 | 3, 9 | bitr4i 187 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 = 𝑋)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 ∀wral 2472 {crab 2476 ‘cfv 5246 (class class class)co 5910 Basecbs 12605 +gcplusg 12682 SubGrpcsubg 13223 NrmSGrpcnsg 13224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-cnex 7953 ax-resscn 7954 ax-1re 7956 ax-addrcl 7959 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-fv 5254 df-ov 5913 df-inn 8973 df-2 9031 df-ndx 12608 df-slot 12609 df-base 12611 df-plusg 12695 df-subg 13226 df-nsg 13227 |
This theorem is referenced by: conjnsg 13337 |
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