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| Mirrors > Home > MPE Home > Th. List > nmznsg | Structured version Visualization version GIF version | ||
| Description: Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-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) |
| nmznsg.4 | ⊢ 𝐻 = (𝐺 ↾s 𝑁) |
| Ref | Expression |
|---|---|
| nmznsg | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
| 2 | elnmz.1 | . . . 4 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} | |
| 3 | nmzsubg.2 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
| 4 | nmzsubg.3 | . . . 4 ⊢ + = (+g‘𝐺) | |
| 5 | 2, 3, 4 | ssnmz 19184 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁) |
| 6 | subgrcl 19149 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
| 7 | 2, 3, 4 | nmzsubg 19183 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺)) |
| 8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) |
| 9 | nmznsg.4 | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑁) | |
| 10 | 9 | subsubg 19167 | . . . 4 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑁))) |
| 11 | 8, 10 | syl 17 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑁))) |
| 12 | 1, 5, 11 | mpbir2and 713 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻)) |
| 13 | 2 | ssrab3 4082 | . . . . . 6 ⊢ 𝑁 ⊆ 𝑋 |
| 14 | 13 | sseli 3979 | . . . . 5 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋) |
| 15 | 2 | nmzbi 19182 | . . . . 5 ⊢ ((𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
| 16 | 14, 15 | sylan2 593 | . . . 4 ⊢ ((𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
| 17 | 16 | rgen2 3199 | . . 3 ⊢ ∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) |
| 18 | 9 | subgbas 19148 | . . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻)) |
| 19 | 8, 18 | syl 17 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻)) |
| 20 | 19 | raleqdv 3326 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))) |
| 21 | 19, 20 | raleqbidv 3346 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))) |
| 22 | 17, 21 | mpbii 233 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
| 23 | eqid 2737 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 24 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝑋 ∈ V |
| 25 | 24, 13 | ssexi 5322 | . . . 4 ⊢ 𝑁 ∈ V |
| 26 | 9, 4 | ressplusg 17334 | . . . 4 ⊢ (𝑁 ∈ V → + = (+g‘𝐻)) |
| 27 | 25, 26 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝐻) |
| 28 | 23, 27 | isnsg 19173 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))) |
| 29 | 12, 22, 28 | sylanbrc 583 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 Grpcgrp 18951 SubGrpcsubg 19138 NrmSGrpcnsg 19139 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 |
| This theorem is referenced by: sylow3lem4 19648 sylow3lem6 19650 |
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