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Theorem subgslw 19525
Description: A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypothesis
Ref Expression
subgslw.1 𝐻 = (𝐺s 𝑆)
Assertion
Ref Expression
subgslw ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

Proof of Theorem subgslw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 slwprm 19518 . . 3 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
213ad2ant2 1132 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝑃 ∈ ℙ)
3 slwsubg 19519 . . . 4 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
433ad2ant2 1132 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5 simp3 1136 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾𝑆)
6 subgslw.1 . . . . 5 𝐻 = (𝐺s 𝑆)
76subsubg 19065 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
873ad2ant1 1131 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
94, 5, 8mpbir2and 709 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
106oveq1i 7421 . . . . . . 7 (𝐻s 𝑥) = ((𝐺s 𝑆) ↾s 𝑥)
11 simpl1 1189 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
126subsubg 19065 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
13123ad2ant1 1131 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
1413simplbda 498 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥𝑆)
15 ressabs 17198 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1611, 14, 15syl2anc 582 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1710, 16eqtrid 2782 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻s 𝑥) = (𝐺s 𝑥))
1817breq2d 5159 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝑃 pGrp (𝐻s 𝑥) ↔ 𝑃 pGrp (𝐺s 𝑥)))
1918anbi2d 627 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ (𝐾𝑥𝑃 pGrp (𝐺s 𝑥))))
20 simpl2 1190 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
2113simprbda 497 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22 eqid 2730 . . . . . 6 (𝐺s 𝑥) = (𝐺s 𝑥)
2322slwispgp 19520 . . . . 5 ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2420, 21, 23syl2anc 582 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2519, 24bitrd 278 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
2625ralrimiva 3144 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
27 isslw 19517 . 2 (𝐾 ∈ (𝑃 pSyl 𝐻) ↔ (𝑃 ∈ ℙ ∧ 𝐾 ∈ (SubGrp‘𝐻) ∧ ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥)))
282, 9, 26, 27syl3anbrc 1341 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wss 3947   class class class wbr 5147  cfv 6542  (class class class)co 7411  cprime 16612  s cress 17177  SubGrpcsubg 19036   pGrp cpgp 19435   pSyl cslw 19436
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-subg 19039  df-slw 19440
This theorem is referenced by:  sylow3lem6  19541
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