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Theorem subgslw 19484
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 19477 . . 3 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
213ad2ant2 1135 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝑃 ∈ ℙ)
3 slwsubg 19478 . . . 4 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
433ad2ant2 1135 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5 simp3 1139 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾𝑆)
6 subgslw.1 . . . . 5 𝐻 = (𝐺s 𝑆)
76subsubg 19029 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
873ad2ant1 1134 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
94, 5, 8mpbir2and 712 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
106oveq1i 7419 . . . . . . 7 (𝐻s 𝑥) = ((𝐺s 𝑆) ↾s 𝑥)
11 simpl1 1192 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
126subsubg 19029 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
13123ad2ant1 1134 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
1413simplbda 501 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥𝑆)
15 ressabs 17194 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1611, 14, 15syl2anc 585 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1710, 16eqtrid 2785 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻s 𝑥) = (𝐺s 𝑥))
1817breq2d 5161 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝑃 pGrp (𝐻s 𝑥) ↔ 𝑃 pGrp (𝐺s 𝑥)))
1918anbi2d 630 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ (𝐾𝑥𝑃 pGrp (𝐺s 𝑥))))
20 simpl2 1193 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
2113simprbda 500 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22 eqid 2733 . . . . . 6 (𝐺s 𝑥) = (𝐺s 𝑥)
2322slwispgp 19479 . . . . 5 ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2420, 21, 23syl2anc 585 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2519, 24bitrd 279 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
2625ralrimiva 3147 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
27 isslw 19476 . 2 (𝐾 ∈ (𝑃 pSyl 𝐻) ↔ (𝑃 ∈ ℙ ∧ 𝐾 ∈ (SubGrp‘𝐻) ∧ ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥)))
282, 9, 26, 27syl3anbrc 1344 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wss 3949   class class class wbr 5149  cfv 6544  (class class class)co 7409  cprime 16608  s cress 17173  SubGrpcsubg 19000   pGrp cpgp 19394   pSyl cslw 19395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-subg 19003  df-slw 19399
This theorem is referenced by:  sylow3lem6  19500
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