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Theorem subgslw 19649
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 19642 . . 3 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
213ad2ant2 1133 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝑃 ∈ ℙ)
3 slwsubg 19643 . . . 4 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
433ad2ant2 1133 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5 simp3 1137 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾𝑆)
6 subgslw.1 . . . . 5 𝐻 = (𝐺s 𝑆)
76subsubg 19180 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
873ad2ant1 1132 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
94, 5, 8mpbir2and 713 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
106oveq1i 7441 . . . . . . 7 (𝐻s 𝑥) = ((𝐺s 𝑆) ↾s 𝑥)
11 simpl1 1190 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
126subsubg 19180 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
13123ad2ant1 1132 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
1413simplbda 499 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥𝑆)
15 ressabs 17295 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1611, 14, 15syl2anc 584 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1710, 16eqtrid 2787 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻s 𝑥) = (𝐺s 𝑥))
1817breq2d 5160 . . . . 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 1191 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
2113simprbda 498 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22 eqid 2735 . . . . . 6 (𝐺s 𝑥) = (𝐺s 𝑥)
2322slwispgp 19644 . . . . 5 ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2420, 21, 23syl2anc 584 . . . 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 3144 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
27 isslw 19641 . 2 (𝐾 ∈ (𝑃 pSyl 𝐻) ↔ (𝑃 ∈ ℙ ∧ 𝐾 ∈ (SubGrp‘𝐻) ∧ ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥)))
282, 9, 26, 27syl3anbrc 1342 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  cprime 16705  s cress 17274  SubGrpcsubg 19151   pGrp cpgp 19559   pSyl cslw 19560
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  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-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-pss 3983  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-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  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-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-subg 19154  df-slw 19564
This theorem is referenced by:  sylow3lem6  19665
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