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Theorem subgslw 18741
 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 18734 . . 3 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
213ad2ant2 1131 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝑃 ∈ ℙ)
3 slwsubg 18735 . . . 4 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
433ad2ant2 1131 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5 simp3 1135 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾𝑆)
6 subgslw.1 . . . . 5 𝐻 = (𝐺s 𝑆)
76subsubg 18302 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾𝑆)))
873ad2ant1 1130 . . 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 7159 . . . . . . 7 (𝐻s 𝑥) = ((𝐺s 𝑆) ↾s 𝑥)
11 simpl1 1188 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
126subsubg 18302 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
13123ad2ant1 1130 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆)))
1413simplbda 503 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥𝑆)
15 ressabs 16563 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1611, 14, 15syl2anc 587 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐺s 𝑆) ↾s 𝑥) = (𝐺s 𝑥))
1710, 16syl5eq 2871 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻s 𝑥) = (𝐺s 𝑥))
1817breq2d 5064 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝑃 pGrp (𝐻s 𝑥) ↔ 𝑃 pGrp (𝐺s 𝑥)))
1918anbi2d 631 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ (𝐾𝑥𝑃 pGrp (𝐺s 𝑥))))
20 simpl2 1189 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
2113simprbda 502 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22 eqid 2824 . . . . . 6 (𝐺s 𝑥) = (𝐺s 𝑥)
2322slwispgp 18736 . . . . 5 ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2420, 21, 23syl2anc 587 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐺s 𝑥)) ↔ 𝐾 = 𝑥))
2519, 24bitrd 282 . . 3 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
2625ralrimiva 3177 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
27 isslw 18733 . 2 (𝐾 ∈ (𝑃 pSyl 𝐻) ↔ (𝑃 ∈ ℙ ∧ 𝐾 ∈ (SubGrp‘𝐻) ∧ ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥)))
282, 9, 26, 27syl3anbrc 1340 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133   ⊆ wss 3919   class class class wbr 5052  ‘cfv 6343  (class class class)co 7149  ℙcprime 16013   ↾s cress 16484  SubGrpcsubg 18273   pGrp cpgp 18654   pSyl cslw 18655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-1cn 10593  ax-addcl 10595 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-nn 11635  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-subg 18276  df-slw 18659 This theorem is referenced by:  sylow3lem6  18757
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