MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgslw Structured version   Visualization version   GIF version

Theorem subgslw 19634
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 19627 . . 3 (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
213ad2ant2 1135 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝑃 ∈ ℙ)
3 slwsubg 19628 . . . 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 19167 . . . 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 713 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
106oveq1i 7441 . . . . . . 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 19167 . . . . . . . . . 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 499 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥𝑆)
15 ressabs 17294 . . . . . . . 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 2789 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻s 𝑥) = (𝐺s 𝑥))
1817breq2d 5155 . . . . 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 498 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22 eqid 2737 . . . . . 6 (𝐺s 𝑥) = (𝐺s 𝑥)
2322slwispgp 19629 . . . . 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 3146 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾𝑥𝑃 pGrp (𝐻s 𝑥)) ↔ 𝐾 = 𝑥))
27 isslw 19626 . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951   class class class wbr 5143  cfv 6561  (class class class)co 7431  cprime 16708  s cress 17274  SubGrpcsubg 19138   pGrp cpgp 19544   pSyl cslw 19545
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-1cn 11213  ax-addcl 11215
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-ral 3062  df-rex 3071  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-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-subg 19141  df-slw 19549
This theorem is referenced by:  sylow3lem6  19650
  Copyright terms: Public domain W3C validator