Metamath Proof Explorer


Theorem subgslw

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)

Ref Expression
Hypothesis subgslw.1 𝐻 = ( 𝐺s 𝑆 )
Assertion subgslw ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → 𝐾 ∈ ( 𝑃 pSyl 𝐻 ) )

Proof

Step Hyp Ref Expression
1 subgslw.1 𝐻 = ( 𝐺s 𝑆 )
2 slwprm ( 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) → 𝑃 ∈ ℙ )
3 2 3ad2ant2 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → 𝑃 ∈ ℙ )
4 slwsubg ( 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) → 𝐾 ∈ ( SubGrp ‘ 𝐺 ) )
5 4 3ad2ant2 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → 𝐾 ∈ ( SubGrp ‘ 𝐺 ) )
6 simp3 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → 𝐾𝑆 )
7 1 subsubg ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾𝑆 ) ) )
8 7 3ad2ant1 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → ( 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝐾 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾𝑆 ) ) )
9 5 6 8 mpbir2and ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → 𝐾 ∈ ( SubGrp ‘ 𝐻 ) )
10 1 oveq1i ( 𝐻s 𝑥 ) = ( ( 𝐺s 𝑆 ) ↾s 𝑥 )
11 simpl1 ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
12 1 subsubg ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥𝑆 ) ) )
13 12 3ad2ant1 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → ( 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ↔ ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥𝑆 ) ) )
14 13 simplbda ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑥𝑆 )
15 ressabs ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥𝑆 ) → ( ( 𝐺s 𝑆 ) ↾s 𝑥 ) = ( 𝐺s 𝑥 ) )
16 11 14 15 syl2anc ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐺s 𝑆 ) ↾s 𝑥 ) = ( 𝐺s 𝑥 ) )
17 10 16 eqtrid ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝐻s 𝑥 ) = ( 𝐺s 𝑥 ) )
18 17 breq2d ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( 𝑃 pGrp ( 𝐻s 𝑥 ) ↔ 𝑃 pGrp ( 𝐺s 𝑥 ) ) )
19 18 anbi2d ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐾𝑥𝑃 pGrp ( 𝐻s 𝑥 ) ) ↔ ( 𝐾𝑥𝑃 pGrp ( 𝐺s 𝑥 ) ) ) )
20 simpl2 ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) )
21 13 simprbda ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) )
22 eqid ( 𝐺s 𝑥 ) = ( 𝐺s 𝑥 )
23 22 slwispgp ( ( 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝐾𝑥𝑃 pGrp ( 𝐺s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) )
24 20 21 23 syl2anc ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐾𝑥𝑃 pGrp ( 𝐺s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) )
25 19 24 bitrd ( ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) ∧ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ) → ( ( 𝐾𝑥𝑃 pGrp ( 𝐻s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) )
26 25 ralrimiva ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → ∀ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ( ( 𝐾𝑥𝑃 pGrp ( 𝐻s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) )
27 isslw ( 𝐾 ∈ ( 𝑃 pSyl 𝐻 ) ↔ ( 𝑃 ∈ ℙ ∧ 𝐾 ∈ ( SubGrp ‘ 𝐻 ) ∧ ∀ 𝑥 ∈ ( SubGrp ‘ 𝐻 ) ( ( 𝐾𝑥𝑃 pGrp ( 𝐻s 𝑥 ) ) ↔ 𝐾 = 𝑥 ) ) )
28 3 9 26 27 syl3anbrc ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐾 ∈ ( 𝑃 pSyl 𝐺 ) ∧ 𝐾𝑆 ) → 𝐾 ∈ ( 𝑃 pSyl 𝐻 ) )