Theorem subgslw 19136

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 = (𝐺 ↾s 𝑆)

Assertion

Ref	Expression
subgslw	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

Proof of Theorem subgslw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		slwprm 19129	. . 3 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
2	1	3ad2ant2 1132	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝑃 ∈ ℙ)
3		slwsubg 19130	. . . 4 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
4	3	3ad2ant2 1132	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5		simp3 1136	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ⊆ 𝑆)
6		subgslw.1	. . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
7	6	subsubg 18693	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
8	7	3ad2ant1 1131	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
9	4, 5, 8	mpbir2and 709	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
10	6	oveq1i 7265	. . . . . . 7 ⊢ (𝐻 ↾s 𝑥) = ((𝐺 ↾s 𝑆) ↾s 𝑥)
11		simpl1 1189	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subsubg 18693	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆)))
13	12	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆)))
14	13	simplbda 499	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ⊆ 𝑆)
15		ressabs 16885	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝑥) = (𝐺 ↾s 𝑥))
16	11, 14, 15	syl2anc 583	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐺 ↾s 𝑆) ↾s 𝑥) = (𝐺 ↾s 𝑥))
17	10, 16	eqtrid 2790	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝑥) = (𝐺 ↾s 𝑥))
18	17	breq2d 5082	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝑃 pGrp (𝐻 ↾s 𝑥) ↔ 𝑃 pGrp (𝐺 ↾s 𝑥)))
19	18	anbi2d 628	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐻 ↾s 𝑥)) ↔ (𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐺 ↾s 𝑥))))
20		simpl2 1190	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
21	13	simprbda 498	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22		eqid 2738	. . . . . 6 ⊢ (𝐺 ↾s 𝑥) = (𝐺 ↾s 𝑥)
23	22	slwispgp 19131	. . . . 5 ⊢ ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐺 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
24	20, 21, 23	syl2anc 583	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐺 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
25	19, 24	bitrd 278	. . 3 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐻 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
26	25	ralrimiva 3107	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐻 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
27		isslw 19128	. 2 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐻) ↔ (𝑃 ∈ ℙ ∧ 𝐾 ∈ (SubGrp‘𝐻) ∧ ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐻 ↾s 𝑥)) ↔ 𝐾 = 𝑥)))
28	2, 9, 26, 27	syl3anbrc 1341	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  ℙcprime 16304   ↾_s cress 16867  SubGrpcsubg 18664   pGrp cpgp 19049   pSyl cslw 19050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-1cn 10860  ax-addcl 10862

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-nn 11904  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-subg 18667  df-slw 19054

This theorem is referenced by:  sylow3lem6  19152