Theorem subgslw 19830

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 19823	. . 3 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
2	1	3ad2ant2 1152	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝑃 ∈ ℙ)
3		slwsubg 19824	. . . 4 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
4	3	3ad2ant2 1152	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5		simp3 1156	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ⊆ 𝑆)
6		subgslw.1	. . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
7	6	subsubg 19360	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
8	7	3ad2ant1 1151	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
9	4, 5, 8	mpbir2and 726	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
10	6	oveq1i 7430	. . . . . . 7 ⊢ (𝐻 ↾s 𝑥) = ((𝐺 ↾s 𝑆) ↾s 𝑥)
11		simpl1 1210	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subsubg 19360	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆)))
13	12	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆)))
14	13	simplbda 505	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ⊆ 𝑆)
15		ressabs 17426	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝑥) = (𝐺 ↾s 𝑥))
16	11, 14, 15	syl2anc 596	. . . . . . 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 2808	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝑥) = (𝐺 ↾s 𝑥))
18	17	breq2d 5115	. . . . 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 642	. . . 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 1211	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
21	13	simprbda 504	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22		eqid 2761	. . . . . 6 ⊢ (𝐺 ↾s 𝑥) = (𝐺 ↾s 𝑥)
23	22	slwispgp 19825	. . . . 5 ⊢ ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐺 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
24	20, 21, 23	syl2anc 596	. . . 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 282	. . 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 3155	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐻 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
27		isslw 19822	. 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 1362	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  ℙcprime 16846   ↾_s cress 17408  SubGrpcsubg 19330   pGrp cpgp 19740   pSyl cslw 19741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-1cn 11258  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-nn 12336  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-subg 19333  df-slw 19745

This theorem is used by:  sylow3lem6  19846