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 = (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 18734	. . 3 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
2	1	3ad2ant2 1130	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝑃 ∈ ℙ)
3		slwsubg 18735	. . . 4 ⊢ (𝐾 ∈ (𝑃 pSyl 𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
4	3	3ad2ant2 1130	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (SubGrp‘𝐺))
5		simp3 1134	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ⊆ 𝑆)
6		subgslw.1	. . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
7	6	subsubg 18302	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
8	7	3ad2ant1 1129	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → (𝐾 ∈ (SubGrp‘𝐻) ↔ (𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐾 ⊆ 𝑆)))
9	4, 5, 8	mpbir2and 711	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (SubGrp‘𝐻))
10	6	oveq1i 7166	. . . . . . 7 ⊢ (𝐻 ↾s 𝑥) = ((𝐺 ↾s 𝑆) ↾s 𝑥)
11		simpl1 1187	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑆 ∈ (SubGrp‘𝐺))
12	6	subsubg 18302	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆)))
13	12	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → (𝑥 ∈ (SubGrp‘𝐻) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆)))
14	13	simplbda 502	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ⊆ 𝑆)
15		ressabs 16563	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝑥) = (𝐺 ↾s 𝑥))
16	11, 14, 15	syl2anc 586	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → ((𝐺 ↾s 𝑆) ↾s 𝑥) = (𝐺 ↾s 𝑥))
17	10, 16	syl5eq 2868	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝑥) = (𝐺 ↾s 𝑥))
18	17	breq2d 5078	. . . . 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 630	. . . 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 1188	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝐾 ∈ (𝑃 pSyl 𝐺))
21	13	simprbda 501	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) ∧ 𝑥 ∈ (SubGrp‘𝐻)) → 𝑥 ∈ (SubGrp‘𝐺))
22		eqid 2821	. . . . . 6 ⊢ (𝐺 ↾s 𝑥) = (𝐺 ↾s 𝑥)
23	22	slwispgp 18736	. . . . 5 ⊢ ((𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐺 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
24	20, 21, 23	syl2anc 586	. . . 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 281	. . 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 3182	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → ∀𝑥 ∈ (SubGrp‘𝐻)((𝐾 ⊆ 𝑥 ∧ 𝑃 pGrp (𝐻 ↾s 𝑥)) ↔ 𝐾 = 𝑥))
27		isslw 18733	. 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 1339	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  ℙcprime 16015   ↾_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 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-1cn 10595  ax-addcl 10597

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-nn 11639  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