Theorem fislw 18040

Description: The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of P dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypothesis

Ref	Expression
fislw.1	|- X = ( Base ` G )

Assertion

Ref	Expression
fislw	|- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( H e. ( P pSyl G ) <-> ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) )

Proof of Theorem fislw

Dummy variables k n p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ H e. ( P pSyl G ) ) -> H e. ( P pSyl G ) )
2		slwsubg 18025	. . . 4 |- ( H e. ( P pSyl G ) -> H e. (SubGrp ` G ) )
3	1, 2	syl 17	. . 3 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ H e. ( P pSyl G ) ) -> H e. (SubGrp ` G ) )
4		fislw.1	. . . 4 |- X = ( Base ` G )
5		simpl2 1065	. . . 4 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ H e. ( P pSyl G ) ) -> X e. Fin )
6	4, 5, 1	slwhash 18039	. . 3 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ H e. ( P pSyl G ) ) -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
7	3, 6	jca 554	. 2 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ H e. ( P pSyl G ) ) -> ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) )
8		simpl3 1066	. . 3 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> P e. Prime )
9		simprl 794	. . 3 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> H e. (SubGrp ` G ) )
10		simpl2 1065	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> X e. Fin )
11	10	adantr 481	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> X e. Fin )
12		simprl 794	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> k e. (SubGrp ` G ) )
13	4	subgss 17595	. . . . . . . . 9 |- ( k e. (SubGrp ` G ) -> k C_ X )
14	12, 13	syl 17	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> k C_ X )
15		ssfi 8180	. . . . . . . 8 |- ( ( X e. Fin /\ k C_ X ) -> k e. Fin )
16	11, 14, 15	syl2anc 693	. . . . . . 7 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> k e. Fin )
17		simprrl 804	. . . . . . 7 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> H C_ k )
18		ssdomg 8001	. . . . . . . . 9 |- ( k e. Fin -> ( H C_ k -> H ~<_ k ) )
19	16, 17, 18	sylc 65	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> H ~<_ k )
20		simprrr 805	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> P pGrp ( G ↾s k ) )
21		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( G ↾s k ) = ( G ↾s k )
22	21	subggrp 17597	. . . . . . . . . . . . . . . . 17 |- ( k e. (SubGrp ` G ) -> ( G ↾s k ) e. Grp )
23	12, 22	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( G ↾s k ) e. Grp )
24	21	subgbas 17598	. . . . . . . . . . . . . . . . . 18 |- ( k e. (SubGrp ` G ) -> k = ( Base ` ( G ↾s k ) ) )
25	12, 24	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> k = ( Base ` ( G ↾s k ) ) )
26	25, 16	eqeltrrd 2702	. . . . . . . . . . . . . . . 16 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( Base ` ( G ↾s k ) ) e. Fin )
27		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( Base ` ( G ↾s k ) ) = ( Base ` ( G ↾s k ) )
28	27	pgpfi 18020	. . . . . . . . . . . . . . . 16 |- ( ( ( G ↾s k ) e. Grp /\ ( Base ` ( G ↾s k ) ) e. Fin ) -> ( P pGrp ( G ↾s k ) <-> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s k ) ) ) = ( P ^ n ) ) ) )
29	23, 26, 28	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pGrp ( G ↾s k ) <-> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s k ) ) ) = ( P ^ n ) ) ) )
30	20, 29	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s k ) ) ) = ( P ^ n ) ) )
31	30	simpld 475	. . . . . . . . . . . . 13 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> P e. Prime )
32		prmnn 15388	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. NN )
33	31, 32	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> P e. NN )
34	33	nnred 11035	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> P e. RR )
35	33	nnge1d 11063	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> 1 <_ P )
36		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( 0g ` G ) = ( 0g ` G )
37	36	subg0cl 17602	. . . . . . . . . . . . . . . . 17 |- ( k e. (SubGrp ` G ) -> ( 0g ` G ) e. k )
38	12, 37	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( 0g ` G ) e. k )
39		ne0i 3921	. . . . . . . . . . . . . . . 16 |- ( ( 0g ` G ) e. k -> k =/= (/) )
40	38, 39	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> k =/= (/) )
41		hashnncl 13157	. . . . . . . . . . . . . . . 16 |- ( k e. Fin -> ( ( # ` k ) e. NN <-> k =/= (/) ) )
42	16, 41	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( ( # ` k ) e. NN <-> k =/= (/) ) )
43	40, 42	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` k ) e. NN )
44	31, 43	pccld 15555	. . . . . . . . . . . . 13 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pCnt ( # ` k ) ) e. NN0 )
45	44	nn0zd 11480	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pCnt ( # ` k ) ) e. ZZ )
46		simpl1 1064	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> G e. Grp )
47	4	grpbn0 17451	. . . . . . . . . . . . . . . . 17 |- ( G e. Grp -> X =/= (/) )
48	46, 47	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> X =/= (/) )
49		hashnncl 13157	. . . . . . . . . . . . . . . . 17 |- ( X e. Fin -> ( ( # ` X ) e. NN <-> X =/= (/) ) )
50	10, 49	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( ( # ` X ) e. NN <-> X =/= (/) ) )
51	48, 50	mpbird 247	. . . . . . . . . . . . . . 15 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( # ` X ) e. NN )
52	8, 51	pccld 15555	. . . . . . . . . . . . . 14 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( P pCnt ( # ` X ) ) e. NN0 )
53	52	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pCnt ( # ` X ) ) e. NN0 )
54	53	nn0zd 11480	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pCnt ( # ` X ) ) e. ZZ )
55	4	lagsubg 17656	. . . . . . . . . . . . . . 15 |- ( ( k e. (SubGrp ` G ) /\ X e. Fin ) -> ( # ` k ) || ( # ` X ) )
56	12, 11, 55	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` k ) || ( # ` X ) )
57	43	nnzd 11481	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` k ) e. ZZ )
58	51	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` X ) e. NN )
59	58	nnzd 11481	. . . . . . . . . . . . . . 15 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` X ) e. ZZ )
60		pc2dvds 15583	. . . . . . . . . . . . . . 15 |- ( ( ( # ` k ) e. ZZ /\ ( # ` X ) e. ZZ ) -> ( ( # ` k ) || ( # ` X ) <-> A. p e. Prime ( p pCnt ( # ` k ) ) <_ ( p pCnt ( # ` X ) ) ) )
61	57, 59, 60	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( ( # ` k ) || ( # ` X ) <-> A. p e. Prime ( p pCnt ( # ` k ) ) <_ ( p pCnt ( # ` X ) ) ) )
62	56, 61	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> A. p e. Prime ( p pCnt ( # ` k ) ) <_ ( p pCnt ( # ` X ) ) )
63		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( p = P -> ( p pCnt ( # ` k ) ) = ( P pCnt ( # ` k ) ) )
64		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( p = P -> ( p pCnt ( # ` X ) ) = ( P pCnt ( # ` X ) ) )
65	63, 64	breq12d 4666	. . . . . . . . . . . . . 14 |- ( p = P -> ( ( p pCnt ( # ` k ) ) <_ ( p pCnt ( # ` X ) ) <-> ( P pCnt ( # ` k ) ) <_ ( P pCnt ( # ` X ) ) ) )
66	65	rspcv 3305	. . . . . . . . . . . . 13 |- ( P e. Prime -> ( A. p e. Prime ( p pCnt ( # ` k ) ) <_ ( p pCnt ( # ` X ) ) -> ( P pCnt ( # ` k ) ) <_ ( P pCnt ( # ` X ) ) ) )
67	31, 62, 66	sylc 65	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pCnt ( # ` k ) ) <_ ( P pCnt ( # ` X ) ) )
68		eluz2 11693	. . . . . . . . . . . 12 |- ( ( P pCnt ( # ` X ) ) e. ( ZZ>= ` ( P pCnt ( # ` k ) ) ) <-> ( ( P pCnt ( # ` k ) ) e. ZZ /\ ( P pCnt ( # ` X ) ) e. ZZ /\ ( P pCnt ( # ` k ) ) <_ ( P pCnt ( # ` X ) ) ) )
69	45, 54, 67, 68	syl3anbrc 1246	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P pCnt ( # ` X ) ) e. ( ZZ>= ` ( P pCnt ( # ` k ) ) ) )
70	34, 35, 69	leexp2ad 13041	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( P ^ ( P pCnt ( # ` k ) ) ) <_ ( P ^ ( P pCnt ( # ` X ) ) ) )
71	30	simprd 479	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> E. n e. NN0 ( # ` ( Base ` ( G ↾s k ) ) ) = ( P ^ n ) )
72	25	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` k ) = ( # ` ( Base ` ( G ↾s k ) ) ) )
73	72	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( ( # ` k ) = ( P ^ n ) <-> ( # ` ( Base ` ( G ↾s k ) ) ) = ( P ^ n ) ) )
74	73	rexbidv 3052	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( E. n e. NN0 ( # ` k ) = ( P ^ n ) <-> E. n e. NN0 ( # ` ( Base ` ( G ↾s k ) ) ) = ( P ^ n ) ) )
75	71, 74	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> E. n e. NN0 ( # ` k ) = ( P ^ n ) )
76		pcprmpw 15587	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( # ` k ) e. NN ) -> ( E. n e. NN0 ( # ` k ) = ( P ^ n ) <-> ( # ` k ) = ( P ^ ( P pCnt ( # ` k ) ) ) ) )
77	31, 43, 76	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( E. n e. NN0 ( # ` k ) = ( P ^ n ) <-> ( # ` k ) = ( P ^ ( P pCnt ( # ` k ) ) ) ) )
78	75, 77	mpbid 222	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` k ) = ( P ^ ( P pCnt ( # ` k ) ) ) )
79		simplrr 801	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
80	70, 78, 79	3brtr4d 4685	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( # ` k ) <_ ( # ` H ) )
81	4	subgss 17595	. . . . . . . . . . . . 13 |- ( H e. (SubGrp ` G ) -> H C_ X )
82	81	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> H C_ X )
83		ssfi 8180	. . . . . . . . . . . 12 |- ( ( X e. Fin /\ H C_ X ) -> H e. Fin )
84	10, 82, 83	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> H e. Fin )
85	84	adantr 481	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> H e. Fin )
86		hashdom 13168	. . . . . . . . . 10 |- ( ( k e. Fin /\ H e. Fin ) -> ( ( # ` k ) <_ ( # ` H ) <-> k ~<_ H ) )
87	16, 85, 86	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> ( ( # ` k ) <_ ( # ` H ) <-> k ~<_ H ) )
88	80, 87	mpbid 222	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> k ~<_ H )
89		sbth 8080	. . . . . . . 8 |- ( ( H ~<_ k /\ k ~<_ H ) -> H ~~ k )
90	19, 88, 89	syl2anc 693	. . . . . . 7 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> H ~~ k )
91		fisseneq 8171	. . . . . . 7 |- ( ( k e. Fin /\ H C_ k /\ H ~~ k ) -> H = k )
92	16, 17, 90, 91	syl3anc 1326	. . . . . 6 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( k e. (SubGrp ` G ) /\ ( H C_ k /\ P pGrp ( G ↾s k ) ) ) ) -> H = k )
93	92	expr 643	. . . . 5 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ k e. (SubGrp ` G ) ) -> ( ( H C_ k /\ P pGrp ( G ↾s k ) ) -> H = k ) )
94		eqid 2622	. . . . . . . . . . . . 13 |- ( G ↾s H ) = ( G ↾s H )
95	94	subgbas 17598	. . . . . . . . . . . 12 |- ( H e. (SubGrp ` G ) -> H = ( Base ` ( G ↾s H ) ) )
96	95	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> H = ( Base ` ( G ↾s H ) ) )
97	96	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( # ` H ) = ( # ` ( Base ` ( G ↾s H ) ) ) )
98		simprr 796	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
99	97, 98	eqtr3d 2658	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
100		oveq2 6658	. . . . . . . . . . 11 |- ( n = ( P pCnt ( # ` X ) ) -> ( P ^ n ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
101	100	eqeq2d 2632	. . . . . . . . . 10 |- ( n = ( P pCnt ( # ` X ) ) -> ( ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ n ) <-> ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) )
102	101	rspcev 3309	. . . . . . . . 9 |- ( ( ( P pCnt ( # ` X ) ) e. NN0 /\ ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) -> E. n e. NN0 ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ n ) )
103	52, 99, 102	syl2anc 693	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> E. n e. NN0 ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ n ) )
104	94	subggrp 17597	. . . . . . . . . 10 |- ( H e. (SubGrp ` G ) -> ( G ↾s H ) e. Grp )
105	104	ad2antrl 764	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( G ↾s H ) e. Grp )
106	96, 84	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( Base ` ( G ↾s H ) ) e. Fin )
107		eqid 2622	. . . . . . . . . 10 |- ( Base ` ( G ↾s H ) ) = ( Base ` ( G ↾s H ) )
108	107	pgpfi 18020	. . . . . . . . 9 |- ( ( ( G ↾s H ) e. Grp /\ ( Base ` ( G ↾s H ) ) e. Fin ) -> ( P pGrp ( G ↾s H ) <-> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ n ) ) ) )
109	105, 106, 108	syl2anc 693	. . . . . . . 8 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( P pGrp ( G ↾s H ) <-> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s H ) ) ) = ( P ^ n ) ) ) )
110	8, 103, 109	mpbir2and 957	. . . . . . 7 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> P pGrp ( G ↾s H ) )
111	110	adantr 481	. . . . . 6 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ k e. (SubGrp ` G ) ) -> P pGrp ( G ↾s H ) )
112		oveq2 6658	. . . . . . . 8 |- ( H = k -> ( G ↾s H ) = ( G ↾s k ) )
113	112	breq2d 4665	. . . . . . 7 |- ( H = k -> ( P pGrp ( G ↾s H ) <-> P pGrp ( G ↾s k ) ) )
114		eqimss 3657	. . . . . . . 8 |- ( H = k -> H C_ k )
115	114	biantrurd 529	. . . . . . 7 |- ( H = k -> ( P pGrp ( G ↾s k ) <-> ( H C_ k /\ P pGrp ( G ↾s k ) ) ) )
116	113, 115	bitrd 268	. . . . . 6 |- ( H = k -> ( P pGrp ( G ↾s H ) <-> ( H C_ k /\ P pGrp ( G ↾s k ) ) ) )
117	111, 116	syl5ibcom 235	. . . . 5 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ k e. (SubGrp ` G ) ) -> ( H = k -> ( H C_ k /\ P pGrp ( G ↾s k ) ) ) )
118	93, 117	impbid 202	. . . 4 |- ( ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ k e. (SubGrp ` G ) ) -> ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) )
119	118	ralrimiva 2966	. . 3 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) )
120		isslw 18023	. . 3 |- ( H e. ( P pSyl G ) <-> ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) )
121	8, 9, 119, 120	syl3anbrc 1246	. 2 |- ( ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) /\ ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> H e. ( P pSyl G ) )
122	7, 121	impbida 877	1 |- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( H e. ( P pSyl G ) <-> ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588   pGrp cpgp 17946   pSyl cslw 17947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-ghm 17658  df-ga 17723  df-od 17948  df-pgp 17950  df-slw 17951

This theorem is referenced by:  sylow3lem1  18042