Theorem slwhash 18039

Description: A sylow subgroup has cardinality equal to the maximum power of P dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
fislw.1	|- X = ( Base ` G )
slwhash.3	|- ( ph -> X e. Fin )
slwhash.4	|- ( ph -> H e. ( P pSyl G ) )

Assertion

Ref	Expression
slwhash	|- ( ph -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )

Proof of Theorem slwhash

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

Step	Hyp	Ref	Expression
1		fislw.1	. . 3 |- X = ( Base ` G )
2		slwhash.4	. . . . 5 |- ( ph -> H e. ( P pSyl G ) )
3		slwsubg 18025	. . . . 5 |- ( H e. ( P pSyl G ) -> H e. (SubGrp ` G ) )
4	2, 3	syl 17	. . . 4 |- ( ph -> H e. (SubGrp ` G ) )
5		subgrcl 17599	. . . 4 |- ( H e. (SubGrp ` G ) -> G e. Grp )
6	4, 5	syl 17	. . 3 |- ( ph -> G e. Grp )
7		slwhash.3	. . 3 |- ( ph -> X e. Fin )
8		slwprm 18024	. . . 4 |- ( H e. ( P pSyl G ) -> P e. Prime )
9	2, 8	syl 17	. . 3 |- ( ph -> P e. Prime )
10	1	grpbn0 17451	. . . . . 6 |- ( G e. Grp -> X =/= (/) )
11	6, 10	syl 17	. . . . 5 |- ( ph -> X =/= (/) )
12		hashnncl 13157	. . . . . 6 |- ( X e. Fin -> ( ( # ` X ) e. NN <-> X =/= (/) ) )
13	7, 12	syl 17	. . . . 5 |- ( ph -> ( ( # ` X ) e. NN <-> X =/= (/) ) )
14	11, 13	mpbird 247	. . . 4 |- ( ph -> ( # ` X ) e. NN )
15	9, 14	pccld 15555	. . 3 |- ( ph -> ( P pCnt ( # ` X ) ) e. NN0 )
16		pcdvds 15568	. . . 4 |- ( ( P e. Prime /\ ( # ` X ) e. NN ) -> ( P ^ ( P pCnt ( # ` X ) ) ) || ( # ` X ) )
17	9, 14, 16	syl2anc 693	. . 3 |- ( ph -> ( P ^ ( P pCnt ( # ` X ) ) ) || ( # ` X ) )
18	1, 6, 7, 9, 15, 17	sylow1 18018	. 2 |- ( ph -> E. k e. (SubGrp ` G ) ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
19	7	adantr 481	. . . 4 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> X e. Fin )
20	4	adantr 481	. . . 4 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> H e. (SubGrp ` G ) )
21		simprl 794	. . . 4 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> k e. (SubGrp ` G ) )
22		eqid 2622	. . . 4 |- ( +g ` G ) = ( +g ` G )
23		eqid 2622	. . . . . . 7 |- ( G ↾s H ) = ( G ↾s H )
24	23	slwpgp 18028	. . . . . 6 |- ( H e. ( P pSyl G ) -> P pGrp ( G ↾s H ) )
25	2, 24	syl 17	. . . . 5 |- ( ph -> P pGrp ( G ↾s H ) )
26	25	adantr 481	. . . 4 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> P pGrp ( G ↾s H ) )
27		simprr 796	. . . 4 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
28		eqid 2622	. . . 4 |- ( -g ` G ) = ( -g ` G )
29	1, 19, 20, 21, 22, 26, 27, 28	sylow2b 18038	. . 3 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> E. g e. X H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) )
30		simprr 796	. . . . . 6 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) )
31	2	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> H e. ( P pSyl G ) )
32	31, 8	syl 17	. . . . . . 7 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> P e. Prime )
33	15	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( P pCnt ( # ` X ) ) e. NN0 )
34	21	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> k e. (SubGrp ` G ) )
35		simprl 794	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> g e. X )
36		eqid 2622	. . . . . . . . . . . . 13 |- ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) = ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) )
37	1, 22, 28, 36	conjsubg 17692	. . . . . . . . . . . 12 |- ( ( k e. (SubGrp ` G ) /\ g e. X ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. (SubGrp ` G ) )
38	34, 35, 37	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. (SubGrp ` G ) )
39		eqid 2622	. . . . . . . . . . . 12 |- ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) = ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) )
40	39	subgbas 17598	. . . . . . . . . . 11 |- ( ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. (SubGrp ` G ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) = ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) )
41	38, 40	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) = ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) )
42	41	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) = ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) )
43	1, 22, 28, 36	conjsubgen 17693	. . . . . . . . . . . 12 |- ( ( k e. (SubGrp ` G ) /\ g e. X ) -> k ~~ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) )
44	34, 35, 43	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> k ~~ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) )
45	7	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> X e. Fin )
46	1	subgss 17595	. . . . . . . . . . . . . 14 |- ( k e. (SubGrp ` G ) -> k C_ X )
47	34, 46	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> k C_ X )
48		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( X e. Fin /\ k C_ X ) -> k e. Fin )
49	45, 47, 48	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> k e. Fin )
50	1	subgss 17595	. . . . . . . . . . . . . 14 |- ( ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. (SubGrp ` G ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) C_ X )
51	38, 50	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) C_ X )
52		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( X e. Fin /\ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) C_ X ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. Fin )
53	45, 51, 52	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. Fin )
54		hashen 13135	. . . . . . . . . . . 12 |- ( ( k e. Fin /\ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. Fin ) -> ( ( # ` k ) = ( # ` ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) <-> k ~~ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
55	49, 53, 54	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( ( # ` k ) = ( # ` ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) <-> k ~~ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
56	44, 55	mpbird 247	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` k ) = ( # ` ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
57		simplrr 801	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
58	56, 57	eqtr3d 2658	. . . . . . . . 9 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
59	42, 58	eqtr3d 2658	. . . . . . . 8 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
60		oveq2 6658	. . . . . . . . . 10 |- ( n = ( P pCnt ( # ` X ) ) -> ( P ^ n ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
61	60	eqeq2d 2632	. . . . . . . . 9 |- ( n = ( P pCnt ( # ` X ) ) -> ( ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ n ) <-> ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) )
62	61	rspcev 3309	. . . . . . . 8 |- ( ( ( P pCnt ( # ` X ) ) e. NN0 /\ ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) -> E. n e. NN0 ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ n ) )
63	33, 59, 62	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> E. n e. NN0 ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ n ) )
64	39	subggrp 17597	. . . . . . . . 9 |- ( ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. (SubGrp ` G ) -> ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) e. Grp )
65	38, 64	syl 17	. . . . . . . 8 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) e. Grp )
66	41, 53	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) e. Fin )
67		eqid 2622	. . . . . . . . 9 |- ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) = ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
68	67	pgpfi 18020	. . . . . . . 8 |- ( ( ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) e. Grp /\ ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) e. Fin ) -> ( P pGrp ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) <-> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ n ) ) ) )
69	65, 66, 68	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( P pGrp ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) <-> ( P e. Prime /\ E. n e. NN0 ( # ` ( Base ` ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) ) = ( P ^ n ) ) ) )
70	32, 63, 69	mpbir2and 957	. . . . . 6 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> P pGrp ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
71	39	slwispgp 18026	. . . . . . 7 |- ( ( H e. ( P pSyl G ) /\ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) e. (SubGrp ` G ) ) -> ( ( H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) /\ P pGrp ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) <-> H = ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
72	31, 38, 71	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( ( H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) /\ P pGrp ( G ↾s ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) <-> H = ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
73	30, 70, 72	mpbi2and 956	. . . . 5 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> H = ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) )
74	73	fveq2d 6195	. . . 4 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` H ) = ( # ` ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) )
75	74, 58	eqtrd 2656	. . 3 |- ( ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) /\ ( g e. X /\ H C_ ran ( x e. k |-> ( ( g ( +g ` G ) x ) ( -g ` G ) g ) ) ) ) -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
76	29, 75	rexlimddv 3035	. 2 |- ( ( ph /\ ( k e. (SubGrp ` G ) /\ ( # ` k ) = ( P ^ ( P pCnt ( # ` X ) ) ) ) ) -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
77	18, 76	rexlimddv 3035	1 |- ( ph -> ( # ` H ) = ( P ^ ( P pCnt ( # ` X ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Fincfn 7955   NNcn 11020   NN0cn0 11292   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   Grpcgrp 17422   -gcsg 17424  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:  fislw  18040  sylow2  18041  sylow3lem4  18045