Theorem pgpssslw 18029

Description: Every P-subgroup is contained in a Sylow P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypotheses

Ref	Expression
pgpssslw.1	|- X = ( Base ` G )
pgpssslw.2	|- S = ( G ↾s H )
pgpssslw.3	|- F = ( x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } |-> ( # ` x ) )

Assertion

Ref	Expression
pgpssslw	|- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> E. k e. ( P pSyl G ) H C_ k )

Distinct variable groups:   x, k, y, G   k, H, x, y   P, k, x, y   k, X, x   k, F   S, k, x, y

Allowed substitution hints:   F( x, y)   X( y)

Proof of Theorem pgpssslw

Dummy variables m w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . . . . . . . 10 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> X e. Fin )
2		elrabi 3359	. . . . . . . . . . 11 |- ( x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } -> x e. (SubGrp ` G ) )
3		pgpssslw.1	. . . . . . . . . . . 12 |- X = ( Base ` G )
4	3	subgss 17595	. . . . . . . . . . 11 |- ( x e. (SubGrp ` G ) -> x C_ X )
5	2, 4	syl 17	. . . . . . . . . 10 |- ( x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } -> x C_ X )
6		ssfi 8180	. . . . . . . . . 10 |- ( ( X e. Fin /\ x C_ X ) -> x e. Fin )
7	1, 5, 6	syl2an 494	. . . . . . . . 9 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> x e. Fin )
8		hashcl 13147	. . . . . . . . 9 |- ( x e. Fin -> ( # ` x ) e. NN0 )
9	7, 8	syl 17	. . . . . . . 8 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( # ` x ) e. NN0 )
10	9	nn0zd 11480	. . . . . . 7 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( # ` x ) e. ZZ )
11		pgpssslw.3	. . . . . . 7 |- F = ( x e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } |-> ( # ` x ) )
12	10, 11	fmptd 6385	. . . . . 6 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> F : { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } --> ZZ )
13		frn 6053	. . . . . 6 |- ( F : { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } --> ZZ -> ran F C_ ZZ )
14	12, 13	syl 17	. . . . 5 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ran F C_ ZZ )
15		fvex 6201	. . . . . . . 8 |- ( # ` x ) e. _V
16	15, 11	fnmpti 6022	. . . . . . 7 |- F Fn { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) }
17		simp1 1061	. . . . . . . 8 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> H e. (SubGrp ` G ) )
18		simp3 1063	. . . . . . . 8 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> P pGrp S )
19		eqimss2 3658	. . . . . . . . . . 11 |- ( y = H -> H C_ y )
20	19	biantrud 528	. . . . . . . . . 10 |- ( y = H -> ( P pGrp ( G ↾s y ) <-> ( P pGrp ( G ↾s y ) /\ H C_ y ) ) )
21		oveq2 6658	. . . . . . . . . . . 12 |- ( y = H -> ( G ↾s y ) = ( G ↾s H ) )
22		pgpssslw.2	. . . . . . . . . . . 12 |- S = ( G ↾s H )
23	21, 22	syl6eqr 2674	. . . . . . . . . . 11 |- ( y = H -> ( G ↾s y ) = S )
24	23	breq2d 4665	. . . . . . . . . 10 |- ( y = H -> ( P pGrp ( G ↾s y ) <-> P pGrp S ) )
25	20, 24	bitr3d 270	. . . . . . . . 9 |- ( y = H -> ( ( P pGrp ( G ↾s y ) /\ H C_ y ) <-> P pGrp S ) )
26	25	elrab 3363	. . . . . . . 8 |- ( H e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } <-> ( H e. (SubGrp ` G ) /\ P pGrp S ) )
27	17, 18, 26	sylanbrc 698	. . . . . . 7 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> H e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } )
28		fnfvelrn 6356	. . . . . . 7 |- ( ( F Fn { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } /\ H e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( F ` H ) e. ran F )
29	16, 27, 28	sylancr 695	. . . . . 6 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ( F ` H ) e. ran F )
30		ne0i 3921	. . . . . 6 |- ( ( F ` H ) e. ran F -> ran F =/= (/) )
31	29, 30	syl 17	. . . . 5 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ran F =/= (/) )
32		hashcl 13147	. . . . . . . 8 |- ( X e. Fin -> ( # ` X ) e. NN0 )
33	1, 32	syl 17	. . . . . . 7 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ( # ` X ) e. NN0 )
34	33	nn0red 11352	. . . . . 6 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ( # ` X ) e. RR )
35		fveq2 6191	. . . . . . . . . . 11 |- ( x = m -> ( # ` x ) = ( # ` m ) )
36		fvex 6201	. . . . . . . . . . 11 |- ( # ` m ) e. _V
37	35, 11, 36	fvmpt 6282	. . . . . . . . . 10 |- ( m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } -> ( F ` m ) = ( # ` m ) )
38	37	adantl 482	. . . . . . . . 9 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( F ` m ) = ( # ` m ) )
39		oveq2 6658	. . . . . . . . . . . . 13 |- ( y = m -> ( G ↾s y ) = ( G ↾s m ) )
40	39	breq2d 4665	. . . . . . . . . . . 12 |- ( y = m -> ( P pGrp ( G ↾s y ) <-> P pGrp ( G ↾s m ) ) )
41		sseq2 3627	. . . . . . . . . . . 12 |- ( y = m -> ( H C_ y <-> H C_ m ) )
42	40, 41	anbi12d 747	. . . . . . . . . . 11 |- ( y = m -> ( ( P pGrp ( G ↾s y ) /\ H C_ y ) <-> ( P pGrp ( G ↾s m ) /\ H C_ m ) ) )
43	42	elrab 3363	. . . . . . . . . 10 |- ( m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } <-> ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) )
44	1	adantr 481	. . . . . . . . . . . 12 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) ) -> X e. Fin )
45	3	subgss 17595	. . . . . . . . . . . . 13 |- ( m e. (SubGrp ` G ) -> m C_ X )
46	45	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) ) -> m C_ X )
47		ssdomg 8001	. . . . . . . . . . . 12 |- ( X e. Fin -> ( m C_ X -> m ~<_ X ) )
48	44, 46, 47	sylc 65	. . . . . . . . . . 11 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) ) -> m ~<_ X )
49		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( X e. Fin /\ m C_ X ) -> m e. Fin )
50	44, 46, 49	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) ) -> m e. Fin )
51		hashdom 13168	. . . . . . . . . . . 12 |- ( ( m e. Fin /\ X e. Fin ) -> ( ( # ` m ) <_ ( # ` X ) <-> m ~<_ X ) )
52	50, 44, 51	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) ) -> ( ( # ` m ) <_ ( # ` X ) <-> m ~<_ X ) )
53	48, 52	mpbird 247	. . . . . . . . . 10 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( m e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s m ) /\ H C_ m ) ) ) -> ( # ` m ) <_ ( # ` X ) )
54	43, 53	sylan2b 492	. . . . . . . . 9 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( # ` m ) <_ ( # ` X ) )
55	38, 54	eqbrtrd 4675	. . . . . . . 8 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( F ` m ) <_ ( # ` X ) )
56	55	ralrimiva 2966	. . . . . . 7 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> A. m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` m ) <_ ( # ` X ) )
57		breq1 4656	. . . . . . . . 9 |- ( w = ( F ` m ) -> ( w <_ ( # ` X ) <-> ( F ` m ) <_ ( # ` X ) ) )
58	57	ralrn 6362	. . . . . . . 8 |- ( F Fn { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } -> ( A. w e. ran F w <_ ( # ` X ) <-> A. m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` m ) <_ ( # ` X ) ) )
59	16, 58	ax-mp 5	. . . . . . 7 |- ( A. w e. ran F w <_ ( # ` X ) <-> A. m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` m ) <_ ( # ` X ) )
60	56, 59	sylibr 224	. . . . . 6 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> A. w e. ran F w <_ ( # ` X ) )
61		breq2 4657	. . . . . . . 8 |- ( z = ( # ` X ) -> ( w <_ z <-> w <_ ( # ` X ) ) )
62	61	ralbidv 2986	. . . . . . 7 |- ( z = ( # ` X ) -> ( A. w e. ran F w <_ z <-> A. w e. ran F w <_ ( # ` X ) ) )
63	62	rspcev 3309	. . . . . 6 |- ( ( ( # ` X ) e. RR /\ A. w e. ran F w <_ ( # ` X ) ) -> E. z e. RR A. w e. ran F w <_ z )
64	34, 60, 63	syl2anc 693	. . . . 5 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> E. z e. RR A. w e. ran F w <_ z )
65		suprzcl 11457	. . . . 5 |- ( ( ran F C_ ZZ /\ ran F =/= (/) /\ E. z e. RR A. w e. ran F w <_ z ) -> sup ( ran F , RR , < ) e. ran F )
66	14, 31, 64, 65	syl3anc 1326	. . . 4 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> sup ( ran F , RR , < ) e. ran F )
67		fvelrnb 6243	. . . . 5 |- ( F Fn { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } -> ( sup ( ran F , RR , < ) e. ran F <-> E. k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` k ) = sup ( ran F , RR , < ) ) )
68	16, 67	ax-mp 5	. . . 4 |- ( sup ( ran F , RR , < ) e. ran F <-> E. k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` k ) = sup ( ran F , RR , < ) )
69	66, 68	sylib 208	. . 3 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> E. k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` k ) = sup ( ran F , RR , < ) )
70		oveq2 6658	. . . . . 6 |- ( y = k -> ( G ↾s y ) = ( G ↾s k ) )
71	70	breq2d 4665	. . . . 5 |- ( y = k -> ( P pGrp ( G ↾s y ) <-> P pGrp ( G ↾s k ) ) )
72		sseq2 3627	. . . . 5 |- ( y = k -> ( H C_ y <-> H C_ k ) )
73	71, 72	anbi12d 747	. . . 4 |- ( y = k -> ( ( P pGrp ( G ↾s y ) /\ H C_ y ) <-> ( P pGrp ( G ↾s k ) /\ H C_ k ) ) )
74	73	rexrab 3370	. . 3 |- ( E. k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ( F ` k ) = sup ( ran F , RR , < ) <-> E. k e. (SubGrp ` G ) ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) )
75	69, 74	sylib 208	. 2 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> E. k e. (SubGrp ` G ) ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) )
76		simpl3 1066	. . . . . . 7 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> P pGrp S )
77		pgpprm 18008	. . . . . . 7 |- ( P pGrp S -> P e. Prime )
78	76, 77	syl 17	. . . . . 6 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> P e. Prime )
79		simprl 794	. . . . . 6 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> k e. (SubGrp ` G ) )
80		zssre 11384	. . . . . . . . . . . . . . . 16 |- ZZ C_ RR
81	14, 80	syl6ss 3615	. . . . . . . . . . . . . . 15 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ran F C_ RR )
82	81	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ran F C_ RR )
83	31	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ran F =/= (/) )
84	64	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> E. z e. RR A. w e. ran F w <_ z )
85		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> m e. (SubGrp ` G ) )
86		simprrr 805	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> P pGrp ( G ↾s m ) )
87		simprrl 804	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> ( P pGrp ( G ↾s k ) /\ H C_ k ) )
88	87	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( P pGrp ( G ↾s k ) /\ H C_ k ) )
89	88	simprd 479	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> H C_ k )
90		simprrl 804	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> k C_ m )
91	89, 90	sstrd 3613	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> H C_ m )
92	86, 91	jca 554	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( P pGrp ( G ↾s m ) /\ H C_ m ) )
93	85, 92, 43	sylanbrc 698	. . . . . . . . . . . . . . . 16 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } )
94	93, 37	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( F ` m ) = ( # ` m ) )
95		fnfvelrn 6356	. . . . . . . . . . . . . . . 16 |- ( ( F Fn { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } /\ m e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } ) -> ( F ` m ) e. ran F )
96	16, 93, 95	sylancr 695	. . . . . . . . . . . . . . 15 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( F ` m ) e. ran F )
97	94, 96	eqeltrrd 2702	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( # ` m ) e. ran F )
98		suprub 10984	. . . . . . . . . . . . . 14 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. z e. RR A. w e. ran F w <_ z ) /\ ( # ` m ) e. ran F ) -> ( # ` m ) <_ sup ( ran F , RR , < ) )
99	82, 83, 84, 97, 98	syl31anc 1329	. . . . . . . . . . . . 13 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( # ` m ) <_ sup ( ran F , RR , < ) )
100		simprrr 805	. . . . . . . . . . . . . . 15 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> ( F ` k ) = sup ( ran F , RR , < ) )
101	100	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( F ` k ) = sup ( ran F , RR , < ) )
102	79	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> k e. (SubGrp ` G ) )
103	73	elrab 3363	. . . . . . . . . . . . . . . 16 |- ( k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } <-> ( k e. (SubGrp ` G ) /\ ( P pGrp ( G ↾s k ) /\ H C_ k ) ) )
104	102, 88, 103	sylanbrc 698	. . . . . . . . . . . . . . 15 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } )
105		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( x = k -> ( # ` x ) = ( # ` k ) )
106		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( # ` k ) e. _V
107	105, 11, 106	fvmpt 6282	. . . . . . . . . . . . . . 15 |- ( k e. { y e. (SubGrp ` G ) | ( P pGrp ( G ↾s y ) /\ H C_ y ) } -> ( F ` k ) = ( # ` k ) )
108	104, 107	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( F ` k ) = ( # ` k ) )
109	101, 108	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> sup ( ran F , RR , < ) = ( # ` k ) )
110	99, 109	breqtrd 4679	. . . . . . . . . . . 12 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( # ` m ) <_ ( # ` k ) )
111		simpll2 1101	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> X e. Fin )
112	45	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> m C_ X )
113	111, 112, 49	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> m e. Fin )
114		ssfi 8180	. . . . . . . . . . . . . 14 |- ( ( m e. Fin /\ k C_ m ) -> k e. Fin )
115	113, 90, 114	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> k e. Fin )
116		hashcl 13147	. . . . . . . . . . . . . 14 |- ( m e. Fin -> ( # ` m ) e. NN0 )
117		hashcl 13147	. . . . . . . . . . . . . 14 |- ( k e. Fin -> ( # ` k ) e. NN0 )
118		nn0re 11301	. . . . . . . . . . . . . . 15 |- ( ( # ` m ) e. NN0 -> ( # ` m ) e. RR )
119		nn0re 11301	. . . . . . . . . . . . . . 15 |- ( ( # ` k ) e. NN0 -> ( # ` k ) e. RR )
120		lenlt 10116	. . . . . . . . . . . . . . 15 |- ( ( ( # ` m ) e. RR /\ ( # ` k ) e. RR ) -> ( ( # ` m ) <_ ( # ` k ) <-> -. ( # ` k ) < ( # ` m ) ) )
121	118, 119, 120	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( ( # ` m ) e. NN0 /\ ( # ` k ) e. NN0 ) -> ( ( # ` m ) <_ ( # ` k ) <-> -. ( # ` k ) < ( # ` m ) ) )
122	116, 117, 121	syl2an 494	. . . . . . . . . . . . 13 |- ( ( m e. Fin /\ k e. Fin ) -> ( ( # ` m ) <_ ( # ` k ) <-> -. ( # ` k ) < ( # ` m ) ) )
123	113, 115, 122	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( ( # ` m ) <_ ( # ` k ) <-> -. ( # ` k ) < ( # ` m ) ) )
124	110, 123	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> -. ( # ` k ) < ( # ` m ) )
125		php3 8146	. . . . . . . . . . . . . 14 |- ( ( m e. Fin /\ k C. m ) -> k ~< m )
126	125	ex 450	. . . . . . . . . . . . 13 |- ( m e. Fin -> ( k C. m -> k ~< m ) )
127	113, 126	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( k C. m -> k ~< m ) )
128		hashsdom 13170	. . . . . . . . . . . . 13 |- ( ( k e. Fin /\ m e. Fin ) -> ( ( # ` k ) < ( # ` m ) <-> k ~< m ) )
129	115, 113, 128	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( ( # ` k ) < ( # ` m ) <-> k ~< m ) )
130	127, 129	sylibrd 249	. . . . . . . . . . 11 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( k C. m -> ( # ` k ) < ( # ` m ) ) )
131	124, 130	mtod 189	. . . . . . . . . 10 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> -. k C. m )
132		sspss 3706	. . . . . . . . . . . 12 |- ( k C_ m <-> ( k C. m \/ k = m ) )
133	90, 132	sylib 208	. . . . . . . . . . 11 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( k C. m \/ k = m ) )
134	133	ord 392	. . . . . . . . . 10 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> ( -. k C. m -> k = m ) )
135	131, 134	mpd 15	. . . . . . . . 9 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ ( m e. (SubGrp ` G ) /\ ( k C_ m /\ P pGrp ( G ↾s m ) ) ) ) -> k = m )
136	135	expr 643	. . . . . . . 8 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ m e. (SubGrp ` G ) ) -> ( ( k C_ m /\ P pGrp ( G ↾s m ) ) -> k = m ) )
137	87	simpld 475	. . . . . . . . . 10 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> P pGrp ( G ↾s k ) )
138	137	adantr 481	. . . . . . . . 9 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ m e. (SubGrp ` G ) ) -> P pGrp ( G ↾s k ) )
139		oveq2 6658	. . . . . . . . . . 11 |- ( k = m -> ( G ↾s k ) = ( G ↾s m ) )
140	139	breq2d 4665	. . . . . . . . . 10 |- ( k = m -> ( P pGrp ( G ↾s k ) <-> P pGrp ( G ↾s m ) ) )
141		eqimss 3657	. . . . . . . . . . 11 |- ( k = m -> k C_ m )
142	141	biantrurd 529	. . . . . . . . . 10 |- ( k = m -> ( P pGrp ( G ↾s m ) <-> ( k C_ m /\ P pGrp ( G ↾s m ) ) ) )
143	140, 142	bitrd 268	. . . . . . . . 9 |- ( k = m -> ( P pGrp ( G ↾s k ) <-> ( k C_ m /\ P pGrp ( G ↾s m ) ) ) )
144	138, 143	syl5ibcom 235	. . . . . . . 8 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ m e. (SubGrp ` G ) ) -> ( k = m -> ( k C_ m /\ P pGrp ( G ↾s m ) ) ) )
145	136, 144	impbid 202	. . . . . . 7 |- ( ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) /\ m e. (SubGrp ` G ) ) -> ( ( k C_ m /\ P pGrp ( G ↾s m ) ) <-> k = m ) )
146	145	ralrimiva 2966	. . . . . 6 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> A. m e. (SubGrp ` G ) ( ( k C_ m /\ P pGrp ( G ↾s m ) ) <-> k = m ) )
147		isslw 18023	. . . . . 6 |- ( k e. ( P pSyl G ) <-> ( P e. Prime /\ k e. (SubGrp ` G ) /\ A. m e. (SubGrp ` G ) ( ( k C_ m /\ P pGrp ( G ↾s m ) ) <-> k = m ) ) )
148	78, 79, 146, 147	syl3anbrc 1246	. . . . 5 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> k e. ( P pSyl G ) )
149	87	simprd 479	. . . . 5 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> H C_ k )
150	148, 149	jca 554	. . . 4 |- ( ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) /\ ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) ) -> ( k e. ( P pSyl G ) /\ H C_ k ) )
151	150	ex 450	. . 3 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ( ( k e. (SubGrp ` G ) /\ ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) ) -> ( k e. ( P pSyl G ) /\ H C_ k ) ) )
152	151	reximdv2 3014	. 2 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> ( E. k e. (SubGrp ` G ) ( ( P pGrp ( G ↾s k ) /\ H C_ k ) /\ ( F ` k ) = sup ( ran F , RR , < ) ) -> E. k e. ( P pSyl G ) H C_ k ) )
153	75, 152	mpd 15	1 |- ( ( H e. (SubGrp ` G ) /\ X e. Fin /\ P pGrp S ) -> E. k e. ( P pSyl G ) H C_ k )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   C. wpss 3575   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   supcsup 8346   RRcr 9935   < clt 10074   <_ cle 10075   NN0cn0 11292   ZZcz 11377   #chash 13117   Primecprime 15385   Basecbs 15857   ↾_s cress 15858  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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-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-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-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-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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-subg 17591  df-pgp 17950  df-slw 17951

This theorem is referenced by:  slwn0  18030