Theorem pgpfac1lem4 18477

Description: Lemma for pgpfac1 18479. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypotheses

Ref	Expression
pgpfac1.k	|- K = (mrCls ` (SubGrp ` G ) )
pgpfac1.s	|- S = ( K ` { A } )
pgpfac1.b	|- B = ( Base ` G )
pgpfac1.o	|- O = ( od ` G )
pgpfac1.e	|- E = (gEx ` G )
pgpfac1.z	|- .0. = ( 0g ` G )
pgpfac1.l	|- .(+) = ( LSSum ` G )
pgpfac1.p	|- ( ph -> P pGrp G )
pgpfac1.g	|- ( ph -> G e. Abel )
pgpfac1.n	|- ( ph -> B e. Fin )
pgpfac1.oe	|- ( ph -> ( O ` A ) = E )
pgpfac1.u	|- ( ph -> U e. (SubGrp ` G ) )
pgpfac1.au	|- ( ph -> A e. U )
pgpfac1.w	|- ( ph -> W e. (SubGrp ` G ) )
pgpfac1.i	|- ( ph -> ( S i^i W ) = { .0. } )
pgpfac1.ss	|- ( ph -> ( S .(+) W ) C_ U )
pgpfac1.2	|- ( ph -> A. w e. (SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) )
pgpfac1.c	|- ( ph -> C e. ( U \ ( S .(+) W ) ) )
pgpfac1.mg	|- .x. = (.g ` G )

Assertion

Ref	Expression
pgpfac1lem4	|- ( ph -> E. t e. (SubGrp ` G ) ( ( S i^i t ) = { .0. } /\ ( S .(+) t ) = U ) )

Distinct variable groups:   t, .0.   w, t, A   t, .(+) , w   t, P, w   t, B   t, G, w   t, U, w   t, C, w   t, S, w   t, W, w   ph, t, w   t, .x. , w   t, K, w

Allowed substitution hints:   B( w)   E( w, t)   O( w, t)   .0. ( w)

Proof of Theorem pgpfac1lem4

Dummy variables k s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac1.k	. . . . . . . 8 |- K = (mrCls ` (SubGrp ` G ) )
2		pgpfac1.s	. . . . . . . 8 |- S = ( K ` { A } )
3		pgpfac1.b	. . . . . . . 8 |- B = ( Base ` G )
4		pgpfac1.o	. . . . . . . 8 |- O = ( od ` G )
5		pgpfac1.e	. . . . . . . 8 |- E = (gEx ` G )
6		pgpfac1.z	. . . . . . . 8 |- .0. = ( 0g ` G )
7		pgpfac1.l	. . . . . . . 8 |- .(+) = ( LSSum ` G )
8		pgpfac1.p	. . . . . . . 8 |- ( ph -> P pGrp G )
9		pgpfac1.g	. . . . . . . 8 |- ( ph -> G e. Abel )
10		pgpfac1.n	. . . . . . . 8 |- ( ph -> B e. Fin )
11		pgpfac1.oe	. . . . . . . 8 |- ( ph -> ( O ` A ) = E )
12		pgpfac1.u	. . . . . . . 8 |- ( ph -> U e. (SubGrp ` G ) )
13		pgpfac1.au	. . . . . . . 8 |- ( ph -> A e. U )
14		pgpfac1.w	. . . . . . . 8 |- ( ph -> W e. (SubGrp ` G ) )
15		pgpfac1.i	. . . . . . . 8 |- ( ph -> ( S i^i W ) = { .0. } )
16		pgpfac1.ss	. . . . . . . 8 |- ( ph -> ( S .(+) W ) C_ U )
17		pgpfac1.2	. . . . . . . 8 |- ( ph -> A. w e. (SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) )
18		pgpfac1.c	. . . . . . . 8 |- ( ph -> C e. ( U \ ( S .(+) W ) ) )
19		pgpfac1.mg	. . . . . . . 8 |- .x. = (.g ` G )
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	pgpfac1lem2 18474	. . . . . . 7 |- ( ph -> ( P .x. C ) e. ( S .(+) W ) )
21		ablgrp 18198	. . . . . . . . . . . 12 |- ( G e. Abel -> G e. Grp )
22	9, 21	syl 17	. . . . . . . . . . 11 |- ( ph -> G e. Grp )
23	3	subgacs 17629	. . . . . . . . . . 11 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` B ) )
24		acsmre 16313	. . . . . . . . . . 11 |- ( (SubGrp ` G ) e. (ACS ` B ) -> (SubGrp ` G ) e. (Moore ` B ) )
25	22, 23, 24	3syl 18	. . . . . . . . . 10 |- ( ph -> (SubGrp ` G ) e. (Moore ` B ) )
26	3	subgss 17595	. . . . . . . . . . . 12 |- ( U e. (SubGrp ` G ) -> U C_ B )
27	12, 26	syl 17	. . . . . . . . . . 11 |- ( ph -> U C_ B )
28	27, 13	sseldd 3604	. . . . . . . . . 10 |- ( ph -> A e. B )
29	1	mrcsncl 16272	. . . . . . . . . 10 |- ( ( (SubGrp ` G ) e. (Moore ` B ) /\ A e. B ) -> ( K ` { A } ) e. (SubGrp ` G ) )
30	25, 28, 29	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( K ` { A } ) e. (SubGrp ` G ) )
31	2, 30	syl5eqel 2705	. . . . . . . 8 |- ( ph -> S e. (SubGrp ` G ) )
32	7	lsmcom 18261	. . . . . . . 8 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) /\ W e. (SubGrp ` G ) ) -> ( S .(+) W ) = ( W .(+) S ) )
33	9, 31, 14, 32	syl3anc 1326	. . . . . . 7 |- ( ph -> ( S .(+) W ) = ( W .(+) S ) )
34	20, 33	eleqtrd 2703	. . . . . 6 |- ( ph -> ( P .x. C ) e. ( W .(+) S ) )
35		eqid 2622	. . . . . . 7 |- ( -g ` G ) = ( -g ` G )
36	35, 7, 14, 31	lsmelvalm 18066	. . . . . 6 |- ( ph -> ( ( P .x. C ) e. ( W .(+) S ) <-> E. w e. W E. s e. S ( P .x. C ) = ( w ( -g ` G ) s ) ) )
37	34, 36	mpbid 222	. . . . 5 |- ( ph -> E. w e. W E. s e. S ( P .x. C ) = ( w ( -g ` G ) s ) )
38		eqid 2622	. . . . . . . . . . 11 |- ( k e. ZZ |-> ( k .x. A ) ) = ( k e. ZZ |-> ( k .x. A ) )
39	3, 19, 38, 1	cycsubg2 17631	. . . . . . . . . 10 |- ( ( G e. Grp /\ A e. B ) -> ( K ` { A } ) = ran ( k e. ZZ |-> ( k .x. A ) ) )
40	22, 28, 39	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( K ` { A } ) = ran ( k e. ZZ |-> ( k .x. A ) ) )
41	2, 40	syl5eq 2668	. . . . . . . 8 |- ( ph -> S = ran ( k e. ZZ |-> ( k .x. A ) ) )
42	41	rexeqdv 3145	. . . . . . 7 |- ( ph -> ( E. s e. S ( P .x. C ) = ( w ( -g ` G ) s ) <-> E. s e. ran ( k e. ZZ |-> ( k .x. A ) ) ( P .x. C ) = ( w ( -g ` G ) s ) ) )
43		ovex 6678	. . . . . . . . 9 |- ( k .x. A ) e. _V
44	43	rgenw 2924	. . . . . . . 8 |- A. k e. ZZ ( k .x. A ) e. _V
45		oveq2 6658	. . . . . . . . . 10 |- ( s = ( k .x. A ) -> ( w ( -g ` G ) s ) = ( w ( -g ` G ) ( k .x. A ) ) )
46	45	eqeq2d 2632	. . . . . . . . 9 |- ( s = ( k .x. A ) -> ( ( P .x. C ) = ( w ( -g ` G ) s ) <-> ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) ) )
47	38, 46	rexrnmpt 6369	. . . . . . . 8 |- ( A. k e. ZZ ( k .x. A ) e. _V -> ( E. s e. ran ( k e. ZZ |-> ( k .x. A ) ) ( P .x. C ) = ( w ( -g ` G ) s ) <-> E. k e. ZZ ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) ) )
48	44, 47	ax-mp 5	. . . . . . 7 |- ( E. s e. ran ( k e. ZZ |-> ( k .x. A ) ) ( P .x. C ) = ( w ( -g ` G ) s ) <-> E. k e. ZZ ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) )
49	42, 48	syl6bb 276	. . . . . 6 |- ( ph -> ( E. s e. S ( P .x. C ) = ( w ( -g ` G ) s ) <-> E. k e. ZZ ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) ) )
50	49	rexbidv 3052	. . . . 5 |- ( ph -> ( E. w e. W E. s e. S ( P .x. C ) = ( w ( -g ` G ) s ) <-> E. w e. W E. k e. ZZ ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) ) )
51	37, 50	mpbid 222	. . . 4 |- ( ph -> E. w e. W E. k e. ZZ ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) )
52		rexcom 3099	. . . 4 |- ( E. w e. W E. k e. ZZ ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) <-> E. k e. ZZ E. w e. W ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) )
53	51, 52	sylib 208	. . 3 |- ( ph -> E. k e. ZZ E. w e. W ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) )
54	22	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> G e. Grp )
55	3	subgss 17595	. . . . . . . . . . 11 |- ( W e. (SubGrp ` G ) -> W C_ B )
56	14, 55	syl 17	. . . . . . . . . 10 |- ( ph -> W C_ B )
57	56	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. ZZ ) -> W C_ B )
58	57	sselda 3603	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> w e. B )
59		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> k e. ZZ )
60	28	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> A e. B )
61	3, 19	mulgcl 17559	. . . . . . . . 9 |- ( ( G e. Grp /\ k e. ZZ /\ A e. B ) -> ( k .x. A ) e. B )
62	54, 59, 60, 61	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> ( k .x. A ) e. B )
63		pgpprm 18008	. . . . . . . . . . 11 |- ( P pGrp G -> P e. Prime )
64		prmz 15389	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ZZ )
65	8, 63, 64	3syl 18	. . . . . . . . . 10 |- ( ph -> P e. ZZ )
66	18	eldifad 3586	. . . . . . . . . . 11 |- ( ph -> C e. U )
67	27, 66	sseldd 3604	. . . . . . . . . 10 |- ( ph -> C e. B )
68	3, 19	mulgcl 17559	. . . . . . . . . 10 |- ( ( G e. Grp /\ P e. ZZ /\ C e. B ) -> ( P .x. C ) e. B )
69	22, 65, 67, 68	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( P .x. C ) e. B )
70	69	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> ( P .x. C ) e. B )
71		eqid 2622	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
72	3, 71, 35	grpsubadd 17503	. . . . . . . 8 |- ( ( G e. Grp /\ ( w e. B /\ ( k .x. A ) e. B /\ ( P .x. C ) e. B ) ) -> ( ( w ( -g ` G ) ( k .x. A ) ) = ( P .x. C ) <-> ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) = w ) )
73	54, 58, 62, 70, 72	syl13anc 1328	. . . . . . 7 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> ( ( w ( -g ` G ) ( k .x. A ) ) = ( P .x. C ) <-> ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) = w ) )
74		eqcom 2629	. . . . . . 7 |- ( ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) <-> ( w ( -g ` G ) ( k .x. A ) ) = ( P .x. C ) )
75		eqcom 2629	. . . . . . 7 |- ( w = ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) <-> ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) = w )
76	73, 74, 75	3bitr4g 303	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ w e. W ) -> ( ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) <-> w = ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) ) )
77	76	rexbidva 3049	. . . . 5 |- ( ( ph /\ k e. ZZ ) -> ( E. w e. W ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) <-> E. w e. W w = ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) ) )
78		risset 3062	. . . . 5 |- ( ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W <-> E. w e. W w = ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) )
79	77, 78	syl6bbr 278	. . . 4 |- ( ( ph /\ k e. ZZ ) -> ( E. w e. W ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) <-> ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) )
80	79	rexbidva 3049	. . 3 |- ( ph -> ( E. k e. ZZ E. w e. W ( P .x. C ) = ( w ( -g ` G ) ( k .x. A ) ) <-> E. k e. ZZ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) )
81	53, 80	mpbid 222	. 2 |- ( ph -> E. k e. ZZ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W )
82	8	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> P pGrp G )
83	9	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> G e. Abel )
84	10	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> B e. Fin )
85	11	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> ( O ` A ) = E )
86	12	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> U e. (SubGrp ` G ) )
87	13	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> A e. U )
88	14	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> W e. (SubGrp ` G ) )
89	15	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> ( S i^i W ) = { .0. } )
90	16	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> ( S .(+) W ) C_ U )
91	17	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> A. w e. (SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) )
92	18	adantr 481	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> C e. ( U \ ( S .(+) W ) ) )
93		simprl 794	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> k e. ZZ )
94		simprr 796	. . 3 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W )
95		eqid 2622	. . 3 |- ( C ( +g ` G ) ( ( k / P ) .x. A ) ) = ( C ( +g ` G ) ( ( k / P ) .x. A ) )
96	1, 2, 3, 4, 5, 6, 7, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 19, 93, 94, 95	pgpfac1lem3 18476	. 2 |- ( ( ph /\ ( k e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( k .x. A ) ) e. W ) ) -> E. t e. (SubGrp ` G ) ( ( S i^i t ) = { .0. } /\ ( S .(+) t ) = U ) )
97	81, 96	rexlimddv 3035	1 |- ( ph -> E. t e. (SubGrp ` G ) ( ( S i^i t ) = { .0. } /\ ( S .(+) t ) = U ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   Fincfn 7955   / cdiv 10684   ZZcz 11377   Primecprime 15385   Basecbs 15857   +g cplusg 15941   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Grpcgrp 17422   -gcsg 17424  ._gcmg 17540  SubGrpcsubg 17588   odcod 17944  gExcgex 17945   pGrp cpgp 17946   LSSumclsm 18049   Abelcabl 18194

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-iin 4523  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-mre 16246  df-mrc 16247  df-acs 16249  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-ga 17723  df-cntz 17750  df-od 17948  df-gex 17949  df-pgp 17950  df-lsm 18051  df-cmn 18195  df-abl 18196

This theorem is referenced by:  pgpfac1lem5  18478