Theorem pgpfac1lem3a 18475

Description: Lemma for pgpfac1 18479. (Contributed by Mario Carneiro, 4-Jun-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 )
pgpfac1.m	|- ( ph -> M e. ZZ )
pgpfac1.mw	|- ( ph -> ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) e. W )

Assertion

Ref	Expression
pgpfac1lem3a	|- ( ph -> ( P || E /\ P || M ) )

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

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

Proof of Theorem pgpfac1lem3a

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pgpfac1.c	. . . 4 |- ( ph -> C e. ( U \ ( S .(+) W ) ) )
2	1	eldifbd 3587	. . 3 |- ( ph -> -. C e. ( S .(+) W ) )
3		pgpfac1.p	. . . . . . . 8 |- ( ph -> P pGrp G )
4		pgpprm 18008	. . . . . . . 8 |- ( P pGrp G -> P e. Prime )
5	3, 4	syl 17	. . . . . . 7 |- ( ph -> P e. Prime )
6		pgpfac1.g	. . . . . . . . 9 |- ( ph -> G e. Abel )
7		ablgrp 18198	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8	6, 7	syl 17	. . . . . . . 8 |- ( ph -> G e. Grp )
9		pgpfac1.n	. . . . . . . 8 |- ( ph -> B e. Fin )
10		pgpfac1.b	. . . . . . . . 9 |- B = ( Base ` G )
11		pgpfac1.e	. . . . . . . . 9 |- E = (gEx ` G )
12	10, 11	gexcl2 18004	. . . . . . . 8 |- ( ( G e. Grp /\ B e. Fin ) -> E e. NN )
13	8, 9, 12	syl2anc 693	. . . . . . 7 |- ( ph -> E e. NN )
14		pceq0 15575	. . . . . . 7 |- ( ( P e. Prime /\ E e. NN ) -> ( ( P pCnt E ) = 0 <-> -. P || E ) )
15	5, 13, 14	syl2anc 693	. . . . . 6 |- ( ph -> ( ( P pCnt E ) = 0 <-> -. P || E ) )
16		oveq2 6658	. . . . . 6 |- ( ( P pCnt E ) = 0 -> ( P ^ ( P pCnt E ) ) = ( P ^ 0 ) )
17	15, 16	syl6bir 244	. . . . 5 |- ( ph -> ( -. P || E -> ( P ^ ( P pCnt E ) ) = ( P ^ 0 ) ) )
18	10	grpbn0 17451	. . . . . . . . . . . . 13 |- ( G e. Grp -> B =/= (/) )
19	8, 18	syl 17	. . . . . . . . . . . 12 |- ( ph -> B =/= (/) )
20		hashnncl 13157	. . . . . . . . . . . . 13 |- ( B e. Fin -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
21	9, 20	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` B ) e. NN <-> B =/= (/) ) )
22	19, 21	mpbird 247	. . . . . . . . . . 11 |- ( ph -> ( # ` B ) e. NN )
23	5, 22	pccld 15555	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( # ` B ) ) e. NN0 )
24	10, 11	gexdvds3 18005	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ B e. Fin ) -> E || ( # ` B ) )
25	8, 9, 24	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> E || ( # ` B ) )
26	10	pgphash 18022	. . . . . . . . . . . 12 |- ( ( P pGrp G /\ B e. Fin ) -> ( # ` B ) = ( P ^ ( P pCnt ( # ` B ) ) ) )
27	3, 9, 26	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( # ` B ) = ( P ^ ( P pCnt ( # ` B ) ) ) )
28	25, 27	breqtrd 4679	. . . . . . . . . 10 |- ( ph -> E || ( P ^ ( P pCnt ( # ` B ) ) ) )
29		oveq2 6658	. . . . . . . . . . . 12 |- ( k = ( P pCnt ( # ` B ) ) -> ( P ^ k ) = ( P ^ ( P pCnt ( # ` B ) ) ) )
30	29	breq2d 4665	. . . . . . . . . . 11 |- ( k = ( P pCnt ( # ` B ) ) -> ( E || ( P ^ k ) <-> E || ( P ^ ( P pCnt ( # ` B ) ) ) ) )
31	30	rspcev 3309	. . . . . . . . . 10 |- ( ( ( P pCnt ( # ` B ) ) e. NN0 /\ E || ( P ^ ( P pCnt ( # ` B ) ) ) ) -> E. k e. NN0 E || ( P ^ k ) )
32	23, 28, 31	syl2anc 693	. . . . . . . . 9 |- ( ph -> E. k e. NN0 E || ( P ^ k ) )
33		pcprmpw2 15586	. . . . . . . . . 10 |- ( ( P e. Prime /\ E e. NN ) -> ( E. k e. NN0 E || ( P ^ k ) <-> E = ( P ^ ( P pCnt E ) ) ) )
34	5, 13, 33	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( E. k e. NN0 E || ( P ^ k ) <-> E = ( P ^ ( P pCnt E ) ) ) )
35	32, 34	mpbid 222	. . . . . . . 8 |- ( ph -> E = ( P ^ ( P pCnt E ) ) )
36	35	eqcomd 2628	. . . . . . 7 |- ( ph -> ( P ^ ( P pCnt E ) ) = E )
37		prmnn 15388	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
38	5, 37	syl 17	. . . . . . . . 9 |- ( ph -> P e. NN )
39	38	nncnd 11036	. . . . . . . 8 |- ( ph -> P e. CC )
40	39	exp0d 13002	. . . . . . 7 |- ( ph -> ( P ^ 0 ) = 1 )
41	36, 40	eqeq12d 2637	. . . . . 6 |- ( ph -> ( ( P ^ ( P pCnt E ) ) = ( P ^ 0 ) <-> E = 1 ) )
42		grpmnd 17429	. . . . . . . 8 |- ( G e. Grp -> G e. Mnd )
43	8, 42	syl 17	. . . . . . 7 |- ( ph -> G e. Mnd )
44	10, 11	gex1 18006	. . . . . . 7 |- ( G e. Mnd -> ( E = 1 <-> B ~~ 1o ) )
45	43, 44	syl 17	. . . . . 6 |- ( ph -> ( E = 1 <-> B ~~ 1o ) )
46	41, 45	bitrd 268	. . . . 5 |- ( ph -> ( ( P ^ ( P pCnt E ) ) = ( P ^ 0 ) <-> B ~~ 1o ) )
47	17, 46	sylibd 229	. . . 4 |- ( ph -> ( -. P || E -> B ~~ 1o ) )
48		pgpfac1.s	. . . . . . . . . . 11 |- S = ( K ` { A } )
49	10	subgacs 17629	. . . . . . . . . . . . . 14 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` B ) )
50	8, 49	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (SubGrp ` G ) e. (ACS ` B ) )
51	50	acsmred 16317	. . . . . . . . . . . 12 |- ( ph -> (SubGrp ` G ) e. (Moore ` B ) )
52		pgpfac1.u	. . . . . . . . . . . . . 14 |- ( ph -> U e. (SubGrp ` G ) )
53	10	subgss 17595	. . . . . . . . . . . . . 14 |- ( U e. (SubGrp ` G ) -> U C_ B )
54	52, 53	syl 17	. . . . . . . . . . . . 13 |- ( ph -> U C_ B )
55		pgpfac1.au	. . . . . . . . . . . . 13 |- ( ph -> A e. U )
56	54, 55	sseldd 3604	. . . . . . . . . . . 12 |- ( ph -> A e. B )
57		pgpfac1.k	. . . . . . . . . . . . 13 |- K = (mrCls ` (SubGrp ` G ) )
58	57	mrcsncl 16272	. . . . . . . . . . . 12 |- ( ( (SubGrp ` G ) e. (Moore ` B ) /\ A e. B ) -> ( K ` { A } ) e. (SubGrp ` G ) )
59	51, 56, 58	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( K ` { A } ) e. (SubGrp ` G ) )
60	48, 59	syl5eqel 2705	. . . . . . . . . 10 |- ( ph -> S e. (SubGrp ` G ) )
61		pgpfac1.w	. . . . . . . . . 10 |- ( ph -> W e. (SubGrp ` G ) )
62		pgpfac1.l	. . . . . . . . . . 11 |- .(+) = ( LSSum ` G )
63	62	lsmsubg2 18262	. . . . . . . . . 10 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) /\ W e. (SubGrp ` G ) ) -> ( S .(+) W ) e. (SubGrp ` G ) )
64	6, 60, 61, 63	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( S .(+) W ) e. (SubGrp ` G ) )
65		pgpfac1.z	. . . . . . . . . 10 |- .0. = ( 0g ` G )
66	65	subg0cl 17602	. . . . . . . . 9 |- ( ( S .(+) W ) e. (SubGrp ` G ) -> .0. e. ( S .(+) W ) )
67	64, 66	syl 17	. . . . . . . 8 |- ( ph -> .0. e. ( S .(+) W ) )
68	67	snssd 4340	. . . . . . 7 |- ( ph -> { .0. } C_ ( S .(+) W ) )
69	68	adantr 481	. . . . . 6 |- ( ( ph /\ B ~~ 1o ) -> { .0. } C_ ( S .(+) W ) )
70	1	eldifad 3586	. . . . . . . . 9 |- ( ph -> C e. U )
71	54, 70	sseldd 3604	. . . . . . . 8 |- ( ph -> C e. B )
72	71	adantr 481	. . . . . . 7 |- ( ( ph /\ B ~~ 1o ) -> C e. B )
73	10, 65	grpidcl 17450	. . . . . . . . 9 |- ( G e. Grp -> .0. e. B )
74	8, 73	syl 17	. . . . . . . 8 |- ( ph -> .0. e. B )
75		en1eqsn 8190	. . . . . . . 8 |- ( ( .0. e. B /\ B ~~ 1o ) -> B = { .0. } )
76	74, 75	sylan 488	. . . . . . 7 |- ( ( ph /\ B ~~ 1o ) -> B = { .0. } )
77	72, 76	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ B ~~ 1o ) -> C e. { .0. } )
78	69, 77	sseldd 3604	. . . . 5 |- ( ( ph /\ B ~~ 1o ) -> C e. ( S .(+) W ) )
79	78	ex 450	. . . 4 |- ( ph -> ( B ~~ 1o -> C e. ( S .(+) W ) ) )
80	47, 79	syld 47	. . 3 |- ( ph -> ( -. P || E -> C e. ( S .(+) W ) ) )
81	2, 80	mt3d 140	. 2 |- ( ph -> P || E )
82		pgpfac1.oe	. . . . 5 |- ( ph -> ( O ` A ) = E )
83	13	nncnd 11036	. . . . . 6 |- ( ph -> E e. CC )
84	38	nnne0d 11065	. . . . . 6 |- ( ph -> P =/= 0 )
85	83, 39, 84	divcan1d 10802	. . . . 5 |- ( ph -> ( ( E / P ) x. P ) = E )
86	82, 85	eqtr4d 2659	. . . 4 |- ( ph -> ( O ` A ) = ( ( E / P ) x. P ) )
87		nndivdvds 14989	. . . . . . . . . . . . 13 |- ( ( E e. NN /\ P e. NN ) -> ( P || E <-> ( E / P ) e. NN ) )
88	13, 38, 87	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( P || E <-> ( E / P ) e. NN ) )
89	81, 88	mpbid 222	. . . . . . . . . . 11 |- ( ph -> ( E / P ) e. NN )
90	89	nnzd 11481	. . . . . . . . . 10 |- ( ph -> ( E / P ) e. ZZ )
91		pgpfac1.m	. . . . . . . . . 10 |- ( ph -> M e. ZZ )
92	90, 91	zmulcld 11488	. . . . . . . . 9 |- ( ph -> ( ( E / P ) x. M ) e. ZZ )
93	56	snssd 4340	. . . . . . . . . . . 12 |- ( ph -> { A } C_ B )
94	51, 57, 93	mrcssidd 16285	. . . . . . . . . . 11 |- ( ph -> { A } C_ ( K ` { A } ) )
95	94, 48	syl6sseqr 3652	. . . . . . . . . 10 |- ( ph -> { A } C_ S )
96		snssg 4327	. . . . . . . . . . 11 |- ( A e. U -> ( A e. S <-> { A } C_ S ) )
97	55, 96	syl 17	. . . . . . . . . 10 |- ( ph -> ( A e. S <-> { A } C_ S ) )
98	95, 97	mpbird 247	. . . . . . . . 9 |- ( ph -> A e. S )
99		pgpfac1.mg	. . . . . . . . . 10 |- .x. = (.g ` G )
100	99	subgmulgcl 17607	. . . . . . . . 9 |- ( ( S e. (SubGrp ` G ) /\ ( ( E / P ) x. M ) e. ZZ /\ A e. S ) -> ( ( ( E / P ) x. M ) .x. A ) e. S )
101	60, 92, 98, 100	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) e. S )
102		prmz 15389	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. ZZ )
103	5, 102	syl 17	. . . . . . . . . . . 12 |- ( ph -> P e. ZZ )
104	10, 99	mulgcl 17559	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ P e. ZZ /\ C e. B ) -> ( P .x. C ) e. B )
105	8, 103, 71, 104	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( P .x. C ) e. B )
106	10, 99	mulgcl 17559	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ M e. ZZ /\ A e. B ) -> ( M .x. A ) e. B )
107	8, 91, 56, 106	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( M .x. A ) e. B )
108		eqid 2622	. . . . . . . . . . . 12 |- ( +g ` G ) = ( +g ` G )
109	10, 99, 108	mulgdi 18232	. . . . . . . . . . 11 |- ( ( G e. Abel /\ ( ( E / P ) e. ZZ /\ ( P .x. C ) e. B /\ ( M .x. A ) e. B ) ) -> ( ( E / P ) .x. ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) ) = ( ( ( E / P ) .x. ( P .x. C ) ) ( +g ` G ) ( ( E / P ) .x. ( M .x. A ) ) ) )
110	6, 90, 105, 107, 109	syl13anc 1328	. . . . . . . . . 10 |- ( ph -> ( ( E / P ) .x. ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) ) = ( ( ( E / P ) .x. ( P .x. C ) ) ( +g ` G ) ( ( E / P ) .x. ( M .x. A ) ) ) )
111	85	oveq1d 6665	. . . . . . . . . . . 12 |- ( ph -> ( ( ( E / P ) x. P ) .x. C ) = ( E .x. C ) )
112	10, 99	mulgass 17579	. . . . . . . . . . . . 13 |- ( ( G e. Grp /\ ( ( E / P ) e. ZZ /\ P e. ZZ /\ C e. B ) ) -> ( ( ( E / P ) x. P ) .x. C ) = ( ( E / P ) .x. ( P .x. C ) ) )
113	8, 90, 103, 71, 112	syl13anc 1328	. . . . . . . . . . . 12 |- ( ph -> ( ( ( E / P ) x. P ) .x. C ) = ( ( E / P ) .x. ( P .x. C ) ) )
114	10, 11, 99, 65	gexid 17996	. . . . . . . . . . . . 13 |- ( C e. B -> ( E .x. C ) = .0. )
115	71, 114	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( E .x. C ) = .0. )
116	111, 113, 115	3eqtr3rd 2665	. . . . . . . . . . 11 |- ( ph -> .0. = ( ( E / P ) .x. ( P .x. C ) ) )
117	10, 99	mulgass 17579	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( ( E / P ) e. ZZ /\ M e. ZZ /\ A e. B ) ) -> ( ( ( E / P ) x. M ) .x. A ) = ( ( E / P ) .x. ( M .x. A ) ) )
118	8, 90, 91, 56, 117	syl13anc 1328	. . . . . . . . . . 11 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) = ( ( E / P ) .x. ( M .x. A ) ) )
119	116, 118	oveq12d 6668	. . . . . . . . . 10 |- ( ph -> ( .0. ( +g ` G ) ( ( ( E / P ) x. M ) .x. A ) ) = ( ( ( E / P ) .x. ( P .x. C ) ) ( +g ` G ) ( ( E / P ) .x. ( M .x. A ) ) ) )
120	10	subgss 17595	. . . . . . . . . . . . 13 |- ( S e. (SubGrp ` G ) -> S C_ B )
121	60, 120	syl 17	. . . . . . . . . . . 12 |- ( ph -> S C_ B )
122	121, 101	sseldd 3604	. . . . . . . . . . 11 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) e. B )
123	10, 108, 65	grplid 17452	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( ( ( E / P ) x. M ) .x. A ) e. B ) -> ( .0. ( +g ` G ) ( ( ( E / P ) x. M ) .x. A ) ) = ( ( ( E / P ) x. M ) .x. A ) )
124	8, 122, 123	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( .0. ( +g ` G ) ( ( ( E / P ) x. M ) .x. A ) ) = ( ( ( E / P ) x. M ) .x. A ) )
125	110, 119, 124	3eqtr2d 2662	. . . . . . . . 9 |- ( ph -> ( ( E / P ) .x. ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) ) = ( ( ( E / P ) x. M ) .x. A ) )
126		pgpfac1.mw	. . . . . . . . . 10 |- ( ph -> ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) e. W )
127	99	subgmulgcl 17607	. . . . . . . . . 10 |- ( ( W e. (SubGrp ` G ) /\ ( E / P ) e. ZZ /\ ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) e. W ) -> ( ( E / P ) .x. ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) ) e. W )
128	61, 90, 126, 127	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( E / P ) .x. ( ( P .x. C ) ( +g ` G ) ( M .x. A ) ) ) e. W )
129	125, 128	eqeltrrd 2702	. . . . . . . 8 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) e. W )
130	101, 129	elind 3798	. . . . . . 7 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) e. ( S i^i W ) )
131		pgpfac1.i	. . . . . . 7 |- ( ph -> ( S i^i W ) = { .0. } )
132	130, 131	eleqtrd 2703	. . . . . 6 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) e. { .0. } )
133		elsni 4194	. . . . . 6 |- ( ( ( ( E / P ) x. M ) .x. A ) e. { .0. } -> ( ( ( E / P ) x. M ) .x. A ) = .0. )
134	132, 133	syl 17	. . . . 5 |- ( ph -> ( ( ( E / P ) x. M ) .x. A ) = .0. )
135		pgpfac1.o	. . . . . . 7 |- O = ( od ` G )
136	10, 135, 99, 65	oddvds 17966	. . . . . 6 |- ( ( G e. Grp /\ A e. B /\ ( ( E / P ) x. M ) e. ZZ ) -> ( ( O ` A ) || ( ( E / P ) x. M ) <-> ( ( ( E / P ) x. M ) .x. A ) = .0. ) )
137	8, 56, 92, 136	syl3anc 1326	. . . . 5 |- ( ph -> ( ( O ` A ) || ( ( E / P ) x. M ) <-> ( ( ( E / P ) x. M ) .x. A ) = .0. ) )
138	134, 137	mpbird 247	. . . 4 |- ( ph -> ( O ` A ) || ( ( E / P ) x. M ) )
139	86, 138	eqbrtrrd 4677	. . 3 |- ( ph -> ( ( E / P ) x. P ) || ( ( E / P ) x. M ) )
140	89	nnne0d 11065	. . . 4 |- ( ph -> ( E / P ) =/= 0 )
141		dvdscmulr 15010	. . . 4 |- ( ( P e. ZZ /\ M e. ZZ /\ ( ( E / P ) e. ZZ /\ ( E / P ) =/= 0 ) ) -> ( ( ( E / P ) x. P ) || ( ( E / P ) x. M ) <-> P || M ) )
142	103, 91, 90, 140, 141	syl112anc 1330	. . 3 |- ( ph -> ( ( ( E / P ) x. P ) || ( ( E / P ) x. M ) <-> P || M ) )
143	139, 142	mpbid 222	. 2 |- ( ph -> P || M )
144	81, 143	jca 554	1 |- ( ph -> ( P || E /\ P || M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   Fincfn 7955   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   Basecbs 15857   +g cplusg 15941   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Mndcmnd 17294   Grpcgrp 17422  ._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:  pgpfac1lem3  18476