Theorem pgpfac1lem1 18473

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 ) )

Assertion

Ref	Expression
pgpfac1lem1	|- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U )

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

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

Proof of Theorem pgpfac1lem1

Step	Hyp	Ref	Expression
1		pgpfac1.ss	. . . 4 |- ( ph -> ( S .(+) W ) C_ U )
2	1	adantr 481	. . 3 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) C_ U )
3		pgpfac1.g	. . . . . 6 |- ( ph -> G e. Abel )
4		ablgrp 18198	. . . . . 6 |- ( G e. Abel -> G e. Grp )
5		pgpfac1.b	. . . . . . 7 |- B = ( Base ` G )
6	5	subgacs 17629	. . . . . 6 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` B ) )
7		acsmre 16313	. . . . . 6 |- ( (SubGrp ` G ) e. (ACS ` B ) -> (SubGrp ` G ) e. (Moore ` B ) )
8	3, 4, 6, 7	4syl 19	. . . . 5 |- ( ph -> (SubGrp ` G ) e. (Moore ` B ) )
9	8	adantr 481	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> (SubGrp ` G ) e. (Moore ` B ) )
10		eldifi 3732	. . . . . 6 |- ( C e. ( U \ ( S .(+) W ) ) -> C e. U )
11	10	adantl 482	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. U )
12	11	snssd 4340	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> { C } C_ U )
13		pgpfac1.u	. . . . 5 |- ( ph -> U e. (SubGrp ` G ) )
14	13	adantr 481	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> U e. (SubGrp ` G ) )
15		pgpfac1.k	. . . . 5 |- K = (mrCls ` (SubGrp ` G ) )
16	15	mrcsscl 16280	. . . 4 |- ( ( (SubGrp ` G ) e. (Moore ` B ) /\ { C } C_ U /\ U e. (SubGrp ` G ) ) -> ( K ` { C } ) C_ U )
17	9, 12, 14, 16	syl3anc 1326	. . 3 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( K ` { C } ) C_ U )
18		pgpfac1.s	. . . . . . 7 |- S = ( K ` { A } )
19	5	subgss 17595	. . . . . . . . . 10 |- ( U e. (SubGrp ` G ) -> U C_ B )
20	13, 19	syl 17	. . . . . . . . 9 |- ( ph -> U C_ B )
21		pgpfac1.au	. . . . . . . . 9 |- ( ph -> A e. U )
22	20, 21	sseldd 3604	. . . . . . . 8 |- ( ph -> A e. B )
23	15	mrcsncl 16272	. . . . . . . 8 |- ( ( (SubGrp ` G ) e. (Moore ` B ) /\ A e. B ) -> ( K ` { A } ) e. (SubGrp ` G ) )
24	8, 22, 23	syl2anc 693	. . . . . . 7 |- ( ph -> ( K ` { A } ) e. (SubGrp ` G ) )
25	18, 24	syl5eqel 2705	. . . . . 6 |- ( ph -> S e. (SubGrp ` G ) )
26		pgpfac1.w	. . . . . 6 |- ( ph -> W e. (SubGrp ` G ) )
27		pgpfac1.l	. . . . . . 7 |- .(+) = ( LSSum ` G )
28	27	lsmsubg2 18262	. . . . . 6 |- ( ( G e. Abel /\ S e. (SubGrp ` G ) /\ W e. (SubGrp ` G ) ) -> ( S .(+) W ) e. (SubGrp ` G ) )
29	3, 25, 26, 28	syl3anc 1326	. . . . 5 |- ( ph -> ( S .(+) W ) e. (SubGrp ` G ) )
30	29	adantr 481	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) e. (SubGrp ` G ) )
31	20	sselda 3603	. . . . . 6 |- ( ( ph /\ C e. U ) -> C e. B )
32	10, 31	sylan2 491	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. B )
33	15	mrcsncl 16272	. . . . 5 |- ( ( (SubGrp ` G ) e. (Moore ` B ) /\ C e. B ) -> ( K ` { C } ) e. (SubGrp ` G ) )
34	9, 32, 33	syl2anc 693	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( K ` { C } ) e. (SubGrp ` G ) )
35	27	lsmlub 18078	. . . 4 |- ( ( ( S .(+) W ) e. (SubGrp ` G ) /\ ( K ` { C } ) e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> ( ( ( S .(+) W ) C_ U /\ ( K ` { C } ) C_ U ) <-> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U ) )
36	30, 34, 14, 35	syl3anc 1326	. . 3 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( S .(+) W ) C_ U /\ ( K ` { C } ) C_ U ) <-> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U ) )
37	2, 17, 36	mpbi2and 956	. 2 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U )
38	27	lsmub1 18071	. . . . . 6 |- ( ( ( S .(+) W ) e. (SubGrp ` G ) /\ ( K ` { C } ) e. (SubGrp ` G ) ) -> ( S .(+) W ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
39	30, 34, 38	syl2anc 693	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
40	27	lsmub2 18072	. . . . . . 7 |- ( ( ( S .(+) W ) e. (SubGrp ` G ) /\ ( K ` { C } ) e. (SubGrp ` G ) ) -> ( K ` { C } ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
41	30, 34, 40	syl2anc 693	. . . . . 6 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( K ` { C } ) C_ ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
42	32	snssd 4340	. . . . . . . 8 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> { C } C_ B )
43	9, 15, 42	mrcssidd 16285	. . . . . . 7 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> { C } C_ ( K ` { C } ) )
44		snssg 4327	. . . . . . . 8 |- ( C e. B -> ( C e. ( K ` { C } ) <-> { C } C_ ( K ` { C } ) ) )
45	32, 44	syl 17	. . . . . . 7 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( C e. ( K ` { C } ) <-> { C } C_ ( K ` { C } ) ) )
46	43, 45	mpbird 247	. . . . . 6 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. ( K ` { C } ) )
47	41, 46	sseldd 3604	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> C e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
48		eldifn 3733	. . . . . 6 |- ( C e. ( U \ ( S .(+) W ) ) -> -. C e. ( S .(+) W ) )
49	48	adantl 482	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> -. C e. ( S .(+) W ) )
50	39, 47, 49	ssnelpssd 3719	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
51	27	lsmub1 18071	. . . . . . . . 9 |- ( ( S e. (SubGrp ` G ) /\ W e. (SubGrp ` G ) ) -> S C_ ( S .(+) W ) )
52	25, 26, 51	syl2anc 693	. . . . . . . 8 |- ( ph -> S C_ ( S .(+) W ) )
53	22	snssd 4340	. . . . . . . . . . 11 |- ( ph -> { A } C_ B )
54	8, 15, 53	mrcssidd 16285	. . . . . . . . . 10 |- ( ph -> { A } C_ ( K ` { A } ) )
55	54, 18	syl6sseqr 3652	. . . . . . . . 9 |- ( ph -> { A } C_ S )
56		snssg 4327	. . . . . . . . . 10 |- ( A e. U -> ( A e. S <-> { A } C_ S ) )
57	21, 56	syl 17	. . . . . . . . 9 |- ( ph -> ( A e. S <-> { A } C_ S ) )
58	55, 57	mpbird 247	. . . . . . . 8 |- ( ph -> A e. S )
59	52, 58	sseldd 3604	. . . . . . 7 |- ( ph -> A e. ( S .(+) W ) )
60	59	adantr 481	. . . . . 6 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> A e. ( S .(+) W ) )
61	39, 60	sseldd 3604	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) )
62	3	adantr 481	. . . . . . 7 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> G e. Abel )
63	27	lsmsubg2 18262	. . . . . . 7 |- ( ( G e. Abel /\ ( S .(+) W ) e. (SubGrp ` G ) /\ ( K ` { C } ) e. (SubGrp ` G ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) e. (SubGrp ` G ) )
64	62, 30, 34, 63	syl3anc 1326	. . . . . 6 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) e. (SubGrp ` G ) )
65		pgpfac1.2	. . . . . . 7 |- ( ph -> A. w e. (SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) )
66	65	adantr 481	. . . . . 6 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> A. w e. (SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) )
67		psseq1 3694	. . . . . . . . 9 |- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( w C. U <-> ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U ) )
68		eleq2 2690	. . . . . . . . 9 |- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( A e. w <-> A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) )
69	67, 68	anbi12d 747	. . . . . . . 8 |- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( ( w C. U /\ A e. w ) <-> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) )
70		psseq2 3695	. . . . . . . . 9 |- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( ( S .(+) W ) C. w <-> ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) )
71	70	notbid 308	. . . . . . . 8 |- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( -. ( S .(+) W ) C. w <-> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) )
72	69, 71	imbi12d 334	. . . . . . 7 |- ( w = ( ( S .(+) W ) .(+) ( K ` { C } ) ) -> ( ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) <-> ( ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) )
73	72	rspcv 3305	. . . . . 6 |- ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) e. (SubGrp ` G ) -> ( A. w e. (SubGrp ` G ) ( ( w C. U /\ A e. w ) -> -. ( S .(+) W ) C. w ) -> ( ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) ) )
74	64, 66, 73	sylc 65	. . . . 5 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U /\ A e. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) )
75	61, 74	mpan2d 710	. . . 4 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U -> -. ( S .(+) W ) C. ( ( S .(+) W ) .(+) ( K ` { C } ) ) ) )
76	50, 75	mt2d 131	. . 3 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> -. ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U )
77		npss 3717	. . 3 |- ( -. ( ( S .(+) W ) .(+) ( K ` { C } ) ) C. U <-> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U ) )
78	76, 77	sylib 208	. 2 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( ( S .(+) W ) .(+) ( K ` { C } ) ) C_ U -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U ) )
79	37, 78	mpd 15	1 |- ( ( ph /\ C e. ( U \ ( S .(+) W ) ) ) -> ( ( S .(+) W ) .(+) ( K ` { C } ) ) = U )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Grpcgrp 17422  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-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

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-iin 4523  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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  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-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196

This theorem is referenced by:  pgpfac1lem2  18474  pgpfac1lem3  18476