Theorem lcvexchlem4 34324

Description: Lemma for lcvexch 34326. (Contributed by NM, 10-Jan-2015.)

Hypotheses

Ref	Expression
lcvexch.s	|- S = ( LSubSp ` W )
lcvexch.p	|- .(+) = ( LSSum ` W )
lcvexch.c	|- C = ( <oLL ` W )
lcvexch.w	|- ( ph -> W e. LMod )
lcvexch.t	|- ( ph -> T e. S )
lcvexch.u	|- ( ph -> U e. S )
lcvexch.f	|- ( ph -> T C ( T .(+) U ) )

Assertion

Ref	Expression
lcvexchlem4	|- ( ph -> ( T i^i U ) C U )

Proof of Theorem lcvexchlem4

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

Step	Hyp	Ref	Expression
1		lcvexch.s	. . . 4 |- S = ( LSubSp ` W )
2		lcvexch.c	. . . 4 |- C = ( <oLL ` W )
3		lcvexch.w	. . . 4 |- ( ph -> W e. LMod )
4		lcvexch.t	. . . 4 |- ( ph -> T e. S )
5		lcvexch.u	. . . . 5 |- ( ph -> U e. S )
6		lcvexch.p	. . . . . 6 |- .(+) = ( LSSum ` W )
7	1, 6	lsmcl 19083	. . . . 5 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T .(+) U ) e. S )
8	3, 4, 5, 7	syl3anc 1326	. . . 4 |- ( ph -> ( T .(+) U ) e. S )
9		lcvexch.f	. . . 4 |- ( ph -> T C ( T .(+) U ) )
10	1, 2, 3, 4, 8, 9	lcvpss 34311	. . 3 |- ( ph -> T C. ( T .(+) U ) )
11	1, 6, 2, 3, 4, 5	lcvexchlem1 34321	. . 3 |- ( ph -> ( T C. ( T .(+) U ) <-> ( T i^i U ) C. U ) )
12	10, 11	mpbid 222	. 2 |- ( ph -> ( T i^i U ) C. U )
13	3	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> W e. LMod )
14	1	lsssssubg 18958	. . . . . . . . 9 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
15	13, 14	syl 17	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> S C_ (SubGrp ` W ) )
16		simp2 1062	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> s e. S )
17	15, 16	sseldd 3604	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> s e. (SubGrp ` W ) )
18	4	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> T e. S )
19	15, 18	sseldd 3604	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> T e. (SubGrp ` W ) )
20	6	lsmub2 18072	. . . . . . 7 |- ( ( s e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) ) -> T C_ ( s .(+) T ) )
21	17, 19, 20	syl2anc 693	. . . . . 6 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> T C_ ( s .(+) T ) )
22	5	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> U e. S )
23	15, 22	sseldd 3604	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> U e. (SubGrp ` W ) )
24		simp3r 1090	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> s C_ U )
25	6	lsmless1 18074	. . . . . . . 8 |- ( ( U e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) /\ s C_ U ) -> ( s .(+) T ) C_ ( U .(+) T ) )
26	23, 19, 24, 25	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( s .(+) T ) C_ ( U .(+) T ) )
27		lmodabl 18910	. . . . . . . . . 10 |- ( W e. LMod -> W e. Abel )
28	3, 27	syl 17	. . . . . . . . 9 |- ( ph -> W e. Abel )
29	3, 14	syl 17	. . . . . . . . . 10 |- ( ph -> S C_ (SubGrp ` W ) )
30	29, 4	sseldd 3604	. . . . . . . . 9 |- ( ph -> T e. (SubGrp ` W ) )
31	29, 5	sseldd 3604	. . . . . . . . 9 |- ( ph -> U e. (SubGrp ` W ) )
32	6	lsmcom 18261	. . . . . . . . 9 |- ( ( W e. Abel /\ T e. (SubGrp ` W ) /\ U e. (SubGrp ` W ) ) -> ( T .(+) U ) = ( U .(+) T ) )
33	28, 30, 31, 32	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
34	33	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
35	26, 34	sseqtr4d 3642	. . . . . 6 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( s .(+) T ) C_ ( T .(+) U ) )
36	9	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> T C ( T .(+) U ) )
37	1, 2, 3, 4, 8	lcvbr3 34310	. . . . . . . . . 10 |- ( ph -> ( T C ( T .(+) U ) <-> ( T C. ( T .(+) U ) /\ A. r e. S ( ( T C_ r /\ r C_ ( T .(+) U ) ) -> ( r = T \/ r = ( T .(+) U ) ) ) ) ) )
38	37	adantr 481	. . . . . . . . 9 |- ( ( ph /\ s e. S ) -> ( T C ( T .(+) U ) <-> ( T C. ( T .(+) U ) /\ A. r e. S ( ( T C_ r /\ r C_ ( T .(+) U ) ) -> ( r = T \/ r = ( T .(+) U ) ) ) ) ) )
39	3	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> W e. LMod )
40		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> s e. S )
41	4	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> T e. S )
42	1, 6	lsmcl 19083	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ s e. S /\ T e. S ) -> ( s .(+) T ) e. S )
43	39, 40, 41, 42	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ s e. S ) -> ( s .(+) T ) e. S )
44		sseq2 3627	. . . . . . . . . . . . . 14 |- ( r = ( s .(+) T ) -> ( T C_ r <-> T C_ ( s .(+) T ) ) )
45		sseq1 3626	. . . . . . . . . . . . . 14 |- ( r = ( s .(+) T ) -> ( r C_ ( T .(+) U ) <-> ( s .(+) T ) C_ ( T .(+) U ) ) )
46	44, 45	anbi12d 747	. . . . . . . . . . . . 13 |- ( r = ( s .(+) T ) -> ( ( T C_ r /\ r C_ ( T .(+) U ) ) <-> ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) ) )
47		eqeq1 2626	. . . . . . . . . . . . . 14 |- ( r = ( s .(+) T ) -> ( r = T <-> ( s .(+) T ) = T ) )
48		eqeq1 2626	. . . . . . . . . . . . . 14 |- ( r = ( s .(+) T ) -> ( r = ( T .(+) U ) <-> ( s .(+) T ) = ( T .(+) U ) ) )
49	47, 48	orbi12d 746	. . . . . . . . . . . . 13 |- ( r = ( s .(+) T ) -> ( ( r = T \/ r = ( T .(+) U ) ) <-> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) )
50	46, 49	imbi12d 334	. . . . . . . . . . . 12 |- ( r = ( s .(+) T ) -> ( ( ( T C_ r /\ r C_ ( T .(+) U ) ) -> ( r = T \/ r = ( T .(+) U ) ) ) <-> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) ) )
51	50	rspcv 3305	. . . . . . . . . . 11 |- ( ( s .(+) T ) e. S -> ( A. r e. S ( ( T C_ r /\ r C_ ( T .(+) U ) ) -> ( r = T \/ r = ( T .(+) U ) ) ) -> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) ) )
52	43, 51	syl 17	. . . . . . . . . 10 |- ( ( ph /\ s e. S ) -> ( A. r e. S ( ( T C_ r /\ r C_ ( T .(+) U ) ) -> ( r = T \/ r = ( T .(+) U ) ) ) -> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) ) )
53	52	adantld 483	. . . . . . . . 9 |- ( ( ph /\ s e. S ) -> ( ( T C. ( T .(+) U ) /\ A. r e. S ( ( T C_ r /\ r C_ ( T .(+) U ) ) -> ( r = T \/ r = ( T .(+) U ) ) ) ) -> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) ) )
54	38, 53	sylbid 230	. . . . . . . 8 |- ( ( ph /\ s e. S ) -> ( T C ( T .(+) U ) -> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) ) )
55	54	3adant3 1081	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( T C ( T .(+) U ) -> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) ) )
56	36, 55	mpd 15	. . . . . 6 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( T C_ ( s .(+) T ) /\ ( s .(+) T ) C_ ( T .(+) U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) ) )
57	21, 35, 56	mp2and 715	. . . . 5 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) )
58		ineq1 3807	. . . . . . 7 |- ( ( s .(+) T ) = T -> ( ( s .(+) T ) i^i U ) = ( T i^i U ) )
59		simp3l 1089	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( T i^i U ) C_ s )
60	1, 6, 2, 13, 18, 22, 16, 59, 24	lcvexchlem2 34322	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( s .(+) T ) i^i U ) = s )
61	60	eqeq1d 2624	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( ( s .(+) T ) i^i U ) = ( T i^i U ) <-> s = ( T i^i U ) ) )
62	58, 61	syl5ib 234	. . . . . 6 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( s .(+) T ) = T -> s = ( T i^i U ) ) )
63		ineq1 3807	. . . . . . 7 |- ( ( s .(+) T ) = ( T .(+) U ) -> ( ( s .(+) T ) i^i U ) = ( ( T .(+) U ) i^i U ) )
64	6	lsmub2 18072	. . . . . . . . . 10 |- ( ( T e. (SubGrp ` W ) /\ U e. (SubGrp ` W ) ) -> U C_ ( T .(+) U ) )
65	19, 23, 64	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> U C_ ( T .(+) U ) )
66		sseqin2 3817	. . . . . . . . 9 |- ( U C_ ( T .(+) U ) <-> ( ( T .(+) U ) i^i U ) = U )
67	65, 66	sylib 208	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( T .(+) U ) i^i U ) = U )
68	60, 67	eqeq12d 2637	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( ( s .(+) T ) i^i U ) = ( ( T .(+) U ) i^i U ) <-> s = U ) )
69	63, 68	syl5ib 234	. . . . . 6 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( s .(+) T ) = ( T .(+) U ) -> s = U ) )
70	62, 69	orim12d 883	. . . . 5 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( ( ( s .(+) T ) = T \/ ( s .(+) T ) = ( T .(+) U ) ) -> ( s = ( T i^i U ) \/ s = U ) ) )
71	57, 70	mpd 15	. . . 4 |- ( ( ph /\ s e. S /\ ( ( T i^i U ) C_ s /\ s C_ U ) ) -> ( s = ( T i^i U ) \/ s = U ) )
72	71	3exp 1264	. . 3 |- ( ph -> ( s e. S -> ( ( ( T i^i U ) C_ s /\ s C_ U ) -> ( s = ( T i^i U ) \/ s = U ) ) ) )
73	72	ralrimiv 2965	. 2 |- ( ph -> A. s e. S ( ( ( T i^i U ) C_ s /\ s C_ U ) -> ( s = ( T i^i U ) \/ s = U ) ) )
74	1	lssincl 18965	. . . 4 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
75	3, 4, 5, 74	syl3anc 1326	. . 3 |- ( ph -> ( T i^i U ) e. S )
76	1, 2, 3, 75, 5	lcvbr3 34310	. 2 |- ( ph -> ( ( T i^i U ) C U <-> ( ( T i^i U ) C. U /\ A. s e. S ( ( ( T i^i U ) C_ s /\ s C_ U ) -> ( s = ( T i^i U ) \/ s = U ) ) ) ) )
77	12, 73, 76	mpbir2and 957	1 |- ( ph -> ( T i^i U ) C U )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   C. wpss 3575   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  SubGrpcsubg 17588   LSSumclsm 18049   Abelcabl 18194   LModclmod 18863   LSubSpclss 18932   <oL_L clcv 34305

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-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lcv 34306

This theorem is referenced by:  lcvexch  34326  lsatcvat3  34339