Theorem lcvexchlem5 32648

Description: Lemma for lcvexch 32649. (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.g	|- ( ph -> ( T i^i U ) C U )

Assertion

Ref	Expression
lcvexchlem5	|- ( ph -> T C ( T .(+) U ) )

Proof of Theorem lcvexchlem5

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	. . . . 5 |- ( ph -> T e. S )
5		lcvexch.u	. . . . 5 |- ( ph -> U e. S )
6	1	lssincl 18236	. . . . 5 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
7	3, 4, 5, 6	syl3anc 1276	. . . 4 |- ( ph -> ( T i^i U ) e. S )
8		lcvexch.g	. . . 4 |- ( ph -> ( T i^i U ) C U )
9	1, 2, 3, 7, 5, 8	lcvpss 32634	. . 3 |- ( ph -> ( T i^i U ) C. U )
10		lcvexch.p	. . . 4 |- .(+) = ( LSSum ` W )
11	1, 10, 2, 3, 4, 5	lcvexchlem1 32644	. . 3 |- ( ph -> ( T C. ( T .(+) U ) <-> ( T i^i U ) C. U ) )
12	9, 11	mpbird 240	. 2 |- ( ph -> T C. ( T .(+) U ) )
13		simp3l 1042	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T C_ s )
14		ssrin 3668	. . . . . . . 8 |- ( T C_ s -> ( T i^i U ) C_ ( s i^i U ) )
15	13, 14	syl 17	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( T i^i U ) C_ ( s i^i U ) )
16		inss2 3664	. . . . . . 7 |- ( s i^i U ) C_ U
17	15, 16	jctir 545	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) )
18	8	3ad2ant1 1035	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( T i^i U ) C U )
19	1, 2, 3, 7, 5	lcvbr3 32633	. . . . . . . . . 10 |- ( ph -> ( ( T i^i U ) C U <-> ( ( T i^i U ) C. U /\ A. r e. S ( ( ( T i^i U ) C_ r /\ r C_ U ) -> ( r = ( T i^i U ) \/ r = U ) ) ) ) )
20	19	adantr 471	. . . . . . . . 9 |- ( ( ph /\ s e. S ) -> ( ( T i^i U ) C U <-> ( ( T i^i U ) C. U /\ A. r e. S ( ( ( T i^i U ) C_ r /\ r C_ U ) -> ( r = ( T i^i U ) \/ r = U ) ) ) ) )
21	3	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> W e. LMod )
22		simpr 467	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> s e. S )
23	5	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> U e. S )
24	1	lssincl 18236	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ s e. S /\ U e. S ) -> ( s i^i U ) e. S )
25	21, 22, 23, 24	syl3anc 1276	. . . . . . . . . . 11 |- ( ( ph /\ s e. S ) -> ( s i^i U ) e. S )
26		sseq2 3465	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( ( T i^i U ) C_ r <-> ( T i^i U ) C_ ( s i^i U ) ) )
27		sseq1 3464	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r C_ U <-> ( s i^i U ) C_ U ) )
28	26, 27	anbi12d 722	. . . . . . . . . . . . 13 |- ( r = ( s i^i U ) -> ( ( ( T i^i U ) C_ r /\ r C_ U ) <-> ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) ) )
29		eqeq1 2465	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = ( T i^i U ) <-> ( s i^i U ) = ( T i^i U ) ) )
30		eqeq1 2465	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = U <-> ( s i^i U ) = U ) )
31	29, 30	orbi12d 721	. . . . . . . . . . . . 13 |- ( r = ( s i^i U ) -> ( ( r = ( T i^i U ) \/ r = U ) <-> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) )
32	28, 31	imbi12d 326	. . . . . . . . . . . 12 |- ( r = ( s i^i U ) -> ( ( ( ( T i^i U ) C_ r /\ r C_ U ) -> ( r = ( T i^i U ) \/ r = U ) ) <-> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) ) )
33	32	rspcv 3157	. . . . . . . . . . 11 |- ( ( s i^i U ) e. S -> ( A. r e. S ( ( ( T i^i U ) C_ r /\ r C_ U ) -> ( r = ( T i^i U ) \/ r = U ) ) -> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) ) )
34	25, 33	syl 17	. . . . . . . . . 10 |- ( ( ph /\ s e. S ) -> ( A. r e. S ( ( ( T i^i U ) C_ r /\ r C_ U ) -> ( r = ( T i^i U ) \/ r = U ) ) -> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) ) )
35	34	adantld 473	. . . . . . . . 9 |- ( ( ph /\ s e. S ) -> ( ( ( T i^i U ) C. U /\ A. r e. S ( ( ( T i^i U ) C_ r /\ r C_ U ) -> ( r = ( T i^i U ) \/ r = U ) ) ) -> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) ) )
36	20, 35	sylbid 223	. . . . . . . 8 |- ( ( ph /\ s e. S ) -> ( ( T i^i U ) C U -> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) ) )
37	36	3adant3 1034	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( T i^i U ) C U -> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) ) )
38	18, 37	mpd 15	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( ( T i^i U ) C_ ( s i^i U ) /\ ( s i^i U ) C_ U ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) ) )
39	17, 38	mpd 15	. . . . 5 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) )
40		oveq1 6321	. . . . . . 7 |- ( ( s i^i U ) = ( T i^i U ) -> ( ( s i^i U ) .(+) T ) = ( ( T i^i U ) .(+) T ) )
41	3	3ad2ant1 1035	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> W e. LMod )
42	4	3ad2ant1 1035	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T e. S )
43	5	3ad2ant1 1035	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> U e. S )
44		simp2 1015	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> s e. S )
45		simp3r 1043	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> s C_ ( T .(+) U ) )
46	1, 10, 2, 41, 42, 43, 44, 13, 45	lcvexchlem3 32646	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) .(+) T ) = s )
47	1	lsssssubg 18229	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
48	3, 47	syl 17	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
49	48, 7	sseldd 3444	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) e. (SubGrp ` W ) )
50	48, 4	sseldd 3444	. . . . . . . . . 10 |- ( ph -> T e. (SubGrp ` W ) )
51		inss1 3663	. . . . . . . . . . 11 |- ( T i^i U ) C_ T
52	51	a1i 11	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) C_ T )
53	10	lsmss1 17364	. . . . . . . . . 10 |- ( ( ( T i^i U ) e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) /\ ( T i^i U ) C_ T ) -> ( ( T i^i U ) .(+) T ) = T )
54	49, 50, 52, 53	syl3anc 1276	. . . . . . . . 9 |- ( ph -> ( ( T i^i U ) .(+) T ) = T )
55	54	3ad2ant1 1035	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( T i^i U ) .(+) T ) = T )
56	46, 55	eqeq12d 2476	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( ( s i^i U ) .(+) T ) = ( ( T i^i U ) .(+) T ) <-> s = T ) )
57	40, 56	syl5ib 227	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = ( T i^i U ) -> s = T ) )
58		oveq1 6321	. . . . . . 7 |- ( ( s i^i U ) = U -> ( ( s i^i U ) .(+) T ) = ( U .(+) T ) )
59		lmodabl 18183	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
60	3, 59	syl 17	. . . . . . . . . 10 |- ( ph -> W e. Abel )
61	48, 5	sseldd 3444	. . . . . . . . . 10 |- ( ph -> U e. (SubGrp ` W ) )
62	10	lsmcom 17544	. . . . . . . . . 10 |- ( ( W e. Abel /\ U e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) ) -> ( U .(+) T ) = ( T .(+) U ) )
63	60, 61, 50, 62	syl3anc 1276	. . . . . . . . 9 |- ( ph -> ( U .(+) T ) = ( T .(+) U ) )
64	63	3ad2ant1 1035	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( U .(+) T ) = ( T .(+) U ) )
65	46, 64	eqeq12d 2476	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( ( s i^i U ) .(+) T ) = ( U .(+) T ) <-> s = ( T .(+) U ) ) )
66	58, 65	syl5ib 227	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = U -> s = ( T .(+) U ) ) )
67	57, 66	orim12d 854	. . . . 5 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( ( s i^i U ) = ( T i^i U ) \/ ( s i^i U ) = U ) -> ( s = T \/ s = ( T .(+) U ) ) ) )
68	39, 67	mpd 15	. . . 4 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( s = T \/ s = ( T .(+) U ) ) )
69	68	3exp 1214	. . 3 |- ( ph -> ( s e. S -> ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) ) )
70	69	ralrimiv 2811	. 2 |- ( ph -> A. s e. S ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) )
71	1, 10	lsmcl 18354	. . . 4 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T .(+) U ) e. S )
72	3, 4, 5, 71	syl3anc 1276	. . 3 |- ( ph -> ( T .(+) U ) e. S )
73	1, 2, 3, 4, 72	lcvbr3 32633	. 2 |- ( ph -> ( T C ( T .(+) U ) <-> ( T C. ( T .(+) U ) /\ A. s e. S ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) ) ) )
74	12, 70, 73	mpbir2and 938	1 |- ( ph -> T C ( T .(+) U ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   A.wral 2748   i^i cin 3414   C_ wss 3415   C. wpss 3416   class class class wbr 4415   ` cfv 5600  (class class class)co 6314  SubGrpcsubg 16859   LSSumclsm 17334   Abelcabl 17479   LModclmod 18139   LSubSpclss 18203   <oL_L clcv 32628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-subg 16862  df-cntz 17019  df-oppg 17045  df-lsm 17336  df-cmn 17480  df-abl 17481  df-mgp 17772  df-ur 17784  df-ring 17830  df-lmod 18141  df-lss 18204  df-lcv 32629

This theorem is referenced by:  lcvexch  32649