Theorem lcvexchlem5 29521

Description: Lemma for lcvexch 29522. (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 15996	. . . . 5 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
7	3, 4, 5, 6	syl3anc 1184	. . . 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 29507	. . 3 |- ( ph -> ( T i^i U ) C. U )
10		lcvexch.p	. . . 4 |- .(+) = ( LSSum ` W )
11	1, 10, 2, 3, 4, 5	lcvexchlem1 29517	. . 3 |- ( ph -> ( T C. ( T .(+) U ) <-> ( T i^i U ) C. U ) )
12	9, 11	mpbird 224	. 2 |- ( ph -> T C. ( T .(+) U ) )
13		simp3l 985	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T C_ s )
14		ssrin 3526	. . . . . . . 8 |- ( T C_ s -> ( T i^i U ) C_ ( s i^i U ) )
15	13, 14	syl 16	. . . . . . 7 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( T i^i U ) C_ ( s i^i U ) )
16		inss2 3522	. . . . . . 7 |- ( s i^i U ) C_ U
17	15, 16	jctir 525	. . . . . 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 978	. . . . . . 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 29506	. . . . . . . . . 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 452	. . . . . . . . 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 452	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> W e. LMod )
22		simpr 448	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> s e. S )
23	5	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> U e. S )
24	1	lssincl 15996	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ s e. S /\ U e. S ) -> ( s i^i U ) e. S )
25	21, 22, 23, 24	syl3anc 1184	. . . . . . . . . . 11 |- ( ( ph /\ s e. S ) -> ( s i^i U ) e. S )
26		sseq2 3330	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( ( T i^i U ) C_ r <-> ( T i^i U ) C_ ( s i^i U ) ) )
27		sseq1 3329	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r C_ U <-> ( s i^i U ) C_ U ) )
28	26, 27	anbi12d 692	. . . . . . . . . . . . 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 2410	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = ( T i^i U ) <-> ( s i^i U ) = ( T i^i U ) ) )
30		eqeq1 2410	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = U <-> ( s i^i U ) = U ) )
31	29, 30	orbi12d 691	. . . . . . . . . . . . 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 312	. . . . . . . . . . . 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 3008	. . . . . . . . . . 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 16	. . . . . . . . . 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 454	. . . . . . . . 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 207	. . . . . . . 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 977	. . . . . . 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 6047	. . . . . . 7 |- ( ( s i^i U ) = ( T i^i U ) -> ( ( s i^i U ) .(+) T ) = ( ( T i^i U ) .(+) T ) )
41	3	3ad2ant1 978	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> W e. LMod )
42	4	3ad2ant1 978	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T e. S )
43	5	3ad2ant1 978	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> U e. S )
44		simp2 958	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> s e. S )
45		simp3r 986	. . . . . . . . 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 29519	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) .(+) T ) = s )
47	1	lsssssubg 15989	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
48	3, 47	syl 16	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
49	48, 7	sseldd 3309	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) e. (SubGrp ` W ) )
50	48, 4	sseldd 3309	. . . . . . . . . 10 |- ( ph -> T e. (SubGrp ` W ) )
51		inss1 3521	. . . . . . . . . . 11 |- ( T i^i U ) C_ T
52	51	a1i 11	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) C_ T )
53	10	lsmss1 15253	. . . . . . . . . 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 1184	. . . . . . . . 9 |- ( ph -> ( ( T i^i U ) .(+) T ) = T )
55	54	3ad2ant1 978	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( T i^i U ) .(+) T ) = T )
56	46, 55	eqeq12d 2418	. . . . . . 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 211	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = ( T i^i U ) -> s = T ) )
58		oveq1 6047	. . . . . . 7 |- ( ( s i^i U ) = U -> ( ( s i^i U ) .(+) T ) = ( U .(+) T ) )
59		lmodabl 15946	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
60	3, 59	syl 16	. . . . . . . . . 10 |- ( ph -> W e. Abel )
61	48, 5	sseldd 3309	. . . . . . . . . 10 |- ( ph -> U e. (SubGrp ` W ) )
62	10	lsmcom 15428	. . . . . . . . . 10 |- ( ( W e. Abel /\ U e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) ) -> ( U .(+) T ) = ( T .(+) U ) )
63	60, 61, 50, 62	syl3anc 1184	. . . . . . . . 9 |- ( ph -> ( U .(+) T ) = ( T .(+) U ) )
64	63	3ad2ant1 978	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( U .(+) T ) = ( T .(+) U ) )
65	46, 64	eqeq12d 2418	. . . . . . 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 211	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = U -> s = ( T .(+) U ) ) )
67	57, 66	orim12d 812	. . . . 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 1152	. . 3 |- ( ph -> ( s e. S -> ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) ) )
70	69	ralrimiv 2748	. 2 |- ( ph -> A. s e. S ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) )
71	1, 10	lsmcl 16110	. . . 4 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T .(+) U ) e. S )
72	3, 4, 5, 71	syl3anc 1184	. . 3 |- ( ph -> ( T .(+) U ) e. S )
73	1, 2, 3, 4, 72	lcvbr3 29506	. 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 889	1 |- ( ph -> T C ( T .(+) U ) )

Syntax hints:   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   i^i cin 3279   C_ wss 3280   C. wpss 3281   class class class wbr 4172   ` cfv 5413  (class class class)co 6040  SubGrpcsubg 14893   LSSumclsm 15223   Abelcabel 15368   LModclmod 15905   LSubSpclss 15963   <oL_L clcv 29501

This theorem is referenced by:  lcvexch  29522

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lcv 29502