Theorem lcvexchlem5 32371

Description: Lemma for lcvexch 32372. (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 17024	. . . . 5 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
7	3, 4, 5, 6	syl3anc 1213	. . . 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 32357	. . 3 |- ( ph -> ( T i^i U ) C. U )
10		lcvexch.p	. . . 4 |- .(+) = ( LSSum ` W )
11	1, 10, 2, 3, 4, 5	lcvexchlem1 32367	. . 3 |- ( ph -> ( T C. ( T .(+) U ) <-> ( T i^i U ) C. U ) )
12	9, 11	mpbird 232	. 2 |- ( ph -> T C. ( T .(+) U ) )
13		simp3l 1011	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T C_ s )
14		ssrin 3572	. . . . . . . 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 3568	. . . . . . 7 |- ( s i^i U ) C_ U
17	15, 16	jctir 535	. . . . . 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 1004	. . . . . . 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 32356	. . . . . . . . . 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 462	. . . . . . . . 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 462	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> W e. LMod )
22		simpr 458	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> s e. S )
23	5	adantr 462	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> U e. S )
24	1	lssincl 17024	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ s e. S /\ U e. S ) -> ( s i^i U ) e. S )
25	21, 22, 23, 24	syl3anc 1213	. . . . . . . . . . 11 |- ( ( ph /\ s e. S ) -> ( s i^i U ) e. S )
26		sseq2 3375	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( ( T i^i U ) C_ r <-> ( T i^i U ) C_ ( s i^i U ) ) )
27		sseq1 3374	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r C_ U <-> ( s i^i U ) C_ U ) )
28	26, 27	anbi12d 705	. . . . . . . . . . . . 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 2447	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = ( T i^i U ) <-> ( s i^i U ) = ( T i^i U ) ) )
30		eqeq1 2447	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = U <-> ( s i^i U ) = U ) )
31	29, 30	orbi12d 704	. . . . . . . . . . . . 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 320	. . . . . . . . . . . 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 3066	. . . . . . . . . . 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 464	. . . . . . . . 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 215	. . . . . . . 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 1003	. . . . . . 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 6097	. . . . . . 7 |- ( ( s i^i U ) = ( T i^i U ) -> ( ( s i^i U ) .(+) T ) = ( ( T i^i U ) .(+) T ) )
41	3	3ad2ant1 1004	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> W e. LMod )
42	4	3ad2ant1 1004	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T e. S )
43	5	3ad2ant1 1004	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> U e. S )
44		simp2 984	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> s e. S )
45		simp3r 1012	. . . . . . . . 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 32369	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) .(+) T ) = s )
47	1	lsssssubg 17017	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
48	3, 47	syl 16	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
49	48, 7	sseldd 3354	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) e. (SubGrp ` W ) )
50	48, 4	sseldd 3354	. . . . . . . . . 10 |- ( ph -> T e. (SubGrp ` W ) )
51		inss1 3567	. . . . . . . . . . 11 |- ( T i^i U ) C_ T
52	51	a1i 11	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) C_ T )
53	10	lsmss1 16156	. . . . . . . . . 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 1213	. . . . . . . . 9 |- ( ph -> ( ( T i^i U ) .(+) T ) = T )
55	54	3ad2ant1 1004	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( T i^i U ) .(+) T ) = T )
56	46, 55	eqeq12d 2455	. . . . . . 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 219	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = ( T i^i U ) -> s = T ) )
58		oveq1 6097	. . . . . . 7 |- ( ( s i^i U ) = U -> ( ( s i^i U ) .(+) T ) = ( U .(+) T ) )
59		lmodabl 16972	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
60	3, 59	syl 16	. . . . . . . . . 10 |- ( ph -> W e. Abel )
61	48, 5	sseldd 3354	. . . . . . . . . 10 |- ( ph -> U e. (SubGrp ` W ) )
62	10	lsmcom 16333	. . . . . . . . . 10 |- ( ( W e. Abel /\ U e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) ) -> ( U .(+) T ) = ( T .(+) U ) )
63	60, 61, 50, 62	syl3anc 1213	. . . . . . . . 9 |- ( ph -> ( U .(+) T ) = ( T .(+) U ) )
64	63	3ad2ant1 1004	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( U .(+) T ) = ( T .(+) U ) )
65	46, 64	eqeq12d 2455	. . . . . . 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 219	. . . . . 6 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) = U -> s = ( T .(+) U ) ) )
67	57, 66	orim12d 829	. . . . 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 1181	. . 3 |- ( ph -> ( s e. S -> ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) ) )
70	69	ralrimiv 2796	. 2 |- ( ph -> A. s e. S ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) )
71	1, 10	lsmcl 17142	. . . 4 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T .(+) U ) e. S )
72	3, 4, 5, 71	syl3anc 1213	. . 3 |- ( ph -> ( T .(+) U ) e. S )
73	1, 2, 3, 4, 72	lcvbr3 32356	. 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 908	1 |- ( ph -> T C ( T .(+) U ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   A.wral 2713   i^i cin 3324   C_ wss 3325   C. wpss 3326   class class class wbr 4289   ` cfv 5415  (class class class)co 6090  SubGrpcsubg 15668   LSSumclsm 16126   Abelcabel 16271   LModclmod 16928   LSubSpclss 16991   <oL_L clcv 32351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-cntz 15828  df-oppg 15854  df-lsm 16128  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lss 16992  df-lcv 32352

This theorem is referenced by:  lcvexch  32372