Theorem lcvexchlem5 34864

Description: Lemma for lcvexch 34865. (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 17737	. . . . 5 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T i^i U ) e. S )
7	3, 4, 5, 6	syl3anc 1228	. . . 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 34850	. . 3 |- ( ph -> ( T i^i U ) C. U )
10		lcvexch.p	. . . 4 |- .(+) = ( LSSum ` W )
11	1, 10, 2, 3, 4, 5	lcvexchlem1 34860	. . 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 1024	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T C_ s )
14		ssrin 3719	. . . . . . . 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 3715	. . . . . . 7 |- ( s i^i U ) C_ U
17	15, 16	jctir 538	. . . . . 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 1017	. . . . . . 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 34849	. . . . . . . . . 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 465	. . . . . . . . 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 465	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> W e. LMod )
22		simpr 461	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> s e. S )
23	5	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ s e. S ) -> U e. S )
24	1	lssincl 17737	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ s e. S /\ U e. S ) -> ( s i^i U ) e. S )
25	21, 22, 23, 24	syl3anc 1228	. . . . . . . . . . 11 |- ( ( ph /\ s e. S ) -> ( s i^i U ) e. S )
26		sseq2 3521	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( ( T i^i U ) C_ r <-> ( T i^i U ) C_ ( s i^i U ) ) )
27		sseq1 3520	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r C_ U <-> ( s i^i U ) C_ U ) )
28	26, 27	anbi12d 710	. . . . . . . . . . . . 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 2461	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = ( T i^i U ) <-> ( s i^i U ) = ( T i^i U ) ) )
30		eqeq1 2461	. . . . . . . . . . . . . 14 |- ( r = ( s i^i U ) -> ( r = U <-> ( s i^i U ) = U ) )
31	29, 30	orbi12d 709	. . . . . . . . . . . . 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 3206	. . . . . . . . . . 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 467	. . . . . . . . 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 1016	. . . . . . 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 6303	. . . . . . 7 |- ( ( s i^i U ) = ( T i^i U ) -> ( ( s i^i U ) .(+) T ) = ( ( T i^i U ) .(+) T ) )
41	3	3ad2ant1 1017	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> W e. LMod )
42	4	3ad2ant1 1017	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> T e. S )
43	5	3ad2ant1 1017	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> U e. S )
44		simp2 997	. . . . . . . . 9 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> s e. S )
45		simp3r 1025	. . . . . . . . 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 34862	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( s i^i U ) .(+) T ) = s )
47	1	lsssssubg 17730	. . . . . . . . . . . 12 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
48	3, 47	syl 16	. . . . . . . . . . 11 |- ( ph -> S C_ (SubGrp ` W ) )
49	48, 7	sseldd 3500	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) e. (SubGrp ` W ) )
50	48, 4	sseldd 3500	. . . . . . . . . 10 |- ( ph -> T e. (SubGrp ` W ) )
51		inss1 3714	. . . . . . . . . . 11 |- ( T i^i U ) C_ T
52	51	a1i 11	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) C_ T )
53	10	lsmss1 16810	. . . . . . . . . 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 1228	. . . . . . . . 9 |- ( ph -> ( ( T i^i U ) .(+) T ) = T )
55	54	3ad2ant1 1017	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( ( T i^i U ) .(+) T ) = T )
56	46, 55	eqeq12d 2479	. . . . . . 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 6303	. . . . . . 7 |- ( ( s i^i U ) = U -> ( ( s i^i U ) .(+) T ) = ( U .(+) T ) )
59		lmodabl 17683	. . . . . . . . . . 11 |- ( W e. LMod -> W e. Abel )
60	3, 59	syl 16	. . . . . . . . . 10 |- ( ph -> W e. Abel )
61	48, 5	sseldd 3500	. . . . . . . . . 10 |- ( ph -> U e. (SubGrp ` W ) )
62	10	lsmcom 16990	. . . . . . . . . 10 |- ( ( W e. Abel /\ U e. (SubGrp ` W ) /\ T e. (SubGrp ` W ) ) -> ( U .(+) T ) = ( T .(+) U ) )
63	60, 61, 50, 62	syl3anc 1228	. . . . . . . . 9 |- ( ph -> ( U .(+) T ) = ( T .(+) U ) )
64	63	3ad2ant1 1017	. . . . . . . 8 |- ( ( ph /\ s e. S /\ ( T C_ s /\ s C_ ( T .(+) U ) ) ) -> ( U .(+) T ) = ( T .(+) U ) )
65	46, 64	eqeq12d 2479	. . . . . . 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 838	. . . . 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 1195	. . 3 |- ( ph -> ( s e. S -> ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) ) )
70	69	ralrimiv 2869	. 2 |- ( ph -> A. s e. S ( ( T C_ s /\ s C_ ( T .(+) U ) ) -> ( s = T \/ s = ( T .(+) U ) ) ) )
71	1, 10	lsmcl 17855	. . . 4 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T .(+) U ) e. S )
72	3, 4, 5, 71	syl3anc 1228	. . 3 |- ( ph -> ( T .(+) U ) e. S )
73	1, 2, 3, 4, 72	lcvbr3 34849	. 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 922	1 |- ( ph -> T C ( T .(+) U ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   i^i cin 3470   C_ wss 3471   C. wpss 3472   class class class wbr 4456   ` cfv 5594  (class class class)co 6296  SubGrpcsubg 16321   LSSumclsm 16780   Abelcabl 16925   LModclmod 17638   LSubSpclss 17704   <oL_L clcv 34844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-lmod 17640  df-lss 17705  df-lcv 34845

This theorem is referenced by:  lcvexch  34865