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Theorem lcvexchlem5 29898
Description: Lemma for lcvexch 29899. (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.
StepHypRef 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 )
61lssincl 16043 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
73, 4, 5, 6syl3anc 1185 . . . 4  |-  ( ph  ->  ( T  i^i  U
)  e.  S )
8 lcvexch.g . . . 4  |-  ( ph  ->  ( T  i^i  U
) C U )
91, 2, 3, 7, 5, 8lcvpss 29884 . . 3  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
10 lcvexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
111, 10, 2, 3, 4, 5lcvexchlem1 29894 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
129, 11mpbird 225 . 2  |-  ( ph  ->  T  C.  ( T 
.(+)  U ) )
13 simp3l 986 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  C_  s
)
14 ssrin 3568 . . . . . . . 8  |-  ( T 
C_  s  ->  ( T  i^i  U )  C_  ( s  i^i  U
) )
1513, 14syl 16 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U )  C_  (
s  i^i  U )
)
16 inss2 3564 . . . . . . 7  |-  ( s  i^i  U )  C_  U
1715, 16jctir 526 . . . . . 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
) )
1883ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U ) C U )
191, 2, 3, 7, 5lcvbr3 29883 . . . . . . . . . 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 ) ) ) ) )
2019adantr 453 . . . . . . . . 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 ) ) ) ) )
213adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
22 simpr 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
235adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  U  e.  S )
241lssincl 16043 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  U  e.  S )  ->  (
s  i^i  U )  e.  S )
2521, 22, 23, 24syl3anc 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  i^i  U )  e.  S )
26 sseq2 3372 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
( T  i^i  U
)  C_  r  <->  ( T  i^i  U )  C_  (
s  i^i  U )
) )
27 sseq1 3371 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  C_  U  <->  ( s  i^i  U )  C_  U
) )
2826, 27anbi12d 693 . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  ( T  i^i  U )  <->  ( s  i^i  U )  =  ( T  i^i  U ) ) )
30 eqeq1 2444 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  U  <->  ( s  i^i  U )  =  U ) )
3129, 30orbi12d 692 . . . . . . . . . . . . 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 ) ) )
3228, 31imbi12d 313 . . . . . . . . . . . 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 ) ) ) )
3332rspcv 3050 . . . . . . . . . . 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 ) ) ) )
3425, 33syl 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 ) ) ) )
3534adantld 455 . . . . . . . . 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 ) ) ) )
3620, 35sylbid 208 . . . . . . . 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 ) ) ) )
37363adant3 978 . . . . . . 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 ) ) ) )
3818, 37mpd 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 ) ) )
3917, 38mpd 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 6090 . . . . . . 7  |-  ( ( s  i^i  U )  =  ( T  i^i  U )  ->  ( (
s  i^i  U )  .(+)  T )  =  ( ( T  i^i  U
)  .(+)  T ) )
4133ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  W  e.  LMod )
4243ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  e.  S )
4353ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  U  e.  S )
44 simp2 959 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  s  e.  S )
45 simp3r 987 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  s  C_  ( T  .(+)  U ) )
461, 10, 2, 41, 42, 43, 44, 13, 45lcvexchlem3 29896 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  .(+)  T )  =  s )
471lsssssubg 16036 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
483, 47syl 16 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4948, 7sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  e.  (SubGrp `  W ) )
5048, 4sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
51 inss1 3563 . . . . . . . . . . 11  |-  ( T  i^i  U )  C_  T
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  C_  T )
5310lsmss1 15300 . . . . . . . . . 10  |-  ( ( ( T  i^i  U
)  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )  /\  ( T  i^i  U
)  C_  T )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5449, 50, 52, 53syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
55543ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5646, 55eqeq12d 2452 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  .(+)  T )  =  ( ( T  i^i  U )  .(+)  T )  <->  s  =  T ) )
5740, 56syl5ib 212 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  ( T  i^i  U )  ->  s  =  T ) )
58 oveq1 6090 . . . . . . 7  |-  ( ( s  i^i  U )  =  U  ->  (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T ) )
59 lmodabl 15993 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
603, 59syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
6148, 5sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
6210lsmcom 15475 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W ) )  -> 
( U  .(+)  T )  =  ( T  .(+)  U ) )
6360, 61, 50, 62syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  T )  =  ( T  .(+)  U ) )
64633ad2ant1 979 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( U  .(+) 
T )  =  ( T  .(+)  U )
)
6546, 64eqeq12d 2452 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T )  <-> 
s  =  ( T 
.(+)  U ) ) )
6658, 65syl5ib 212 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  U  ->  s  =  ( T  .(+)  U ) ) )
6757, 66orim12d 813 . . . . 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 ) ) ) )
6839, 67mpd 15 . . . 4  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) )
69683exp 1153 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) ) )
7069ralrimiv 2790 . 2  |-  ( ph  ->  A. s  e.  S  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) )
711, 10lsmcl 16157 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
723, 4, 5, 71syl3anc 1185 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
731, 2, 3, 4, 72lcvbr3 29883 . 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 )
) ) ) ) )
7412, 70, 73mpbir2and 890 1  |-  ( ph  ->  T C ( T 
.(+)  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322    C. wpss 3323   class class class wbr 4214   ` cfv 5456  (class class class)co 6083  SubGrpcsubg 14940   LSSumclsm 15270   Abelcabel 15415   LModclmod 15952   LSubSpclss 16010    <oLL clcv 29878
This theorem is referenced by:  lcvexch  29899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-oppg 15144  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011  df-lcv 29879
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