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Theorem lcvexchlem5 32683
Description: Lemma for lcvexch 32684. (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 17046 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
73, 4, 5, 6syl3anc 1218 . . . 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 32669 . . 3  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
10 lcvexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
111, 10, 2, 3, 4, 5lcvexchlem1 32679 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
129, 11mpbird 232 . 2  |-  ( ph  ->  T  C.  ( T  .(+) 
U ) )
13 simp3l 1016 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  C_  s
)
14 ssrin 3575 . . . . . . . 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 3571 . . . . . . 7  |-  ( s  i^i  U )  C_  U
1715, 16jctir 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
) )
1883ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U ) C U )
191, 2, 3, 7, 5lcvbr3 32668 . . . . . . . . . 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 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 ) ) ) ) )
213adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
22 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
235adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  U  e.  S )
241lssincl 17046 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  U  e.  S )  ->  (
s  i^i  U )  e.  S )
2521, 22, 23, 24syl3anc 1218 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  i^i  U )  e.  S )
26 sseq2 3378 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
( T  i^i  U
)  C_  r  <->  ( T  i^i  U )  C_  (
s  i^i  U )
) )
27 sseq1 3377 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  C_  U  <->  ( s  i^i  U )  C_  U
) )
2826, 27anbi12d 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 2449 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  ( T  i^i  U )  <->  ( s  i^i  U )  =  ( T  i^i  U ) ) )
30 eqeq1 2449 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  i^i 
U )  ->  (
r  =  U  <->  ( s  i^i  U )  =  U ) )
3129, 30orbi12d 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 ) ) )
3228, 31imbi12d 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 ) ) ) )
3332rspcv 3069 . . . . . . . . . . 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 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 ) ) ) )
3620, 35sylbid 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 ) ) ) )
37363adant3 1008 . . . . . . 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 6098 . . . . . . 7  |-  ( ( s  i^i  U )  =  ( T  i^i  U )  ->  ( (
s  i^i  U )  .(+)  T )  =  ( ( T  i^i  U
)  .(+)  T ) )
4133ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  W  e.  LMod )
4243ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  e.  S )
4353ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  U  e.  S )
44 simp2 989 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  s  e.  S )
45 simp3r 1017 . . . . . . . . 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 32681 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  .(+)  T )  =  s )
471lsssssubg 17039 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
483, 47syl 16 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4948, 7sseldd 3357 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  e.  (SubGrp `  W ) )
5048, 4sseldd 3357 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
51 inss1 3570 . . . . . . . . . . 11  |-  ( T  i^i  U )  C_  T
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  C_  T )
5310lsmss1 16163 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ph  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
55543ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5646, 55eqeq12d 2457 . . . . . . 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 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 6098 . . . . . . 7  |-  ( ( s  i^i  U )  =  U  ->  (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T ) )
59 lmodabl 16992 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
603, 59syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
6148, 5sseldd 3357 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
6210lsmcom 16340 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W ) )  -> 
( U  .(+)  T )  =  ( T  .(+)  U ) )
6360, 61, 50, 62syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  T )  =  ( T  .(+)  U ) )
64633ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( U  .(+) 
T )  =  ( T  .(+)  U )
)
6546, 64eqeq12d 2457 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T )  <-> 
s  =  ( T 
.(+)  U ) ) )
6658, 65syl5ib 219 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  =  U  ->  s  =  ( T  .(+)  U ) ) )
6757, 66orim12d 834 . . . . 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 1186 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) ) )
7069ralrimiv 2798 . 2  |-  ( ph  ->  A. s  e.  S  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) )
711, 10lsmcl 17164 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
723, 4, 5, 71syl3anc 1218 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
731, 2, 3, 4, 72lcvbr3 32668 . 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 913 1  |-  ( ph  ->  T C ( T 
.(+)  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715    i^i cin 3327    C_ wss 3328    C. wpss 3329   class class class wbr 4292   ` cfv 5418  (class class class)co 6091  SubGrpcsubg 15675   LSSumclsm 16133   Abelcabel 16278   LModclmod 16948   LSubSpclss 17013    <oLL clcv 32663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-oppg 15861  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014  df-lcv 32664
This theorem is referenced by:  lcvexch  32684
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