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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.
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 17024 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
73, 4, 5, 6syl3anc 1213 . . . 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 32357 . . 3  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
10 lcvexch.p . . . 4  |-  .(+)  =  (
LSSum `  W )
111, 10, 2, 3, 4, 5lcvexchlem1 32367 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
129, 11mpbird 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
) )
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 3568 . . . . . . 7  |-  ( s  i^i  U )  C_  U
1715, 16jctir 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
) )
1883ad2ant1 1004 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( T  i^i  U ) C U )
191, 2, 3, 7, 5lcvbr3 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 ) ) ) ) )
2019adantr 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 ) ) ) ) )
213adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
22 simpr 458 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
235adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  U  e.  S )
241lssincl 17024 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  U  e.  S )  ->  (
s  i^i  U )  e.  S )
2521, 22, 23, 24syl3anc 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
) )
2826, 27anbi12d 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 ) )
3129, 30orbi12d 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 ) ) )
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 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 ) ) ) )
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 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 ) ) ) )
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 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 ) ) ) )
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 6097 . . . . . . 7  |-  ( ( s  i^i  U )  =  ( T  i^i  U )  ->  ( (
s  i^i  U )  .(+)  T )  =  ( ( T  i^i  U
)  .(+)  T ) )
4133ad2ant1 1004 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  W  e.  LMod )
4243ad2ant1 1004 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  T  e.  S )
4353ad2ant1 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 ) )
461, 10, 2, 41, 42, 43, 44, 13, 45lcvexchlem3 32369 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( (
s  i^i  U )  .(+)  T )  =  s )
471lsssssubg 17017 . . . . . . . . . . . 12  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
483, 47syl 16 . . . . . . . . . . 11  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
4948, 7sseldd 3354 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  e.  (SubGrp `  W ) )
5048, 4sseldd 3354 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
51 inss1 3567 . . . . . . . . . . 11  |-  ( T  i^i  U )  C_  T
5251a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( T  i^i  U
)  C_  T )
5310lsmss1 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 )
5449, 50, 52, 53syl3anc 1213 . . . . . . . . 9  |-  ( ph  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
55543ad2ant1 1004 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( ( T  i^i  U )  .(+)  T )  =  T )
5646, 55eqeq12d 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 ) )
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 6097 . . . . . . 7  |-  ( ( s  i^i  U )  =  U  ->  (
( s  i^i  U
)  .(+)  T )  =  ( U  .(+)  T ) )
59 lmodabl 16972 . . . . . . . . . . 11  |-  ( W  e.  LMod  ->  W  e. 
Abel )
603, 59syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  Abel )
6148, 5sseldd 3354 . . . . . . . . . 10  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
6210lsmcom 16333 . . . . . . . . . 10  |-  ( ( W  e.  Abel  /\  U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W ) )  -> 
( U  .(+)  T )  =  ( T  .(+)  U ) )
6360, 61, 50, 62syl3anc 1213 . . . . . . . . 9  |-  ( ph  ->  ( U  .(+)  T )  =  ( T  .(+)  U ) )
64633ad2ant1 1004 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( U  .(+) 
T )  =  ( T  .(+)  U )
)
6546, 64eqeq12d 2455 . . . . . . 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 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 ) ) ) )
6839, 67mpd 15 . . . 4  |-  ( (
ph  /\  s  e.  S  /\  ( T  C_  s  /\  s  C_  ( T  .(+)  U ) ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) )
69683exp 1181 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) ) )
7069ralrimiv 2796 . 2  |-  ( ph  ->  A. s  e.  S  ( ( T  C_  s  /\  s  C_  ( T  .(+)  U ) )  ->  ( s  =  T  \/  s  =  ( T  .(+)  U ) ) ) )
711, 10lsmcl 17142 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
723, 4, 5, 71syl3anc 1213 . . 3  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
731, 2, 3, 4, 72lcvbr3 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 )
) ) ) ) )
7412, 70, 73mpbir2and 908 1  |-  ( ph  ->  T C ( T 
.(+)  U ) )
Colors of variables: wff setvar class
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    <oLL 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
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