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Theorem lcvexchlem4 32647
Description: Lemma for lcvexch 32649. (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.f  |-  ( ph  ->  T C ( T 
.(+)  U ) )
Assertion
Ref Expression
lcvexchlem4  |-  ( ph  ->  ( T  i^i  U
) C U )

Proof of Theorem lcvexchlem4
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 . . . 4  |-  ( ph  ->  T  e.  S )
5 lcvexch.u . . . . 5  |-  ( ph  ->  U  e.  S )
6 lcvexch.p . . . . . 6  |-  .(+)  =  (
LSSum `  W )
71, 6lsmcl 18354 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
83, 4, 5, 7syl3anc 1276 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
9 lcvexch.f . . . 4  |-  ( ph  ->  T C ( T 
.(+)  U ) )
101, 2, 3, 4, 8, 9lcvpss 32634 . . 3  |-  ( ph  ->  T  C.  ( T  .(+) 
U ) )
111, 6, 2, 3, 4, 5lcvexchlem1 32644 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
1210, 11mpbid 215 . 2  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
1333ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  W  e.  LMod )
141lsssssubg 18229 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
1513, 14syl 17 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  S  C_  (SubGrp `  W )
)
16 simp2 1015 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  S )
1715, 16sseldd 3444 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  (SubGrp `  W )
)
1843ad2ant1 1035 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  S )
1915, 18sseldd 3444 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  (SubGrp `  W )
)
206lsmub2 17357 . . . . . . 7  |-  ( ( s  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )
)  ->  T  C_  (
s  .(+)  T ) )
2117, 19, 20syl2anc 671 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  C_  ( s  .(+)  T ) )
2253ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  S )
2315, 22sseldd 3444 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  (SubGrp `  W )
)
24 simp3r 1043 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  C_  U )
256lsmless1 17359 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )  /\  s  C_  U )  ->  ( s  .(+)  T )  C_  ( U  .(+) 
T ) )
2623, 19, 24, 25syl3anc 1276 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( U  .(+)  T ) )
27 lmodabl 18183 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Abel )
283, 27syl 17 . . . . . . . . 9  |-  ( ph  ->  W  e.  Abel )
293, 14syl 17 . . . . . . . . . 10  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
3029, 4sseldd 3444 . . . . . . . . 9  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
3129, 5sseldd 3444 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
326lsmcom 17544 . . . . . . . . 9  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )
3328, 30, 31, 32syl3anc 1276 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
34333ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3526, 34sseqtr4d 3480 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( T  .(+)  U ) )
3693ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T C ( T  .(+)  U ) )
371, 2, 3, 4, 8lcvbr3 32633 . . . . . . . . . 10  |-  ( ph  ->  ( T C ( T  .(+)  U )  <->  ( T  C.  ( T  .(+) 
U )  /\  A. r  e.  S  (
( T  C_  r  /\  r  C_  ( T 
.(+)  U ) )  -> 
( r  =  T  \/  r  =  ( T  .(+)  U )
) ) ) ) )
3837adantr 471 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S )  ->  ( T C ( T  .(+)  U )  <->  ( T  C.  ( T  .(+)  U )  /\  A. r  e.  S  ( ( T 
C_  r  /\  r  C_  ( T  .(+)  U ) )  ->  ( r  =  T  \/  r  =  ( T  .(+)  U ) ) ) ) ) )
393adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
40 simpr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
414adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  T  e.  S )
421, 6lsmcl 18354 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  T  e.  S )  ->  (
s  .(+)  T )  e.  S )
4339, 40, 41, 42syl3anc 1276 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  .(+)  T )  e.  S )
44 sseq2 3465 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( T  C_  r  <->  T  C_  ( s 
.(+)  T ) ) )
45 sseq1 3464 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  C_  ( T  .(+)  U )  <-> 
( s  .(+)  T ) 
C_  ( T  .(+)  U ) ) )
4644, 45anbi12d 722 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( ( T  C_  r  /\  r  C_  ( T  .(+)  U ) )  <->  ( T  C_  ( s  .(+)  T )  /\  ( s  .(+)  T )  C_  ( T  .(+) 
U ) ) ) )
47 eqeq1 2465 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  T  <->  ( s  .(+)  T )  =  T ) )
48 eqeq1 2465 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  ( T  .(+)  U )  <->  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) )
4947, 48orbi12d 721 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( (
r  =  T  \/  r  =  ( T  .(+) 
U ) )  <->  ( (
s  .(+)  T )  =  T  \/  ( s 
.(+)  T )  =  ( T  .(+)  U )
) ) )
5046, 49imbi12d 326 . . . . . . . . . . . 12  |-  ( r  =  ( s  .(+)  T )  ->  ( (
( T  C_  r  /\  r  C_  ( T 
.(+)  U ) )  -> 
( r  =  T  \/  r  =  ( T  .(+)  U )
) )  <->  ( ( T  C_  ( s  .(+)  T )  /\  ( s 
.(+)  T )  C_  ( T  .(+)  U ) )  ->  ( ( s 
.(+)  T )  =  T  \/  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) ) ) )
5150rspcv 3157 . . . . . . . . . . 11  |-  ( ( s  .(+)  T )  e.  S  ->  ( A. r  e.  S  (
( T  C_  r  /\  r  C_  ( T 
.(+)  U ) )  -> 
( r  =  T  \/  r  =  ( T  .(+)  U )
) )  ->  (
( T  C_  (
s  .(+)  T )  /\  ( s  .(+)  T ) 
C_  ( T  .(+)  U ) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) ) ) )
5243, 51syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  S )  ->  ( A. r  e.  S  ( ( T  C_  r  /\  r  C_  ( T  .(+)  U ) )  ->  ( r  =  T  \/  r  =  ( T  .(+)  U ) ) )  ->  (
( T  C_  (
s  .(+)  T )  /\  ( s  .(+)  T ) 
C_  ( T  .(+)  U ) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) ) ) )
5352adantld 473 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S )  ->  (
( T  C.  ( T  .(+)  U )  /\  A. r  e.  S  ( ( T  C_  r  /\  r  C_  ( T 
.(+)  U ) )  -> 
( r  =  T  \/  r  =  ( T  .(+)  U )
) ) )  -> 
( ( T  C_  ( s  .(+)  T )  /\  ( s  .(+)  T )  C_  ( T  .(+) 
U ) )  -> 
( ( s  .(+)  T )  =  T  \/  ( s  .(+)  T )  =  ( T  .(+)  U ) ) ) ) )
5438, 53sylbid 223 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S )  ->  ( T C ( T  .(+)  U )  ->  ( ( T  C_  ( s  .(+)  T )  /\  ( s 
.(+)  T )  C_  ( T  .(+)  U ) )  ->  ( ( s 
.(+)  T )  =  T  \/  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) ) ) )
55543adant3 1034 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  ( T C ( T  .(+)  U )  ->  ( ( T  C_  ( s  .(+)  T )  /\  ( s 
.(+)  T )  C_  ( T  .(+)  U ) )  ->  ( ( s 
.(+)  T )  =  T  \/  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) ) ) )
5636, 55mpd 15 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( T  C_  (
s  .(+)  T )  /\  ( s  .(+)  T ) 
C_  ( T  .(+)  U ) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) ) )
5721, 35, 56mp2and 690 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) )
58 ineq1 3638 . . . . . . 7  |-  ( ( s  .(+)  T )  =  T  ->  ( ( s  .(+)  T )  i^i  U )  =  ( T  i^i  U ) )
59 simp3l 1042 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  ( T  i^i  U )  C_  s )
601, 6, 2, 13, 18, 22, 16, 59, 24lcvexchlem2 32645 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  i^i  U )  =  s )
6160eqeq1d 2463 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( ( s  .(+)  T )  i^i  U )  =  ( T  i^i  U )  <->  s  =  ( T  i^i  U ) ) )
6258, 61syl5ib 227 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  ->  s  =  ( T  i^i  U ) ) )
63 ineq1 3638 . . . . . . 7  |-  ( ( s  .(+)  T )  =  ( T  .(+)  U )  ->  ( (
s  .(+)  T )  i^i 
U )  =  ( ( T  .(+)  U )  i^i  U ) )
646lsmub2 17357 . . . . . . . . . 10  |-  ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  ( T  .(+)  U ) )
6519, 23, 64syl2anc 671 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  C_  ( T  .(+)  U ) )
66 dfss1 3648 . . . . . . . . 9  |-  ( U 
C_  ( T  .(+)  U )  <->  ( ( T 
.(+)  U )  i^i  U
)  =  U )
6765, 66sylib 201 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( T  .(+)  U )  i^i  U )  =  U )
6860, 67eqeq12d 2476 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( ( s  .(+)  T )  i^i  U )  =  ( ( T 
.(+)  U )  i^i  U
)  <->  s  =  U ) )
6963, 68syl5ib 227 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  ( T  .(+)  U )  ->  s  =  U ) )
7062, 69orim12d 854 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( ( s  .(+)  T )  =  T  \/  ( s  .(+)  T )  =  ( T  .(+)  U ) )  ->  (
s  =  ( T  i^i  U )  \/  s  =  U ) ) )
7157, 70mpd 15 . . . 4  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  =  ( T  i^i  U )  \/  s  =  U ) )
72713exp 1214 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) ) )
7372ralrimiv 2811 . 2  |-  ( ph  ->  A. s  e.  S  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) )
741lssincl 18236 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
753, 4, 5, 74syl3anc 1276 . . 3  |-  ( ph  ->  ( T  i^i  U
)  e.  S )
761, 2, 3, 75, 5lcvbr3 32633 . 2  |-  ( ph  ->  ( ( T  i^i  U ) C U  <->  ( ( T  i^i  U )  C.  U  /\  A. s  e.  S  ( ( ( T  i^i  U ) 
C_  s  /\  s  C_  U )  ->  (
s  =  ( T  i^i  U )  \/  s  =  U ) ) ) ) )
7712, 73, 76mpbir2and 938 1  |-  ( ph  ->  ( T  i^i  U
) C U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   A.wral 2748    i^i cin 3414    C_ wss 3415    C. wpss 3416   class class class wbr 4415   ` cfv 5600  (class class class)co 6314  SubGrpcsubg 16859   LSSumclsm 17334   Abelcabl 17479   LModclmod 18139   LSubSpclss 18203    <oLL clcv 32628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-subg 16862  df-cntz 17019  df-lsm 17336  df-cmn 17480  df-abl 17481  df-mgp 17772  df-ur 17784  df-ring 17830  df-lmod 18141  df-lss 18204  df-lcv 32629
This theorem is referenced by:  lcvexch  32649  lsatcvat3  32662
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