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Theorem lcvexchlem4 32778
Description: Lemma for lcvexch 32780. (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 17186 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
83, 4, 5, 7syl3anc 1218 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
9 lcvexch.f . . . 4  |-  ( ph  ->  T C ( T 
.(+)  U ) )
101, 2, 3, 4, 8, 9lcvpss 32765 . . 3  |-  ( ph  ->  T  C.  ( T  .(+) 
U ) )
111, 6, 2, 3, 4, 5lcvexchlem1 32775 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
1210, 11mpbid 210 . 2  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
1333ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  W  e.  LMod )
141lsssssubg 17061 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
1513, 14syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  S  C_  (SubGrp `  W )
)
16 simp2 989 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  S )
1715, 16sseldd 3378 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  (SubGrp `  W )
)
1843ad2ant1 1009 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  S )
1915, 18sseldd 3378 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  (SubGrp `  W )
)
206lsmub2 16177 . . . . . . 7  |-  ( ( s  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )
)  ->  T  C_  (
s  .(+)  T ) )
2117, 19, 20syl2anc 661 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  C_  ( s  .(+)  T ) )
2253ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  S )
2315, 22sseldd 3378 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  (SubGrp `  W )
)
24 simp3r 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  C_  U )
256lsmless1 16179 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )  /\  s  C_  U )  ->  ( s  .(+)  T )  C_  ( U  .(+) 
T ) )
2623, 19, 24, 25syl3anc 1218 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( U  .(+)  T ) )
27 lmodabl 17014 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Abel )
283, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  Abel )
293, 14syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
3029, 4sseldd 3378 . . . . . . . . 9  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
3129, 5sseldd 3378 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
326lsmcom 16361 . . . . . . . . 9  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )
3328, 30, 31, 32syl3anc 1218 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
34333ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3526, 34sseqtr4d 3414 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( T  .(+)  U ) )
3693ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T C ( T  .(+)  U ) )
371, 2, 3, 4, 8lcvbr3 32764 . . . . . . . . . 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 465 . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
40 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
414adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  T  e.  S )
421, 6lsmcl 17186 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  T  e.  S )  ->  (
s  .(+)  T )  e.  S )
4339, 40, 41, 42syl3anc 1218 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  .(+)  T )  e.  S )
44 sseq2 3399 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( T  C_  r  <->  T  C_  ( s 
.(+)  T ) ) )
45 sseq1 3398 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  C_  ( T  .(+)  U )  <-> 
( s  .(+)  T ) 
C_  ( T  .(+)  U ) ) )
4644, 45anbi12d 710 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( ( T  C_  r  /\  r  C_  ( T  .(+)  U ) )  <->  ( T  C_  ( s  .(+)  T )  /\  ( s  .(+)  T )  C_  ( T  .(+) 
U ) ) ) )
47 eqeq1 2449 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  T  <->  ( s  .(+)  T )  =  T ) )
48 eqeq1 2449 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  ( T  .(+)  U )  <->  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) )
4947, 48orbi12d 709 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( (
r  =  T  \/  r  =  ( T  .(+) 
U ) )  <->  ( (
s  .(+)  T )  =  T  \/  ( s 
.(+)  T )  =  ( T  .(+)  U )
) ) )
5046, 49imbi12d 320 . . . . . . . . . . . 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 3090 . . . . . . . . . . 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 16 . . . . . . . . . 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 467 . . . . . . . . 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 215 . . . . . . . 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 1008 . . . . . . 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 679 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) )
58 ineq1 3566 . . . . . . 7  |-  ( ( s  .(+)  T )  =  T  ->  ( ( s  .(+)  T )  i^i  U )  =  ( T  i^i  U ) )
59 simp3l 1016 . . . . . . . . 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 32776 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  i^i  U )  =  s )
6160eqeq1d 2451 . . . . . . 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 219 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  ->  s  =  ( T  i^i  U ) ) )
63 ineq1 3566 . . . . . . 7  |-  ( ( s  .(+)  T )  =  ( T  .(+)  U )  ->  ( (
s  .(+)  T )  i^i 
U )  =  ( ( T  .(+)  U )  i^i  U ) )
646lsmub2 16177 . . . . . . . . . 10  |-  ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  ( T  .(+)  U ) )
6519, 23, 64syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  C_  ( T  .(+)  U ) )
66 dfss1 3576 . . . . . . . . 9  |-  ( U 
C_  ( T  .(+)  U )  <->  ( ( T 
.(+)  U )  i^i  U
)  =  U )
6765, 66sylib 196 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( T  .(+)  U )  i^i  U )  =  U )
6860, 67eqeq12d 2457 . . . . . . 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 219 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  ( T  .(+)  U )  ->  s  =  U ) )
7062, 69orim12d 834 . . . . 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 1186 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) ) )
7372ralrimiv 2819 . 2  |-  ( ph  ->  A. s  e.  S  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) )
741lssincl 17068 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
753, 4, 5, 74syl3anc 1218 . . 3  |-  ( ph  ->  ( T  i^i  U
)  e.  S )
761, 2, 3, 75, 5lcvbr3 32764 . 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 913 1  |-  ( ph  ->  ( T  i^i  U
) C 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 2736    i^i cin 3348    C_ wss 3349    C. wpss 3350   class class class wbr 4313   ` cfv 5439  (class class class)co 6112  SubGrpcsubg 15696   LSSumclsm 16154   Abelcabel 16299   LModclmod 16970   LSubSpclss 17035    <oLL clcv 32759
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-lcv 32760
This theorem is referenced by:  lcvexch  32780  lsatcvat3  32793
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