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Theorem lcvexchlem4 29835
Description: Lemma for lcvexch 29837. (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 16155 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
83, 4, 5, 7syl3anc 1184 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
9 lcvexch.f . . . 4  |-  ( ph  ->  T C ( T 
.(+)  U ) )
101, 2, 3, 4, 8, 9lcvpss 29822 . . 3  |-  ( ph  ->  T  C.  ( T 
.(+)  U ) )
111, 6, 2, 3, 4, 5lcvexchlem1 29832 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
1210, 11mpbid 202 . 2  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
1333ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  W  e.  LMod )
141lsssssubg 16034 . . . . . . . . 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 958 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  S )
1715, 16sseldd 3349 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  (SubGrp `  W )
)
1843ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  S )
1915, 18sseldd 3349 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  (SubGrp `  W )
)
206lsmub2 15291 . . . . . . 7  |-  ( ( s  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )
)  ->  T  C_  (
s  .(+)  T ) )
2117, 19, 20syl2anc 643 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  C_  ( s  .(+)  T ) )
2253ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  S )
2315, 22sseldd 3349 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  (SubGrp `  W )
)
24 simp3r 986 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  C_  U )
256lsmless1 15293 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )  /\  s  C_  U )  ->  ( s  .(+)  T )  C_  ( U  .(+) 
T ) )
2623, 19, 24, 25syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( U  .(+)  T ) )
27 lmodabl 15991 . . . . . . . . . 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 3349 . . . . . . . . 9  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
3129, 5sseldd 3349 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
326lsmcom 15473 . . . . . . . . 9  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )
3328, 30, 31, 32syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
34333ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3526, 34sseqtr4d 3385 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( T  .(+)  U ) )
3693ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T C ( T  .(+)  U ) )
371, 2, 3, 4, 8lcvbr3 29821 . . . . . . . . . 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 452 . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
40 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
414adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  T  e.  S )
421, 6lsmcl 16155 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  T  e.  S )  ->  (
s  .(+)  T )  e.  S )
4339, 40, 41, 42syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  .(+)  T )  e.  S )
44 sseq2 3370 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( T  C_  r  <->  T  C_  ( s 
.(+)  T ) ) )
45 sseq1 3369 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  C_  ( T  .(+)  U )  <-> 
( s  .(+)  T ) 
C_  ( T  .(+)  U ) ) )
4644, 45anbi12d 692 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( ( T  C_  r  /\  r  C_  ( T  .(+)  U ) )  <->  ( T  C_  ( s  .(+)  T )  /\  ( s  .(+)  T )  C_  ( T  .(+) 
U ) ) ) )
47 eqeq1 2442 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  T  <->  ( s  .(+)  T )  =  T ) )
48 eqeq1 2442 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  ( T  .(+)  U )  <->  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) )
4947, 48orbi12d 691 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( (
r  =  T  \/  r  =  ( T  .(+) 
U ) )  <->  ( (
s  .(+)  T )  =  T  \/  ( s 
.(+)  T )  =  ( T  .(+)  U )
) ) )
5046, 49imbi12d 312 . . . . . . . . . . . 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 3048 . . . . . . . . . . 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 454 . . . . . . . . 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 207 . . . . . . . 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 977 . . . . . . 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 661 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) )
58 ineq1 3535 . . . . . . 7  |-  ( ( s  .(+)  T )  =  T  ->  ( ( s  .(+)  T )  i^i  U )  =  ( T  i^i  U ) )
59 simp3l 985 . . . . . . . . 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 29833 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  i^i  U )  =  s )
6160eqeq1d 2444 . . . . . . 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 211 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  ->  s  =  ( T  i^i  U ) ) )
63 ineq1 3535 . . . . . . 7  |-  ( ( s  .(+)  T )  =  ( T  .(+)  U )  ->  ( (
s  .(+)  T )  i^i 
U )  =  ( ( T  .(+)  U )  i^i  U ) )
646lsmub2 15291 . . . . . . . . . 10  |-  ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  ( T  .(+)  U ) )
6519, 23, 64syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  C_  ( T  .(+)  U ) )
66 dfss1 3545 . . . . . . . . 9  |-  ( U 
C_  ( T  .(+)  U )  <->  ( ( T 
.(+)  U )  i^i  U
)  =  U )
6765, 66sylib 189 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( T  .(+)  U )  i^i  U )  =  U )
6860, 67eqeq12d 2450 . . . . . . 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 211 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  ( T  .(+)  U )  ->  s  =  U ) )
7062, 69orim12d 812 . . . . 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 1152 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) ) )
7372ralrimiv 2788 . 2  |-  ( ph  ->  A. s  e.  S  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) )
741lssincl 16041 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
753, 4, 5, 74syl3anc 1184 . . 3  |-  ( ph  ->  ( T  i^i  U
)  e.  S )
761, 2, 3, 75, 5lcvbr3 29821 . 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 889 1  |-  ( ph  ->  ( T  i^i  U
) C U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320    C. wpss 3321   class class class wbr 4212   ` cfv 5454  (class class class)co 6081  SubGrpcsubg 14938   LSSumclsm 15268   Abelcabel 15413   LModclmod 15950   LSubSpclss 16008    <oLL clcv 29816
This theorem is referenced by:  lcvexch  29837  lsatcvat3  29850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lcv 29817
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