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Theorem lcvexchlem4 29045
Description: Lemma for lcvexch 29047. (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 15885 . . . . 5  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
83, 4, 5, 7syl3anc 1182 . . . 4  |-  ( ph  ->  ( T  .(+)  U )  e.  S )
9 lcvexch.f . . . 4  |-  ( ph  ->  T C ( T 
.(+)  U ) )
101, 2, 3, 4, 8, 9lcvpss 29032 . . 3  |-  ( ph  ->  T  C.  ( T 
.(+)  U ) )
111, 6, 2, 3, 4, 5lcvexchlem1 29042 . . 3  |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U
) )
1210, 11mpbid 201 . 2  |-  ( ph  ->  ( T  i^i  U
)  C.  U )
1333ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  W  e.  LMod )
141lsssssubg 15764 . . . . . . . . 9  |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
1513, 14syl 15 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  S  C_  (SubGrp `  W )
)
16 simp2 956 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  S )
1715, 16sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  e.  (SubGrp `  W )
)
1843ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  S )
1915, 18sseldd 3215 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  (SubGrp `  W )
)
206lsmub2 15017 . . . . . . 7  |-  ( ( s  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )
)  ->  T  C_  (
s  .(+)  T ) )
2117, 19, 20syl2anc 642 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  C_  ( s  .(+)  T ) )
2253ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  S )
2315, 22sseldd 3215 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  e.  (SubGrp `  W )
)
24 simp3r 984 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  s  C_  U )
256lsmless1 15019 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  W )  /\  T  e.  (SubGrp `  W )  /\  s  C_  U )  ->  ( s  .(+)  T )  C_  ( U  .(+) 
T ) )
2623, 19, 24, 25syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( U  .(+)  T ) )
27 lmodabl 15721 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  W  e. 
Abel )
283, 27syl 15 . . . . . . . . 9  |-  ( ph  ->  W  e.  Abel )
293, 14syl 15 . . . . . . . . . 10  |-  ( ph  ->  S  C_  (SubGrp `  W
) )
3029, 4sseldd 3215 . . . . . . . . 9  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
3129, 5sseldd 3215 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
326lsmcom 15199 . . . . . . . . 9  |-  ( ( W  e.  Abel  /\  T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W ) )  -> 
( T  .(+)  U )  =  ( U  .(+)  T ) )
3328, 30, 31, 32syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
34333ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
3526, 34sseqtr4d 3249 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  .(+)  T )  C_  ( T  .(+)  U ) )
3693ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T C ( T  .(+)  U ) )
371, 2, 3, 4, 8lcvbr3 29031 . . . . . . . . . 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 451 . . . . . . . . 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 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  W  e.  LMod )
40 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  s  e.  S )
414adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  S )  ->  T  e.  S )
421, 6lsmcl 15885 . . . . . . . . . . . 12  |-  ( ( W  e.  LMod  /\  s  e.  S  /\  T  e.  S )  ->  (
s  .(+)  T )  e.  S )
4339, 40, 41, 42syl3anc 1182 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  S )  ->  (
s  .(+)  T )  e.  S )
44 sseq2 3234 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( T  C_  r  <->  T  C_  ( s 
.(+)  T ) ) )
45 sseq1 3233 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  C_  ( T  .(+)  U )  <-> 
( s  .(+)  T ) 
C_  ( T  .(+)  U ) ) )
4644, 45anbi12d 691 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( ( T  C_  r  /\  r  C_  ( T  .(+)  U ) )  <->  ( T  C_  ( s  .(+)  T )  /\  ( s  .(+)  T )  C_  ( T  .(+) 
U ) ) ) )
47 eqeq1 2322 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  T  <->  ( s  .(+)  T )  =  T ) )
48 eqeq1 2322 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  ( T  .(+)  U )  <->  ( s  .(+)  T )  =  ( T 
.(+)  U ) ) )
4947, 48orbi12d 690 . . . . . . . . . . . . 13  |-  ( r  =  ( s  .(+)  T )  ->  ( (
r  =  T  \/  r  =  ( T  .(+) 
U ) )  <->  ( (
s  .(+)  T )  =  T  \/  ( s 
.(+)  T )  =  ( T  .(+)  U )
) ) )
5046, 49imbi12d 311 . . . . . . . . . . . 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 2914 . . . . . . . . . . 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 15 . . . . . . . . . 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 453 . . . . . . . . 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 206 . . . . . . . 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 975 . . . . . . 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 14 . . . . . 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 660 . . . . 5  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  \/  (
s  .(+)  T )  =  ( T  .(+)  U ) ) )
58 ineq1 3397 . . . . . . 7  |-  ( ( s  .(+)  T )  =  T  ->  ( ( s  .(+)  T )  i^i  U )  =  ( T  i^i  U ) )
59 simp3l 983 . . . . . . . . 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 29043 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  i^i  U )  =  s )
6160eqeq1d 2324 . . . . . . 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 210 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  T  ->  s  =  ( T  i^i  U ) ) )
63 ineq1 3397 . . . . . . 7  |-  ( ( s  .(+)  T )  =  ( T  .(+)  U )  ->  ( (
s  .(+)  T )  i^i 
U )  =  ( ( T  .(+)  U )  i^i  U ) )
646lsmub2 15017 . . . . . . . . . 10  |-  ( ( T  e.  (SubGrp `  W )  /\  U  e.  (SubGrp `  W )
)  ->  U  C_  ( T  .(+)  U ) )
6519, 23, 64syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  U  C_  ( T  .(+)  U ) )
66 dfss1 3407 . . . . . . . . 9  |-  ( U 
C_  ( T  .(+)  U )  <->  ( ( T 
.(+)  U )  i^i  U
)  =  U )
6765, 66sylib 188 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( T  .(+)  U )  i^i  U )  =  U )
6860, 67eqeq12d 2330 . . . . . . 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 210 . . . . . 6  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  =  ( T  .(+)  U )  ->  s  =  U ) )
7062, 69orim12d 811 . . . . 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 14 . . . 4  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
s  =  ( T  i^i  U )  \/  s  =  U ) )
72713exp 1150 . . 3  |-  ( ph  ->  ( s  e.  S  ->  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) ) )
7372ralrimiv 2659 . 2  |-  ( ph  ->  A. s  e.  S  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) )
741lssincl 15771 . . . 4  |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
753, 4, 5, 74syl3anc 1182 . . 3  |-  ( ph  ->  ( T  i^i  U
)  e.  S )
761, 2, 3, 75, 5lcvbr3 29031 . 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 888 1  |-  ( ph  ->  ( T  i^i  U
) C U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577    i^i cin 3185    C_ wss 3186    C. wpss 3187   class class class wbr 4060   ` cfv 5292  (class class class)co 5900  SubGrpcsubg 14664   LSSumclsm 14994   Abelcabel 15139   LModclmod 15676   LSubSpclss 15738    <oLL clcv 29026
This theorem is referenced by:  lcvexch  29047  lsatcvat3  29060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-0g 13453  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-sbg 14540  df-subg 14667  df-cntz 14842  df-lsm 14996  df-cmn 15140  df-abl 15141  df-mgp 15375  df-rng 15389  df-ur 15391  df-lmod 15678  df-lss 15739  df-lcv 29027
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