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Theorem lcvexchlem4 29227
Description: Lemma for lcvexch 29229. (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 15836 . . . . 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 29214 . . 3  |-  ( ph  ->  T  C.  ( T 
.(+)  U ) )
111, 6, 2, 3, 4, 5lcvexchlem1 29224 . . 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 15715 . . . . . . . . 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 3181 . . . . . . 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 3181 . . . . . . 7  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  T  e.  (SubGrp `  W )
)
206lsmub2 14968 . . . . . . 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 3181 . . . . . . . 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 14970 . . . . . . . 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 15672 . . . . . . . . . 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 3181 . . . . . . . . 9  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
3129, 5sseldd 3181 . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
326lsmcom 15150 . . . . . . . . 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 3215 . . . . . 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 29213 . . . . . . . . . 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 15836 . . . . . . . . . . . 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 3200 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( T  C_  r  <->  T  C_  ( s 
.(+)  T ) ) )
45 sseq1 3199 . . . . . . . . . . . . . 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 2289 . . . . . . . . . . . . . 14  |-  ( r  =  ( s  .(+)  T )  ->  ( r  =  T  <->  ( s  .(+)  T )  =  T ) )
48 eqeq1 2289 . . . . . . . . . . . . . 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 2880 . . . . . . . . . . 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 3363 . . . . . . 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 29225 . . . . . . . 8  |-  ( (
ph  /\  s  e.  S  /\  ( ( T  i^i  U )  C_  s  /\  s  C_  U
) )  ->  (
( s  .(+)  T )  i^i  U )  =  s )
6160eqeq1d 2291 . . . . . . 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 3363 . . . . . . 7  |-  ( ( s  .(+)  T )  =  ( T  .(+)  U )  ->  ( (
s  .(+)  T )  i^i 
U )  =  ( ( T  .(+)  U )  i^i  U ) )
646lsmub2 14968 . . . . . . . . . 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 3373 . . . . . . . . 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 2297 . . . . . . 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 2625 . 2  |-  ( ph  ->  A. s  e.  S  ( ( ( T  i^i  U )  C_  s  /\  s  C_  U
)  ->  ( s  =  ( T  i^i  U )  \/  s  =  U ) ) )
741lssincl 15722 . . . 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 29213 . 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 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152    C. wpss 3153   class class class wbr 4023   ` cfv 5255  (class class class)co 5858  SubGrpcsubg 14615   LSSumclsm 14945   Abelcabel 15090   LModclmod 15627   LSubSpclss 15689    <oLL clcv 29208
This theorem is referenced by:  lcvexch  29229  lsatcvat3  29242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lcv 29209
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