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Theorem islshpcv 32538
Description: Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
Hypotheses
Ref Expression
islshpcv.v  |-  V  =  ( Base `  W
)
islshpcv.s  |-  S  =  ( LSubSp `  W )
islshpcv.h  |-  H  =  (LSHyp `  W )
islshpcv.c  |-  C  =  (  <oLL  `  W )
islshpcv.w  |-  ( ph  ->  W  e.  LVec )
Assertion
Ref Expression
islshpcv  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U C V ) ) )

Proof of Theorem islshpcv
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 islshpcv.v . . 3  |-  V  =  ( Base `  W
)
2 islshpcv.s . . 3  |-  S  =  ( LSubSp `  W )
3 eqid 2438 . . 3  |-  ( LSSum `  W )  =  (
LSSum `  W )
4 islshpcv.h . . 3  |-  H  =  (LSHyp `  W )
5 eqid 2438 . . 3  |-  (LSAtoms `  W
)  =  (LSAtoms `  W
)
6 islshpcv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
7 lveclmod 17164 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 16 . . 3  |-  ( ph  ->  W  e.  LMod )
91, 2, 3, 4, 5, 8islshpat 32502 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V ) ) )
10 simp12 1019 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  e.  S )
111, 2lssss 16995 . . . . . . . . . . . 12  |-  ( U  e.  S  ->  U  C_  V )
1210, 11syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  C_  V )
13 simp13 1020 . . . . . . . . . . 11  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  =/=  V )
14 df-pss 3339 . . . . . . . . . . 11  |-  ( U 
C.  V  <->  ( U  C_  V  /\  U  =/= 
V ) )
1512, 13, 14sylanbrc 664 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  C.  V )
16 psseq2 3439 . . . . . . . . . . 11  |-  ( ( U ( LSSum `  W
) q )  =  V  ->  ( U  C.  ( U ( LSSum `  W ) q )  <-> 
U  C.  V )
)
17163ad2ant3 1011 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U  C.  ( U ( LSSum `  W
) q )  <->  U  C.  V
) )
1815, 17mpbird 232 . . . . . . . . 9  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  C.  ( U (
LSSum `  W ) q ) )
19 islshpcv.c . . . . . . . . . 10  |-  C  =  (  <oLL  `  W )
2063ad2ant1 1009 . . . . . . . . . . 11  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  W  e.  LVec )
21203ad2ant1 1009 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  W  e.  LVec )
22 simp2 989 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
q  e.  (LSAtoms `  W
) )
232, 3, 5, 19, 21, 10, 22lcv2 32527 . . . . . . . . 9  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U  C.  ( U ( LSSum `  W
) q )  <->  U C
( U ( LSSum `  W ) q ) ) )
2418, 23mpbid 210 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U C ( U (
LSSum `  W ) q ) )
25 simp3 990 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U ( LSSum `  W ) q )  =  V )
2624, 25breqtrd 4311 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U C V )
2710, 26jca 532 . . . . . 6  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U  e.  S  /\  U C V ) )
2827rexlimdv3a 2838 . . . . 5  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  ( E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V  -> 
( U  e.  S  /\  U C V ) ) )
29283exp 1186 . . . 4  |-  ( ph  ->  ( U  e.  S  ->  ( U  =/=  V  ->  ( E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V  ->  ( U  e.  S  /\  U C V ) ) ) ) )
30293impd 1201 . . 3  |-  ( ph  ->  ( ( U  e.  S  /\  U  =/= 
V  /\  E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V )  -> 
( U  e.  S  /\  U C V ) ) )
31 simprl 755 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  e.  S )
326adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  W  e.  LVec )
338adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  W  e.  LMod )
341, 2lss1 16997 . . . . . . . 8  |-  ( W  e.  LMod  ->  V  e.  S )
3533, 34syl 16 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  V  e.  S )
36 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U C V )
372, 19, 32, 31, 35, 36lcvpss 32509 . . . . . 6  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  C.  V )
3837pssned 3449 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  =/=  V )
392, 3, 5, 19, 33, 31, 35, 36lcvat 32515 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V )
4031, 38, 393jca 1168 . . . 4  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  -> 
( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V ) )
4140ex 434 . . 3  |-  ( ph  ->  ( ( U  e.  S  /\  U C V )  ->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V ) ) )
4230, 41impbid 191 . 2  |-  ( ph  ->  ( ( U  e.  S  /\  U  =/= 
V  /\  E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V )  <->  ( U  e.  S  /\  U C V ) ) )
439, 42bitrd 253 1  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U C V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711    C_ wss 3323    C. wpss 3324   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   LSSumclsm 16124   LModclmod 16926   LSubSpclss 16990   LVecclvec 17160  LSAtomsclsa 32459  LSHypclsh 32460    <oLL clcv 32503
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-drng 16812  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lvec 17161  df-lsatoms 32461  df-lshyp 32462  df-lcv 32504
This theorem is referenced by:  l1cvpat  32539  lshpat  32541
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