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Theorem islshpcv 34251
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 2467 . . 3  |-  ( LSSum `  W )  =  (
LSSum `  W )
4 islshpcv.h . . 3  |-  H  =  (LSHyp `  W )
5 eqid 2467 . . 3  |-  (LSAtoms `  W
)  =  (LSAtoms `  W
)
6 islshpcv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
7 lveclmod 17623 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 16 . . 3  |-  ( ph  ->  W  e.  LMod )
91, 2, 3, 4, 5, 8islshpat 34215 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V ) ) )
10 simp12 1027 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  e.  S )
111, 2lssss 17454 . . . . . . . . . . . 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 1028 . . . . . . . . . . 11  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  =/=  V )
14 df-pss 3497 . . . . . . . . . . 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 3597 . . . . . . . . . . 11  |-  ( ( U ( LSSum `  W
) q )  =  V  ->  ( U  C.  ( U ( LSSum `  W ) q )  <-> 
U  C.  V )
)
17163ad2ant3 1019 . . . . . . . . . 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 1017 . . . . . . . . . . 11  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  W  e.  LVec )
21203ad2ant1 1017 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  W  e.  LVec )
22 simp2 997 . . . . . . . . . 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 34240 . . . . . . . . 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 998 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U ( LSSum `  W ) q )  =  V )
2624, 25breqtrd 4477 . . . . . . 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 2961 . . . . 5  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  ( E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V  -> 
( U  e.  S  /\  U C V ) ) )
29283exp 1195 . . . 4  |-  ( ph  ->  ( U  e.  S  ->  ( U  =/=  V  ->  ( E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V  ->  ( U  e.  S  /\  U C V ) ) ) ) )
30293impd 1210 . . 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 17456 . . . . . . . 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 34222 . . . . . 6  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  C.  V )
3837pssned 3607 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  =/=  V )
392, 3, 5, 19, 33, 31, 35, 36lcvat 34228 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V )
4031, 38, 393jca 1176 . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818    C_ wss 3481    C. wpss 3482   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   LSSumclsm 16527   LModclmod 17383   LSubSpclss 17449   LVecclvec 17619  LSAtomsclsa 34172  LSHypclsh 34173    <oLL clcv 34216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-cntz 16227  df-lsm 16529  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-lsatoms 34174  df-lshyp 34175  df-lcv 34217
This theorem is referenced by:  l1cvpat  34252  lshpat  34254
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