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Theorem islshpcv 29865
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 2296 . . 3  |-  ( LSSum `  W )  =  (
LSSum `  W )
4 islshpcv.h . . 3  |-  H  =  (LSHyp `  W )
5 eqid 2296 . . 3  |-  (LSAtoms `  W
)  =  (LSAtoms `  W
)
6 islshpcv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
7 lveclmod 15875 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 15 . . 3  |-  ( ph  ->  W  e.  LMod )
91, 2, 3, 4, 5, 8islshpat 29829 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V ) ) )
10 simp12 986 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  e.  S )
111, 2lssss 15710 . . . . . . . . . . . 12  |-  ( U  e.  S  ->  U  C_  V )
1210, 11syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  C_  V )
13 simp13 987 . . . . . . . . . . 11  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  =/=  V )
14 df-pss 3181 . . . . . . . . . . 11  |-  ( U 
C.  V  <->  ( U  C_  V  /\  U  =/= 
V ) )
1512, 13, 14sylanbrc 645 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  C.  V )
16 psseq2 3277 . . . . . . . . . . 11  |-  ( ( U ( LSSum `  W
) q )  =  V  ->  ( U  C.  ( U ( LSSum `  W ) q )  <-> 
U  C.  V )
)
17163ad2ant3 978 . . . . . . . . . 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 223 . . . . . . . . 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 976 . . . . . . . . . . 11  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  W  e.  LVec )
21203ad2ant1 976 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  W  e.  LVec )
22 simp2 956 . . . . . . . . . 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 29854 . . . . . . . . 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 201 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U C ( U (
LSSum `  W ) q ) )
25 simp3 957 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U ( LSSum `  W ) q )  =  V )
2624, 25breqtrd 4063 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U C V )
2710, 26jca 518 . . . . . 6  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U  e.  S  /\  U C V ) )
2827rexlimdv3a 2682 . . . . 5  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  ( E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V  -> 
( U  e.  S  /\  U C V ) ) )
29283exp 1150 . . . 4  |-  ( ph  ->  ( U  e.  S  ->  ( U  =/=  V  ->  ( E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V  ->  ( U  e.  S  /\  U C V ) ) ) ) )
30293impd 1165 . . 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 732 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  e.  S )
326adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  W  e.  LVec )
338adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  W  e.  LMod )
341, 2lss1 15712 . . . . . . . 8  |-  ( W  e.  LMod  ->  V  e.  S )
3533, 34syl 15 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  V  e.  S )
36 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U C V )
372, 19, 32, 31, 35, 36lcvpss 29836 . . . . . 6  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  C.  V )
38 pssne 3285 . . . . . 6  |-  ( U 
C.  V  ->  U  =/=  V )
3937, 38syl 15 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  =/=  V )
402, 3, 5, 19, 33, 31, 35, 36lcvat 29842 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V )
4131, 39, 403jca 1132 . . . 4  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  -> 
( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V ) )
4241ex 423 . . 3  |-  ( ph  ->  ( ( U  e.  S  /\  U C V )  ->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V ) ) )
4330, 42impbid 183 . 2  |-  ( ph  ->  ( ( U  e.  S  /\  U  =/= 
V  /\  E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V )  <->  ( U  e.  S  /\  U C V ) ) )
449, 43bitrd 244 1  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U C V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165    C. wpss 3166   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LVecclvec 15871  LSAtomsclsa 29786  LSHypclsh 29787    <oLL clcv 29830
This theorem is referenced by:  l1cvpat  29866  lshpat  29868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lcv 29831
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