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Theorem islshpcv 29243
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 2283 . . 3  |-  ( LSSum `  W )  =  (
LSSum `  W )
4 islshpcv.h . . 3  |-  H  =  (LSHyp `  W )
5 eqid 2283 . . 3  |-  (LSAtoms `  W
)  =  (LSAtoms `  W
)
6 islshpcv.w . . . 4  |-  ( ph  ->  W  e.  LVec )
7 lveclmod 15859 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 15 . . 3  |-  ( ph  ->  W  e.  LMod )
91, 2, 3, 4, 5, 8islshpat 29207 . 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 15694 . . . . . . . . . . . 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 3168 . . . . . . . . . . 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 3264 . . . . . . . . . . 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 29232 . . . . . . . . 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 4047 . . . . . . 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 2669 . . . . 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 15696 . . . . . . . 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 29214 . . . . . 6  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  C.  V )
38 pssne 3272 . . . . . 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 29220 . . . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152    C. wpss 3153   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LVecclvec 15855  LSAtomsclsa 29164  LSHypclsh 29165    <oLL clcv 29208
This theorem is referenced by:  l1cvpat  29244  lshpat  29246
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-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-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  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-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209
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