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Theorem islshpcv 29913
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 16180 . . . 4  |-  ( W  e.  LVec  ->  W  e. 
LMod )
86, 7syl 16 . . 3  |-  ( ph  ->  W  e.  LMod )
91, 2, 3, 4, 5, 8islshpat 29877 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V ) ) )
10 simp12 989 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  e.  S )
111, 2lssss 16015 . . . . . . . . . . . 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 990 . . . . . . . . . . 11  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  =/=  V )
14 df-pss 3338 . . . . . . . . . . 11  |-  ( U 
C.  V  <->  ( U  C_  V  /\  U  =/= 
V ) )
1512, 13, 14sylanbrc 647 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U  C.  V )
16 psseq2 3437 . . . . . . . . . . 11  |-  ( ( U ( LSSum `  W
) q )  =  V  ->  ( U  C.  ( U ( LSSum `  W ) q )  <-> 
U  C.  V )
)
17163ad2ant3 981 . . . . . . . . . 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 225 . . . . . . . . 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 979 . . . . . . . . . . 11  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  W  e.  LVec )
21203ad2ant1 979 . . . . . . . . . 10  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  W  e.  LVec )
22 simp2 959 . . . . . . . . . 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 29902 . . . . . . . . 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 203 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U C ( U (
LSSum `  W ) q ) )
25 simp3 960 . . . . . . . 8  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U ( LSSum `  W ) q )  =  V )
2624, 25breqtrd 4238 . . . . . . 7  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  ->  U C V )
2710, 26jca 520 . . . . . 6  |-  ( ( ( ph  /\  U  e.  S  /\  U  =/= 
V )  /\  q  e.  (LSAtoms `  W )  /\  ( U ( LSSum `  W ) q )  =  V )  -> 
( U  e.  S  /\  U C V ) )
2827rexlimdv3a 2834 . . . . 5  |-  ( (
ph  /\  U  e.  S  /\  U  =/=  V
)  ->  ( E. q  e.  (LSAtoms `  W
) ( U (
LSSum `  W ) q )  =  V  -> 
( U  e.  S  /\  U C V ) ) )
29283exp 1153 . . . 4  |-  ( ph  ->  ( U  e.  S  ->  ( U  =/=  V  ->  ( E. q  e.  (LSAtoms `  W )
( U ( LSSum `  W ) q )  =  V  ->  ( U  e.  S  /\  U C V ) ) ) ) )
30293impd 1168 . . 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 734 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  e.  S )
326adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  W  e.  LVec )
338adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  W  e.  LMod )
341, 2lss1 16017 . . . . . . . 8  |-  ( W  e.  LMod  ->  V  e.  S )
3533, 34syl 16 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  V  e.  S )
36 simprr 735 . . . . . . 7  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U C V )
372, 19, 32, 31, 35, 36lcvpss 29884 . . . . . 6  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  C.  V )
3837pssned 3447 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  U  =/=  V )
392, 3, 5, 19, 33, 31, 35, 36lcvat 29890 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  ->  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V )
4031, 38, 393jca 1135 . . . 4  |-  ( (
ph  /\  ( U  e.  S  /\  U C V ) )  -> 
( U  e.  S  /\  U  =/=  V  /\  E. q  e.  (LSAtoms `  W ) ( U ( LSSum `  W )
q )  =  V ) )
4140ex 425 . . 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 185 . 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 246 1  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U C V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    C_ wss 3322    C. wpss 3323   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   LSSumclsm 15270   LModclmod 15952   LSubSpclss 16010   LVecclvec 16176  LSAtomsclsa 29834  LSHypclsh 29835    <oLL clcv 29878
This theorem is referenced by:  l1cvpat  29914  lshpat  29916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-drng 15839  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lvec 16177  df-lsatoms 29836  df-lshyp 29837  df-lcv 29879
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