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Theorem islshpsm 29792
Description: Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
islshpsm.v  |-  V  =  ( Base `  W
)
islshpsm.n  |-  N  =  ( LSpan `  W )
islshpsm.s  |-  S  =  ( LSubSp `  W )
islshpsm.p  |-  .(+)  =  (
LSSum `  W )
islshpsm.h  |-  H  =  (LSHyp `  W )
islshpsm.w  |-  ( ph  ->  W  e.  LMod )
Assertion
Ref Expression
islshpsm  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) ) )
Distinct variable groups:    v, S    v, U    v, V    v, W    ph, v
Allowed substitution hints:    .(+) ( v)    H( v)    N( v)

Proof of Theorem islshpsm
StepHypRef Expression
1 islshpsm.w . . 3  |-  ( ph  ->  W  e.  LMod )
2 islshpsm.v . . . 4  |-  V  =  ( Base `  W
)
3 islshpsm.n . . . 4  |-  N  =  ( LSpan `  W )
4 islshpsm.s . . . 4  |-  S  =  ( LSubSp `  W )
5 islshpsm.h . . . 4  |-  H  =  (LSHyp `  W )
62, 3, 4, 5islshp 29791 . . 3  |-  ( W  e.  LMod  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) ) )
71, 6syl 15 . 2  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) ) )
81ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  W  e.  LMod )
9 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  U  e.  S )
104, 3lspid 15755 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
118, 9, 10syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( N `  U )  =  U )
1211uneq1d 3341 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( ( N `  U )  u.  ( N `  {
v } ) )  =  ( U  u.  ( N `  { v } ) ) )
1312fveq2d 5545 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( N `  ( ( N `  U )  u.  ( N `  { v } ) ) )  =  ( N `  ( U  u.  ( N `  { v } ) ) ) )
142, 4lssss 15710 . . . . . . . . . 10  |-  ( U  e.  S  ->  U  C_  V )
159, 14syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  U  C_  V
)
16 snssi 3775 . . . . . . . . . 10  |-  ( v  e.  V  ->  { v }  C_  V )
1716adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  { v }  C_  V )
182, 3lspun 15760 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  C_  V  /\  { v }  C_  V )  ->  ( N `  ( U  u.  { v } ) )  =  ( N `  (
( N `  U
)  u.  ( N `
 { v } ) ) ) )
198, 15, 17, 18syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( N `  ( U  u.  {
v } ) )  =  ( N `  ( ( N `  U )  u.  ( N `  { v } ) ) ) )
202, 4, 3lspcl 15749 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  {
v }  C_  V
)  ->  ( N `  { v } )  e.  S )
218, 17, 20syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( N `  { v } )  e.  S )
22 islshpsm.p . . . . . . . . . 10  |-  .(+)  =  (
LSSum `  W )
234, 3, 22lsmsp 15855 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  ( N `  { v } )  e.  S
)  ->  ( U  .(+) 
( N `  {
v } ) )  =  ( N `  ( U  u.  ( N `  { v } ) ) ) )
248, 9, 21, 23syl3anc 1182 . . . . . . . 8  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( U  .(+) 
( N `  {
v } ) )  =  ( N `  ( U  u.  ( N `  { v } ) ) ) )
2513, 19, 243eqtr4rd 2339 . . . . . . 7  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( U  .(+) 
( N `  {
v } ) )  =  ( N `  ( U  u.  { v } ) ) )
2625eqeq1d 2304 . . . . . 6  |-  ( ( ( ph  /\  ( U  e.  S  /\  U  =/=  V ) )  /\  v  e.  V
)  ->  ( ( U  .(+)  ( N `  { v } ) )  =  V  <->  ( N `  ( U  u.  {
v } ) )  =  V ) )
2726rexbidva 2573 . . . . 5  |-  ( (
ph  /\  ( U  e.  S  /\  U  =/= 
V ) )  -> 
( E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V  <->  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) )
2827pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( ( U  e.  S  /\  U  =/=  V )  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V )  <-> 
( ( U  e.  S  /\  U  =/= 
V )  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) ) )
2928bicomd 192 . . 3  |-  ( ph  ->  ( ( ( U  e.  S  /\  U  =/=  V )  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V )  <-> 
( ( U  e.  S  /\  U  =/= 
V )  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) ) )
30 df-3an 936 . . 3  |-  ( ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V )  <-> 
( ( U  e.  S  /\  U  =/= 
V )  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) )
31 df-3an 936 . . 3  |-  ( ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V )  <-> 
( ( U  e.  S  /\  U  =/= 
V )  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  V ) )
3229, 30, 313bitr4g 279 . 2  |-  ( ph  ->  ( ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V )  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( U  .(+) 
( N `  {
v } ) )  =  V ) ) )
337, 32bitrd 244 1  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `  { v } ) )  =  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    u. cun 3163    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744  LSHypclsh 29787
This theorem is referenced by:  lshpnelb  29796  lshpcmp  29800  islshpat  29829  lshpkrex  29930  dochshpncl  32196
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-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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  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-lmod 15645  df-lss 15706  df-lsp 15745  df-lshyp 29789
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