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Theorem lshpcmp 29103
Description: If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
Hypotheses
Ref Expression
lshpcmp.h  |-  H  =  (LSHyp `  W )
lshpcmp.w  |-  ( ph  ->  W  e.  LVec )
lshpcmp.t  |-  ( ph  ->  T  e.  H )
lshpcmp.u  |-  ( ph  ->  U  e.  H )
Assertion
Ref Expression
lshpcmp  |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )

Proof of Theorem lshpcmp
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 lshpcmp.h . . . . 5  |-  H  =  (LSHyp `  W )
3 lshpcmp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 16105 . . . . . 6  |-  ( W  e.  LVec  ->  W  e. 
LMod )
53, 4syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
6 lshpcmp.u . . . . 5  |-  ( ph  ->  U  e.  H )
71, 2, 5, 6lshpne 29097 . . . 4  |-  ( ph  ->  U  =/=  ( Base `  W ) )
8 eqid 2387 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
98, 2, 5, 6lshplss 29096 . . . . . . 7  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
101, 8lssss 15940 . . . . . . 7  |-  ( U  e.  ( LSubSp `  W
)  ->  U  C_  ( Base `  W ) )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  U  C_  ( Base `  W ) )
12 lshpcmp.t . . . . . . . . 9  |-  ( ph  ->  T  e.  H )
13 eqid 2387 . . . . . . . . . 10  |-  ( LSpan `  W )  =  (
LSpan `  W )
14 eqid 2387 . . . . . . . . . 10  |-  ( LSSum `  W )  =  (
LSSum `  W )
151, 13, 8, 14, 2, 5islshpsm 29095 . . . . . . . . 9  |-  ( ph  ->  ( T  e.  H  <->  ( T  e.  ( LSubSp `  W )  /\  T  =/=  ( Base `  W
)  /\  E. v  e.  ( Base `  W
) ( T (
LSSum `  W ) ( ( LSpan `  W ) `  { v } ) )  =  ( Base `  W ) ) ) )
1612, 15mpbid 202 . . . . . . . 8  |-  ( ph  ->  ( T  e.  (
LSubSp `  W )  /\  T  =/=  ( Base `  W
)  /\  E. v  e.  ( Base `  W
) ( T (
LSSum `  W ) ( ( LSpan `  W ) `  { v } ) )  =  ( Base `  W ) ) )
1716simp3d 971 . . . . . . 7  |-  ( ph  ->  E. v  e.  (
Base `  W )
( T ( LSSum `  W ) ( (
LSpan `  W ) `  { v } ) )  =  ( Base `  W ) )
18 id 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( ph  /\  v  e.  ( Base `  W ) ) )
1918adantrr 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  ( T
( LSSum `  W )
( ( LSpan `  W
) `  { v } ) )  =  ( Base `  W
) ) )  -> 
( ph  /\  v  e.  ( Base `  W
) ) )
203adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
218, 2, 5, 12lshplss 29096 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  ( LSubSp `  W ) )
2221adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  ( LSubSp `  W )
)
239adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  ( LSubSp `  W )
)
24 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
251, 8, 13, 14, 20, 22, 23, 24lsmcv 16140 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  ( Base `  W
) )  /\  T  C.  U  /\  U  C_  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) ) )  ->  U  =  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) ) )
2619, 25syl3an1 1217 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  =  ( Base `  W
) ) )  /\  T  C.  U  /\  U  C_  ( T ( LSSum `  W ) ( (
LSpan `  W ) `  { v } ) ) )  ->  U  =  ( T (
LSSum `  W ) ( ( LSpan `  W ) `  { v } ) ) )
27263expia 1155 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  =  ( Base `  W
) ) )  /\  T  C.  U )  -> 
( U  C_  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  ->  U  =  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) ) ) )
28 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  =  ( Base `  W
) ) )  /\  T  C.  U )  -> 
( T ( LSSum `  W ) ( (
LSpan `  W ) `  { v } ) )  =  ( Base `  W ) )
2928sseq2d 3319 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  =  ( Base `  W
) ) )  /\  T  C.  U )  -> 
( U  C_  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  <-> 
U  C_  ( Base `  W ) ) )
3028eqeq2d 2398 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  =  ( Base `  W
) ) )  /\  T  C.  U )  -> 
( U  =  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  <-> 
U  =  ( Base `  W ) ) )
3127, 29, 303imtr3d 259 . . . . . . . . 9  |-  ( ( ( ph  /\  (
v  e.  ( Base `  W )  /\  ( T ( LSSum `  W
) ( ( LSpan `  W ) `  {
v } ) )  =  ( Base `  W
) ) )  /\  T  C.  U )  -> 
( U  C_  ( Base `  W )  ->  U  =  ( Base `  W ) ) )
3231exp42 595 . . . . . . . 8  |-  ( ph  ->  ( v  e.  (
Base `  W )  ->  ( ( T (
LSSum `  W ) ( ( LSpan `  W ) `  { v } ) )  =  ( Base `  W )  ->  ( T  C.  U  ->  ( U  C_  ( Base `  W
)  ->  U  =  ( Base `  W )
) ) ) ) )
3332rexlimdv 2772 . . . . . . 7  |-  ( ph  ->  ( E. v  e.  ( Base `  W
) ( T (
LSSum `  W ) ( ( LSpan `  W ) `  { v } ) )  =  ( Base `  W )  ->  ( T  C.  U  ->  ( U  C_  ( Base `  W
)  ->  U  =  ( Base `  W )
) ) ) )
3417, 33mpd 15 . . . . . 6  |-  ( ph  ->  ( T  C.  U  ->  ( U  C_  ( Base `  W )  ->  U  =  ( Base `  W ) ) ) )
3511, 34mpid 39 . . . . 5  |-  ( ph  ->  ( T  C.  U  ->  U  =  ( Base `  W ) ) )
3635necon3ad 2586 . . . 4  |-  ( ph  ->  ( U  =/=  ( Base `  W )  ->  -.  T  C.  U ) )
377, 36mpd 15 . . 3  |-  ( ph  ->  -.  T  C.  U
)
38 df-pss 3279 . . . . 5  |-  ( T 
C.  U  <->  ( T  C_  U  /\  T  =/= 
U ) )
3938simplbi2 609 . . . 4  |-  ( T 
C_  U  ->  ( T  =/=  U  ->  T  C.  U ) )
4039necon1bd 2618 . . 3  |-  ( T 
C_  U  ->  ( -.  T  C.  U  ->  T  =  U )
)
4137, 40syl5com 28 . 2  |-  ( ph  ->  ( T  C_  U  ->  T  =  U ) )
42 eqimss 3343 . 2  |-  ( T  =  U  ->  T  C_  U )
4341, 42impbid1 195 1  |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    C_ wss 3263    C. wpss 3264   {csn 3757   ` cfv 5394  (class class class)co 6020   Basecbs 13396   LSSumclsm 15195   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974   LVecclvec 16101  LSHypclsh 29090
This theorem is referenced by:  lshpinN  29104  lfl1dim  29236  lfl1dim2N  29237  lkrpssN  29278  dochlkr  31500  dochsatshpb  31567  lcfl9a  31620  lclkrlem2e  31626  lclkrlem2g  31628  lclkrlem2s  31640  lcfrlem25  31682  lcfrlem35  31692  hdmaplkr  32031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-0g 13654  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-cntz 15043  df-lsm 15197  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-drng 15764  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lvec 16102  df-lshyp 29092
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