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Theorem lshpcmp 32633
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 2443 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 lshpcmp.h . . . . 5  |-  H  =  (LSHyp `  W )
3 lshpcmp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 17187 . . . . . 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 32627 . . . 4  |-  ( ph  ->  U  =/=  ( Base `  W ) )
8 eqid 2443 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
98, 2, 5, 6lshplss 32626 . . . . . . 7  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
101, 8lssss 17018 . . . . . . 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 2443 . . . . . . . . . 10  |-  ( LSpan `  W )  =  (
LSpan `  W )
14 eqid 2443 . . . . . . . . . 10  |-  ( LSSum `  W )  =  (
LSSum `  W )
151, 13, 8, 14, 2, 5islshpsm 32625 . . . . . . . . 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 210 . . . . . . . 8  |-  ( ph  ->  ( T  e.  (
LSubSp `  W )  /\  T  =/=  ( Base `  W
)  /\  E. v  e.  ( Base `  W
) ( T (
LSSum `  W ) ( ( LSpan `  W ) `  { v } ) )  =  ( Base `  W ) ) )
1716simp3d 1002 . . . . . . 7  |-  ( ph  ->  E. v  e.  (
Base `  W )
( T ( LSSum `  W ) ( (
LSpan `  W ) `  { v } ) )  =  ( Base `  W ) )
18 id 22 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  ( ph  /\  v  e.  ( Base `  W ) ) )
1918adantrr 716 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  ( T
( LSSum `  W )
( ( LSpan `  W
) `  { v } ) )  =  ( Base `  W
) ) )  -> 
( ph  /\  v  e.  ( Base `  W
) ) )
203adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
218, 2, 5, 12lshplss 32626 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  ( LSubSp `  W ) )
2221adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  ( LSubSp `  W )
)
239adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  ( LSubSp `  W )
)
24 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
251, 8, 13, 14, 20, 22, 23, 24lsmcv 17222 . . . . . . . . . . . 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 1251 . . . . . . . . . . 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 1189 . . . . . . . . . 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 760 . . . . . . . . . . 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 3384 . . . . . . . . . 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 2454 . . . . . . . . . 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 267 . . . . . . . . 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 611 . . . . . . . 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 2840 . . . . . . 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 41 . . . . 5  |-  ( ph  ->  ( T  C.  U  ->  U  =  ( Base `  W ) ) )
3635necon3ad 2644 . . . 4  |-  ( ph  ->  ( U  =/=  ( Base `  W )  ->  -.  T  C.  U ) )
377, 36mpd 15 . . 3  |-  ( ph  ->  -.  T  C.  U
)
38 df-pss 3344 . . . . 5  |-  ( T 
C.  U  <->  ( T  C_  U  /\  T  =/= 
U ) )
3938simplbi2 625 . . . 4  |-  ( T 
C_  U  ->  ( T  =/=  U  ->  T  C.  U ) )
4039necon1bd 2679 . . 3  |-  ( T 
C_  U  ->  ( -.  T  C.  U  ->  T  =  U )
)
4137, 40syl5com 30 . 2  |-  ( ph  ->  ( T  C_  U  ->  T  =  U ) )
42 eqimss 3408 . 2  |-  ( T  =  U  ->  T  C_  U )
4341, 42impbid1 203 1  |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716    C_ wss 3328    C. wpss 3329   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174   LSSumclsm 16133   LModclmod 16948   LSubSpclss 17013   LSpanclspn 17052   LVecclvec 17183  LSHypclsh 32620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184  df-lshyp 32622
This theorem is referenced by:  lshpinN  32634  lfl1dim  32766  lfl1dim2N  32767  lkrpssN  32808  dochlkr  35030  dochsatshpb  35097  lcfl9a  35150  lclkrlem2e  35156  lclkrlem2g  35158  lclkrlem2s  35170  lcfrlem25  35212  lcfrlem35  35222  hdmaplkr  35561
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