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Theorem lshpcmp 29713
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 2435 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 lshpcmp.h . . . . 5  |-  H  =  (LSHyp `  W )
3 lshpcmp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 16170 . . . . . 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 29707 . . . 4  |-  ( ph  ->  U  =/=  ( Base `  W ) )
8 eqid 2435 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
98, 2, 5, 6lshplss 29706 . . . . . . 7  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
101, 8lssss 16005 . . . . . . 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 2435 . . . . . . . . . 10  |-  ( LSpan `  W )  =  (
LSpan `  W )
14 eqid 2435 . . . . . . . . . 10  |-  ( LSSum `  W )  =  (
LSSum `  W )
151, 13, 8, 14, 2, 5islshpsm 29705 . . . . . . . . 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 29706 . . . . . . . . . . . . . 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 16205 . . . . . . . . . . . 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 3368 . . . . . . . . . 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 2446 . . . . . . . . . 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 2821 . . . . . . 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 2634 . . . 4  |-  ( ph  ->  ( U  =/=  ( Base `  W )  ->  -.  T  C.  U ) )
377, 36mpd 15 . . 3  |-  ( ph  ->  -.  T  C.  U
)
38 df-pss 3328 . . . . 5  |-  ( T 
C.  U  <->  ( T  C_  U  /\  T  =/= 
U ) )
3938simplbi2 609 . . . 4  |-  ( T 
C_  U  ->  ( T  =/=  U  ->  T  C.  U ) )
4039necon1bd 2666 . . 3  |-  ( T 
C_  U  ->  ( -.  T  C.  U  ->  T  =  U )
)
4137, 40syl5com 28 . 2  |-  ( ph  ->  ( T  C_  U  ->  T  =  U ) )
42 eqimss 3392 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    C_ wss 3312    C. wpss 3313   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461   LSSumclsm 15260   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   LVecclvec 16166  LSHypclsh 29700
This theorem is referenced by:  lshpinN  29714  lfl1dim  29846  lfl1dim2N  29847  lkrpssN  29888  dochlkr  32110  dochsatshpb  32177  lcfl9a  32230  lclkrlem2e  32236  lclkrlem2g  32238  lclkrlem2s  32250  lcfrlem25  32292  lcfrlem35  32302  hdmaplkr  32641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lshyp 29702
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