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Theorem lshpcmp 33660
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 2460 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 lshpcmp.h . . . . 5  |-  H  =  (LSHyp `  W )
3 lshpcmp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 17528 . . . . . 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 33654 . . . 4  |-  ( ph  ->  U  =/=  ( Base `  W ) )
8 eqid 2460 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
98, 2, 5, 6lshplss 33653 . . . . . . 7  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
101, 8lssss 17359 . . . . . . 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 2460 . . . . . . . . . 10  |-  ( LSpan `  W )  =  (
LSpan `  W )
14 eqid 2460 . . . . . . . . . 10  |-  ( LSSum `  W )  =  (
LSSum `  W )
151, 13, 8, 14, 2, 5islshpsm 33652 . . . . . . . . 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 1005 . . . . . . 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 33653 . . . . . . . . . . . . . 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 17563 . . . . . . . . . . . 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 1256 . . . . . . . . . . 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 1193 . . . . . . . . . 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 3525 . . . . . . . . . 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 2474 . . . . . . . . . 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 2946 . . . . . . 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 2670 . . . 4  |-  ( ph  ->  ( U  =/=  ( Base `  W )  ->  -.  T  C.  U ) )
377, 36mpd 15 . . 3  |-  ( ph  ->  -.  T  C.  U
)
38 df-pss 3485 . . . . 5  |-  ( T 
C.  U  <->  ( T  C_  U  /\  T  =/= 
U ) )
3938simplbi2 625 . . . 4  |-  ( T 
C_  U  ->  ( T  =/=  U  ->  T  C.  U ) )
4039necon1bd 2678 . . 3  |-  ( T 
C_  U  ->  ( -.  T  C.  U  ->  T  =  U )
)
4137, 40syl5com 30 . 2  |-  ( ph  ->  ( T  C_  U  ->  T  =  U ) )
42 eqimss 3549 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808    C_ wss 3469    C. wpss 3470   {csn 4020   ` cfv 5579  (class class class)co 6275   Basecbs 14479   LSSumclsm 16443   LModclmod 17288   LSubSpclss 17354   LSpanclspn 17393   LVecclvec 17524  LSHypclsh 33647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-0g 14686  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-drng 17174  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lvec 17525  df-lshyp 33649
This theorem is referenced by:  lshpinN  33661  lfl1dim  33793  lfl1dim2N  33794  lkrpssN  33835  dochlkr  36057  dochsatshpb  36124  lcfl9a  36177  lclkrlem2e  36183  lclkrlem2g  36185  lclkrlem2s  36197  lcfrlem25  36239  lcfrlem35  36249  hdmaplkr  36588
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