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Theorem lshpcmp 35129
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 2454 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 lshpcmp.h . . . . 5  |-  H  =  (LSHyp `  W )
3 lshpcmp.w . . . . . 6  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 17950 . . . . . 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 35123 . . . 4  |-  ( ph  ->  U  =/=  ( Base `  W ) )
8 eqid 2454 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
98, 2, 5, 6lshplss 35122 . . . . . . 7  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
101, 8lssss 17781 . . . . . . 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 2454 . . . . . . . . . 10  |-  ( LSpan `  W )  =  (
LSpan `  W )
14 eqid 2454 . . . . . . . . . 10  |-  ( LSSum `  W )  =  (
LSSum `  W )
151, 13, 8, 14, 2, 5islshpsm 35121 . . . . . . . . 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 1008 . . . . . . 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 714 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( v  e.  ( Base `  W
)  /\  ( T
( LSSum `  W )
( ( LSpan `  W
) `  { v } ) )  =  ( Base `  W
) ) )  -> 
( ph  /\  v  e.  ( Base `  W
) ) )
203adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  W  e.  LVec )
218, 2, 5, 12lshplss 35122 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  ( LSubSp `  W ) )
2221adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  T  e.  ( LSubSp `  W )
)
239adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  U  e.  ( LSubSp `  W )
)
24 simpr 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( Base `  W )
)  ->  v  e.  ( Base `  W )
)
251, 8, 13, 14, 20, 22, 23, 24lsmcv 17985 . . . . . . . . . . . 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 1259 . . . . . . . . . . 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 1196 . . . . . . . . . 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 3517 . . . . . . . . . 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 2468 . . . . . . . . . 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 609 . . . . . . . 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 2944 . . . . . . 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 2664 . . . 4  |-  ( ph  ->  ( U  =/=  ( Base `  W )  ->  -.  T  C.  U ) )
377, 36mpd 15 . . 3  |-  ( ph  ->  -.  T  C.  U
)
38 df-pss 3477 . . . . 5  |-  ( T 
C.  U  <->  ( T  C_  U  /\  T  =/= 
U ) )
3938simplbi2 623 . . . 4  |-  ( T 
C_  U  ->  ( T  =/=  U  ->  T  C.  U ) )
4039necon1bd 2672 . . 3  |-  ( T 
C_  U  ->  ( -.  T  C.  U  ->  T  =  U )
)
4137, 40syl5com 30 . 2  |-  ( ph  ->  ( T  C_  U  ->  T  =  U ) )
42 eqimss 3541 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    C_ wss 3461    C. wpss 3462   {csn 4016   ` cfv 5570  (class class class)co 6270   Basecbs 14719   LSSumclsm 16856   LModclmod 17710   LSubSpclss 17776   LSpanclspn 17815   LVecclvec 17946  LSHypclsh 35116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-subg 16400  df-cntz 16557  df-lsm 16858  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-drng 17596  df-lmod 17712  df-lss 17777  df-lsp 17816  df-lvec 17947  df-lshyp 35118
This theorem is referenced by:  lshpinN  35130  lfl1dim  35262  lfl1dim2N  35263  lkrpssN  35304  dochlkr  37528  dochsatshpb  37595  lcfl9a  37648  lclkrlem2e  37654  lclkrlem2g  37656  lclkrlem2s  37668  lcfrlem25  37710  lcfrlem35  37720  hdmaplkr  38059
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