Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dib1dim2 Structured version   Unicode version

Theorem dib1dim2 36321
Description: Two expressions for a 1-dimensional subspace of vector space H (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 24-Feb-2014.)
Hypotheses
Ref Expression
dib1dim2.b  |-  B  =  ( Base `  K
)
dib1dim2.h  |-  H  =  ( LHyp `  K
)
dib1dim2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dib1dim2.r  |-  R  =  ( ( trL `  K
) `  W )
dib1dim2.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
dib1dim2.u  |-  U  =  ( ( DVecH `  K
) `  W )
dib1dim2.i  |-  I  =  ( ( DIsoB `  K
) `  W )
dib1dim2.n  |-  N  =  ( LSpan `  U )
Assertion
Ref Expression
dib1dim2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { <. F ,  O >. } ) )
Distinct variable groups:    B, h    h, K    T, h    h, W
Allowed substitution hints:    R( h)    U( h)    F( h)    H( h)    I( h)    N( h)    O( h)

Proof of Theorem dib1dim2
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2826 . . 3  |-  { u  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  |  E. v  e.  ( ( TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. }  =  { u  |  ( u  e.  ( T  X.  ( (
TEndo `  K ) `  W ) )  /\  E. v  e.  ( (
TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. ) }
2 dib1dim2.b . . . 4  |-  B  =  ( Base `  K
)
3 dib1dim2.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dib1dim2.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 dib1dim2.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
6 eqid 2467 . . . 4  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dib1dim2.o . . . 4  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
8 dib1dim2.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
92, 3, 4, 5, 6, 7, 8dib1dim 36318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
u  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  |  E. v  e.  ( ( TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. } )
10 dib1dim2.u . . . . . . . 8  |-  U  =  ( ( DVecH `  K
) `  W )
11 eqid 2467 . . . . . . . 8  |-  (Scalar `  U )  =  (Scalar `  U )
12 eqid 2467 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
133, 6, 10, 11, 12dvhbase 36236 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
1413adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( Base `  (Scalar `  U )
)  =  ( (
TEndo `  K ) `  W ) )
1514rexeqdv 3070 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. v  e.  ( Base `  (Scalar `  U )
) u  =  ( v ( .s `  U ) <. F ,  O >. )  <->  E. v  e.  ( ( TEndo `  K
) `  W )
u  =  ( v ( .s `  U
) <. F ,  O >. ) ) )
16 simpll 753 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpr 461 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  v  e.  ( ( TEndo `  K
) `  W )
)
18 simplr 754 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  F  e.  T )
192, 3, 4, 6, 7tendo0cl 35942 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
2019ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  O  e.  ( ( TEndo `  K
) `  W )
)
21 eqid 2467 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
223, 4, 6, 10, 21dvhopvsca 36255 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( v  e.  ( ( TEndo `  K
) `  W )  /\  F  e.  T  /\  O  e.  (
( TEndo `  K ) `  W ) ) )  ->  ( v ( .s `  U )
<. F ,  O >. )  =  <. ( v `  F ) ,  ( v  o.  O )
>. )
2316, 17, 18, 20, 22syl13anc 1230 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v
( .s `  U
) <. F ,  O >. )  =  <. (
v `  F ) ,  ( v  o.  O ) >. )
242, 3, 4, 6, 7tendo0mulr 35979 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W ) )  -> 
( v  o.  O
)  =  O )
2524adantlr 714 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v  o.  O )  =  O )
2625opeq2d 4226 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  <. ( v `
 F ) ,  ( v  o.  O
) >.  =  <. (
v `  F ) ,  O >. )
2723, 26eqtrd 2508 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v
( .s `  U
) <. F ,  O >. )  =  <. (
v `  F ) ,  O >. )
2827eqeq2d 2481 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( u  =  ( v ( .s `  U )
<. F ,  O >. )  <-> 
u  =  <. (
v `  F ) ,  O >. ) )
2928rexbidva 2975 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. v  e.  ( ( TEndo `  K ) `  W ) u  =  ( v ( .s
`  U ) <. F ,  O >. )  <->  E. v  e.  (
( TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. ) )
303, 4, 6tendocl 35919 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
v `  F )  e.  T )
31303expa 1196 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  /\  F  e.  T )  ->  (
v `  F )  e.  T )
3231an32s 802 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v `  F )  e.  T
)
33 opelxpi 5037 . . . . . . . . 9  |-  ( ( ( v `  F
)  e.  T  /\  O  e.  ( ( TEndo `  K ) `  W ) )  ->  <. ( v `  F
) ,  O >.  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) )
3432, 20, 33syl2anc 661 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  <. ( v `
 F ) ,  O >.  e.  ( T  X.  ( ( TEndo `  K ) `  W
) ) )
35 eleq1a 2550 . . . . . . . 8  |-  ( <.
( v `  F
) ,  O >.  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  -> 
( u  =  <. ( v `  F ) ,  O >.  ->  u  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) ) )
3634, 35syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( u  =  <. ( v `  F ) ,  O >.  ->  u  e.  ( T  X.  ( (
TEndo `  K ) `  W ) ) ) )
3736rexlimdva 2959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. v  e.  ( ( TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >.  ->  u  e.  ( T  X.  ( ( TEndo `  K
) `  W )
) ) )
3837pm4.71rd 635 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. v  e.  ( ( TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >.  <->  (
u  e.  ( T  X.  ( ( TEndo `  K ) `  W
) )  /\  E. v  e.  ( ( TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. ) ) )
3915, 29, 383bitrd 279 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. v  e.  ( Base `  (Scalar `  U )
) u  =  ( v ( .s `  U ) <. F ,  O >. )  <->  ( u  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  /\  E. v  e.  ( (
TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. ) ) )
4039abbidv 2603 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  { u  |  E. v  e.  (
Base `  (Scalar `  U
) ) u  =  ( v ( .s
`  U ) <. F ,  O >. ) }  =  { u  |  ( u  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  /\  E. v  e.  ( (
TEndo `  K ) `  W ) u  = 
<. ( v `  F
) ,  O >. ) } )
411, 9, 403eqtr4a 2534 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
u  |  E. v  e.  ( Base `  (Scalar `  U ) ) u  =  ( v ( .s `  U )
<. F ,  O >. ) } )
42 simpl 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
433, 10, 42dvhlmod 36263 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  U  e.  LMod )
44 simpr 461 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  T )
4519adantr 465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  O  e.  ( ( TEndo `  K
) `  W )
)
46 eqid 2467 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
473, 4, 6, 10, 46dvhelvbasei 36241 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  O  e.  ( ( TEndo `  K
) `  W )
) )  ->  <. F ,  O >.  e.  ( Base `  U ) )
4842, 44, 45, 47syl12anc 1226 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  <. F ,  O >.  e.  ( Base `  U ) )
49 dib1dim2.n . . . 4  |-  N  =  ( LSpan `  U )
5011, 12, 46, 21, 49lspsn 17519 . . 3  |-  ( ( U  e.  LMod  /\  <. F ,  O >.  e.  (
Base `  U )
)  ->  ( N `  { <. F ,  O >. } )  =  {
u  |  E. v  e.  ( Base `  (Scalar `  U ) ) u  =  ( v ( .s `  U )
<. F ,  O >. ) } )
5143, 48, 50syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( N `  { <. F ,  O >. } )  =  {
u  |  E. v  e.  ( Base `  (Scalar `  U ) ) u  =  ( v ( .s `  U )
<. F ,  O >. ) } )
5241, 51eqtr4d 2511 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { <. F ,  O >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   {crab 2821   {csn 4033   <.cop 4039    |-> cmpt 4511    _I cid 4796    X. cxp 5003    |` cres 5007    o. ccom 5009   ` cfv 5594  (class class class)co 6295   Basecbs 14507  Scalarcsca 14575   .scvsca 14576   LModclmod 17383   LSpanclspn 17488   HLchlt 34503   LHypclh 35136   LTrncltrn 35253   trLctrl 35310   TEndoctendo 35904   DVecHcdvh 36231   DIsoBcdib 36291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-riotaBAD 34112
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-undef 7014  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-0g 14714  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-lmod 17385  df-lss 17450  df-lsp 17489  df-lvec 17620  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-llines 34650  df-lplanes 34651  df-lvols 34652  df-lines 34653  df-psubsp 34655  df-pmap 34656  df-padd 34948  df-lhyp 35140  df-laut 35141  df-ldil 35256  df-ltrn 35257  df-trl 35311  df-tendo 35907  df-edring 35909  df-disoa 36182  df-dvech 36232  df-dib 36292
This theorem is referenced by:  cdlemn2a  36349  dih1dimb  36393  dih1dimatlem  36482
  Copyright terms: Public domain W3C validator