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

Theorem dib1dim2 34736
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 2746 . . 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 2451 . . . 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 34733 . . 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 2451 . . . . . . . 8  |-  (Scalar `  U )  =  (Scalar `  U )
12 eqid 2451 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
133, 6, 10, 11, 12dvhbase 34651 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
1413adantr 467 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( Base `  (Scalar `  U )
)  =  ( (
TEndo `  K ) `  W ) )
1514rexeqdv 2994 . . . . 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 760 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpr 463 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  v  e.  ( ( TEndo `  K
) `  W )
)
18 simplr 762 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  F  e.  T )
192, 3, 4, 6, 7tendo0cl 34357 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
2019ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  O  e.  ( ( TEndo `  K
) `  W )
)
21 eqid 2451 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
223, 4, 6, 10, 21dvhopvsca 34670 . . . . . . . . 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 1270 . . . . . . . 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 34394 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W ) )  -> 
( v  o.  O
)  =  O )
2524adantlr 721 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v  o.  O )  =  O )
2625opeq2d 4173 . . . . . . . 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 2485 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v
( .s `  U
) <. F ,  O >. )  =  <. (
v `  F ) ,  O >. )
2827eqeq2d 2461 . . . . . 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 2898 . . . . 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 34334 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
v `  F )  e.  T )
31303expa 1208 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  /\  F  e.  T )  ->  (
v `  F )  e.  T )
3231an32s 813 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v `  F )  e.  T
)
33 opelxpi 4866 . . . . . . . . 9  |-  ( ( ( v `  F
)  e.  T  /\  O  e.  ( ( TEndo `  K ) `  W ) )  ->  <. ( v `  F
) ,  O >.  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) )
3432, 20, 33syl2anc 667 . . . . . . . 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 2524 . . . . . . . 8  |-  ( <.
( v `  F
) ,  O >.  e.  ( T  X.  (
( TEndo `  K ) `  W ) )  -> 
( u  =  <. ( v `  F ) ,  O >.  ->  u  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) ) )
3634, 35syl 17 . . . . . . 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 2879 . . . . . 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 641 . . . . 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 283 . . . 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 2569 . . 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 2511 . 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 459 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
433, 10, 42dvhlmod 34678 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  U  e.  LMod )
44 simpr 463 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  T )
4519adantr 467 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  O  e.  ( ( TEndo `  K
) `  W )
)
46 eqid 2451 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
473, 4, 6, 10, 46dvhelvbasei 34656 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  O  e.  ( ( TEndo `  K
) `  W )
) )  ->  <. F ,  O >.  e.  ( Base `  U ) )
4842, 44, 45, 47syl12anc 1266 . . 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 18225 . . 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 667 . 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 2488 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 371    = wceq 1444    e. wcel 1887   {cab 2437   E.wrex 2738   {crab 2741   {csn 3968   <.cop 3974    |-> cmpt 4461    _I cid 4744    X. cxp 4832    |` cres 4836    o. ccom 4838   ` cfv 5582  (class class class)co 6290   Basecbs 15121  Scalarcsca 15193   .scvsca 15194   LModclmod 18091   LSpanclspn 18194   HLchlt 32916   LHypclh 33549   LTrncltrn 33666   trLctrl 33724   TEndoctendo 34319   DVecHcdvh 34646   DIsoBcdib 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-riotaBAD 32525
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-undef 7020  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-0g 15340  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mgp 17724  df-ur 17736  df-ring 17782  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-drng 17977  df-lmod 18093  df-lss 18156  df-lsp 18195  df-lvec 18326  df-oposet 32742  df-ol 32744  df-oml 32745  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917  df-llines 33063  df-lplanes 33064  df-lvols 33065  df-lines 33066  df-psubsp 33068  df-pmap 33069  df-padd 33361  df-lhyp 33553  df-laut 33554  df-ldil 33669  df-ltrn 33670  df-trl 33725  df-tendo 34322  df-edring 34324  df-disoa 34597  df-dvech 34647  df-dib 34707
This theorem is referenced by:  cdlemn2a  34764  dih1dimb  34808  dih1dimatlem  34897
  Copyright terms: Public domain W3C validator