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Theorem dib1dim2 31903
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 2706 . . 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 2435 . . . 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 31900 . . 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 2435 . . . . . . . 8  |-  (Scalar `  U )  =  (Scalar `  U )
12 eqid 2435 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
133, 6, 10, 11, 12dvhbase 31818 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
1413adantr 452 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( Base `  (Scalar `  U )
)  =  ( (
TEndo `  K ) `  W ) )
1514rexeqdv 2903 . . . . 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 731 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpr 448 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  v  e.  ( ( TEndo `  K
) `  W )
)
18 simplr 732 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  F  e.  T )
192, 3, 4, 6, 7tendo0cl 31524 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
2019ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  O  e.  ( ( TEndo `  K
) `  W )
)
21 eqid 2435 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
223, 4, 6, 10, 21dvhopvsca 31837 . . . . . . . . 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 1186 . . . . . . . 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 31561 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W ) )  -> 
( v  o.  O
)  =  O )
2524adantlr 696 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v  o.  O )  =  O )
2625opeq2d 3983 . . . . . . . 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 2467 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v
( .s `  U
) <. F ,  O >. )  =  <. (
v `  F ) ,  O >. )
2827eqeq2d 2446 . . . . . 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 2714 . . . . 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 31501 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
v `  F )  e.  T )
31303expa 1153 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  /\  F  e.  T )  ->  (
v `  F )  e.  T )
3231an32s 780 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v `  F )  e.  T
)
33 opelxpi 4902 . . . . . . . . 9  |-  ( ( ( v `  F
)  e.  T  /\  O  e.  ( ( TEndo `  K ) `  W ) )  ->  <. ( v `  F
) ,  O >.  e.  ( T  X.  (
( TEndo `  K ) `  W ) ) )
3432, 20, 33syl2anc 643 . . . . . . . 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 2504 . . . . . . . 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 2822 . . . . . 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 617 . . . . 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 271 . . . 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 2549 . . 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 2493 . 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 444 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
433, 10, 42dvhlmod 31845 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  U  e.  LMod )
44 simpr 448 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F  e.  T )
4519adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  O  e.  ( ( TEndo `  K
) `  W )
)
46 eqid 2435 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
473, 4, 6, 10, 46dvhelvbasei 31823 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  O  e.  ( ( TEndo `  K
) `  W )
) )  ->  <. F ,  O >.  e.  ( Base `  U ) )
4842, 44, 45, 47syl12anc 1182 . . 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 16070 . . 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 643 . 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 2470 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( N `  { <. F ,  O >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   {crab 2701   {csn 3806   <.cop 3809    e. cmpt 4258    _I cid 4485    X. cxp 4868    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   LModclmod 15942   LSpanclspn 16039   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   TEndoctendo 31486   DVecHcdvh 31813   DIsoBcdib 31873
This theorem is referenced by:  cdlemn2a  31931  dih1dimb  31975  dih1dimatlem  32064
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-fal 1329  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-iin 4088  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-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  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-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489  df-edring 31491  df-disoa 31764  df-dvech 31814  df-dib 31874
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