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Theorem dib1dim2 31285
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 2660 . . 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 2389 . . . 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 31282 . . 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 2389 . . . . . . . 8  |-  (Scalar `  U )  =  (Scalar `  U )
12 eqid 2389 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
133, 6, 10, 11, 12dvhbase 31200 . . . . . . 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 2856 . . . . 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 30906 . . . . . . . . . 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 2389 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
223, 4, 6, 10, 21dvhopvsca 31219 . . . . . . . . 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 30943 . . . . . . . . . 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 3935 . . . . . . . 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 2421 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v
( .s `  U
) <. F ,  O >. )  =  <. (
v `  F ) ,  O >. )
2827eqeq2d 2400 . . . . . 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 2668 . . . . 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 30883 . . . . . . . . . . 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 4852 . . . . . . . . 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 2458 . . . . . . . 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 2775 . . . . . 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 2503 . . 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 2447 . 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 31227 . . 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 2389 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
473, 4, 6, 10, 46dvhelvbasei 31205 . . . 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 16007 . . 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 2424 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 1649    e. wcel 1717   {cab 2375   E.wrex 2652   {crab 2655   {csn 3759   <.cop 3762    e. cmpt 4209    _I cid 4436    X. cxp 4818    |` cres 4822    o. ccom 4824   ` cfv 5396  (class class class)co 6022   Basecbs 13398  Scalarcsca 13461   .scvsca 13462   LModclmod 15879   LSpanclspn 15976   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   trLctrl 30274   TEndoctendo 30868   DVecHcdvh 31195   DIsoBcdib 31255
This theorem is referenced by:  cdlemn2a  31313  dih1dimb  31357  dih1dimatlem  31446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-tpos 6417  df-undef 6481  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-0g 13656  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-dvr 15717  df-drng 15766  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lvec 16104  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tendo 30871  df-edring 30873  df-disoa 31146  df-dvech 31196  df-dib 31256
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