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Theorem dib1dim2 34811
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 2723 . . 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 2442 . . . 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 34808 . . 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 2442 . . . . . . . 8  |-  (Scalar `  U )  =  (Scalar `  U )
12 eqid 2442 . . . . . . . 8  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
133, 6, 10, 11, 12dvhbase 34726 . . . . . . 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 2923 . . . . 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 34432 . . . . . . . . . 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 2442 . . . . . . . . . 10  |-  ( .s
`  U )  =  ( .s `  U
)
223, 4, 6, 10, 21dvhopvsca 34745 . . . . . . . . 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 1220 . . . . . . . 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 34469 . . . . . . . . . 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 4065 . . . . . . . 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 2474 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  v  e.  ( ( TEndo `  K
) `  W )
)  ->  ( v
( .s `  U
) <. F ,  O >. )  =  <. (
v `  F ) ,  O >. )
2827eqeq2d 2453 . . . . . 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 2731 . . . . 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 34409 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  v  e.  ( ( TEndo `  K ) `  W )  /\  F  e.  T )  ->  (
v `  F )  e.  T )
31303expa 1187 . . . . . . . . . 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 4870 . . . . . . . . 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 2511 . . . . . . . 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 2840 . . . . . 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 2556 . . 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 2500 . 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 34753 . . 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 2442 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
473, 4, 6, 10, 46dvhelvbasei 34731 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  O  e.  ( ( TEndo `  K
) `  W )
) )  ->  <. F ,  O >.  e.  ( Base `  U ) )
4842, 44, 45, 47syl12anc 1216 . . 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 17082 . . 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 2477 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 1369    e. wcel 1756   {cab 2428   E.wrex 2715   {crab 2718   {csn 3876   <.cop 3882    e. cmpt 4349    _I cid 4630    X. cxp 4837    |` cres 4841    o. ccom 4843   ` cfv 5417  (class class class)co 6090   Basecbs 14173  Scalarcsca 14240   .scvsca 14241   LModclmod 16947   LSpanclspn 17051   HLchlt 32993   LHypclh 33626   LTrncltrn 33743   trLctrl 33800   TEndoctendo 34394   DVecHcdvh 34721   DIsoBcdib 34781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-riotaBAD 32602
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-undef 6791  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-0g 14379  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-mnd 15414  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mgp 16591  df-ur 16603  df-rng 16646  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-drng 16833  df-lmod 16949  df-lss 17013  df-lsp 17052  df-lvec 17183  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-lplanes 33141  df-lvols 33142  df-lines 33143  df-psubsp 33145  df-pmap 33146  df-padd 33438  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801  df-tendo 34397  df-edring 34399  df-disoa 34672  df-dvech 34722  df-dib 34782
This theorem is referenced by:  cdlemn2a  34839  dih1dimb  34883  dih1dimatlem  34972
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