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Theorem dib1dim 35962
Description: Two expressions for the 1-dimensional subspaces of vector space H. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dib1dim.b  |-  B  =  ( Base `  K
)
dib1dim.h  |-  H  =  ( LHyp `  K
)
dib1dim.t  |-  T  =  ( ( LTrn `  K
) `  W )
dib1dim.r  |-  R  =  ( ( trL `  K
) `  W )
dib1dim.e  |-  E  =  ( ( TEndo `  K
) `  W )
dib1dim.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
dib1dim.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dib1dim  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
g  e.  ( T  X.  E )  |  E. s  e.  E  g  =  <. ( s `
 F ) ,  O >. } )
Distinct variable groups:    B, h    g, s, E    g, F, s    H, s    h, s, K    g, O, s    R, s    g, h, T, s    h, W, s
Allowed substitution hints:    B( g, s)    R( g, h)    E( h)    F( h)    H( g, h)    I(
g, h, s)    K( g)    O( h)    W( g)

Proof of Theorem dib1dim
Dummy variables  f 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dib1dim.b . . . . 5  |-  B  =  ( Base `  K
)
3 dib1dim.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dib1dim.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
5 dib1dim.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
62, 3, 4, 5trlcl 34960 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
7 eqid 2467 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 34980 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F ) ( le
`  K ) W )
9 dib1dim.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
10 eqid 2467 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
11 dib1dim.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
122, 7, 3, 4, 9, 10, 11dibval2 35941 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R `
 F )  e.  B  /\  ( R `
 F ) ( le `  K ) W ) )  -> 
( I `  ( R `  F )
)  =  ( ( ( ( DIsoA `  K
) `  W ) `  ( R `  F
) )  X.  { O } ) )
131, 6, 8, 12syl12anc 1226 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( ( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  X. 
{ O } ) )
14 relxp 5108 . . . 4  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  X. 
{ O } )
15 opelxp 5028 . . . . 5  |-  ( <.
f ,  t >.  e.  ( ( ( (
DIsoA `  K ) `  W ) `  ( R `  F )
)  X.  { O } )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  /\  t  e.  { O } ) )
16 dib1dim.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
173, 4, 5, 16, 10dia1dim 35858 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( DIsoA `  K ) `  W ) `  ( R `  F )
)  =  { f  |  E. s  e.  E  f  =  ( s `  F ) } )
1817abeq2d 2593 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  <->  E. s  e.  E  f  =  ( s `  F
) ) )
1918anbi1d 704 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  ( R `  F )
)  /\  t  e.  { O } )  <->  ( E. s  e.  E  f  =  ( s `  F )  /\  t  e.  { O } ) ) )
203, 4, 16tendocl 35563 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
21203expa 1196 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E )  /\  F  e.  T )  ->  (
s `  F )  e.  T )
2221an32s 802 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
s `  F )  e.  T )
232, 3, 4, 16, 9tendo0cl 35586 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
2423ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  O  e.  E )
2522, 24jca 532 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  O  e.  E )
)
26 eleq1 2539 . . . . . . . . . . 11  |-  ( f  =  ( s `  F )  ->  (
f  e.  T  <->  ( s `  F )  e.  T
) )
27 eleq1 2539 . . . . . . . . . . 11  |-  ( t  =  O  ->  (
t  e.  E  <->  O  e.  E ) )
2826, 27bi2anan9 871 . . . . . . . . . 10  |-  ( ( f  =  ( s `
 F )  /\  t  =  O )  ->  ( ( f  e.  T  /\  t  e.  E )  <->  ( (
s `  F )  e.  T  /\  O  e.  E ) ) )
2925, 28syl5ibrcom 222 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( f  =  ( s `  F )  /\  t  =  O )  ->  ( f  e.  T  /\  t  e.  E ) ) )
3029rexlimdva 2955 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  E  (
f  =  ( s `
 F )  /\  t  =  O )  ->  ( f  e.  T  /\  t  e.  E
) ) )
3130pm4.71rd 635 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. s  e.  E  (
f  =  ( s `
 F )  /\  t  =  O )  <->  ( ( f  e.  T  /\  t  e.  E
)  /\  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) ) ) )
32 elsn 4041 . . . . . . . . 9  |-  ( t  e.  { O }  <->  t  =  O )
3332anbi2i 694 . . . . . . . 8  |-  ( ( E. s  e.  E  f  =  ( s `  F )  /\  t  e.  { O } )  <-> 
( E. s  e.  E  f  =  ( s `  F )  /\  t  =  O ) )
34 r19.41v 3014 . . . . . . . 8  |-  ( E. s  e.  E  ( f  =  ( s `
 F )  /\  t  =  O )  <->  ( E. s  e.  E  f  =  ( s `  F )  /\  t  =  O ) )
3533, 34bitr4i 252 . . . . . . 7  |-  ( ( E. s  e.  E  f  =  ( s `  F )  /\  t  e.  { O } )  <->  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) )
36 df-3an 975 . . . . . . 7  |-  ( ( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `
 F )  /\  t  =  O )
)  <->  ( ( f  e.  T  /\  t  e.  E )  /\  E. s  e.  E  (
f  =  ( s `
 F )  /\  t  =  O )
) )
3731, 35, 363bitr4g 288 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( E. s  e.  E  f  =  ( s `  F )  /\  t  e.  { O } )  <-> 
( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) ) ) )
3819, 37bitrd 253 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  ( R `  F )
)  /\  t  e.  { O } )  <->  ( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) ) ) )
3915, 38syl5bb 257 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( <. f ,  t >.  e.  ( ( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  X. 
{ O } )  <-> 
( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) ) ) )
4014, 39opabbi2dv 5150 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( ( DIsoA `  K
) `  W ) `  ( R `  F
) )  X.  { O } )  =  { <. f ,  t >.  |  ( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) ) } )
4113, 40eqtrd 2508 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  { <. f ,  t >.  |  ( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) ) } )
42 eqeq1 2471 . . . . 5  |-  ( g  =  <. f ,  t
>.  ->  ( g  = 
<. ( s `  F
) ,  O >.  <->  <. f ,  t >.  =  <. ( s `  F ) ,  O >. )
)
43 vex 3116 . . . . . 6  |-  f  e. 
_V
44 vex 3116 . . . . . 6  |-  t  e. 
_V
4543, 44opth 4721 . . . . 5  |-  ( <.
f ,  t >.  =  <. ( s `  F ) ,  O >.  <-> 
( f  =  ( s `  F )  /\  t  =  O ) )
4642, 45syl6bb 261 . . . 4  |-  ( g  =  <. f ,  t
>.  ->  ( g  = 
<. ( s `  F
) ,  O >.  <->  (
f  =  ( s `
 F )  /\  t  =  O )
) )
4746rexbidv 2973 . . 3  |-  ( g  =  <. f ,  t
>.  ->  ( E. s  e.  E  g  =  <. ( s `  F
) ,  O >.  <->  E. s  e.  E  (
f  =  ( s `
 F )  /\  t  =  O )
) )
4847rabxp 5035 . 2  |-  { g  e.  ( T  X.  E )  |  E. s  e.  E  g  =  <. ( s `  F ) ,  O >. }  =  { <. f ,  t >.  |  ( f  e.  T  /\  t  e.  E  /\  E. s  e.  E  ( f  =  ( s `
 F )  /\  t  =  O )
) }
4941, 48syl6eqr 2526 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  {
g  e.  ( T  X.  E )  |  E. s  e.  E  g  =  <. ( s `
 F ) ,  O >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   {csn 4027   <.cop 4033   class class class wbr 4447   {copab 4504    |-> cmpt 4505    _I cid 4790    X. cxp 4997    |` cres 5001   ` cfv 5586   Basecbs 14483   lecple 14555   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954   TEndoctendo 35548   DIsoAcdia 35825   DIsoBcdib 35935
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551  df-disoa 35826  df-dib 35936
This theorem is referenced by:  dib1dim2  35965
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