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Theorem dib1dim 34804
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 464 . . . 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 33801 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  B
)
7 eqid 2471 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
87, 3, 4, 5trlle 33821 . . . 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 2471 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
11 dib1dim.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
122, 7, 3, 4, 9, 10, 11dibval2 34783 . . . 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 1290 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( I `  ( R `  F
) )  =  ( ( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  X. 
{ O } ) )
14 relxp 4947 . . . 4  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  ( R `  F ) )  X. 
{ O } )
15 opelxp 4869 . . . . 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 34700 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( (
( DIsoA `  K ) `  W ) `  ( R `  F )
)  =  { f  |  E. s  e.  E  f  =  ( s `  F ) } )
1817abeq2d 2582 . . . . . . 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 719 . . . . . 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 34405 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E  /\  F  e.  T
)  ->  ( s `  F )  e.  T
)
21203expa 1231 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  s  e.  E )  /\  F  e.  T )  ->  (
s `  F )  e.  T )
2221an32s 821 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
s `  F )  e.  T )
232, 3, 4, 16, 9tendo0cl 34428 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  E )
2423ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  O  e.  E )
2522, 24jca 541 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  s  e.  E )  ->  (
( s `  F
)  e.  T  /\  O  e.  E )
)
26 eleq1 2537 . . . . . . . . . . 11  |-  ( f  =  ( s `  F )  ->  (
f  e.  T  <->  ( s `  F )  e.  T
) )
27 eleq1 2537 . . . . . . . . . . 11  |-  ( t  =  O  ->  (
t  e.  E  <->  O  e.  E ) )
2826, 27bi2anan9 890 . . . . . . . . . 10  |-  ( ( f  =  ( s `
 F )  /\  t  =  O )  ->  ( ( f  e.  T  /\  t  e.  E )  <->  ( (
s `  F )  e.  T  /\  O  e.  E ) ) )
2925, 28syl5ibrcom 230 . . . . . . . . 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 2871 . . . . . . . 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 647 . . . . . . 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 3973 . . . . . . . . 9  |-  ( t  e.  { O }  <->  t  =  O )
3332anbi2i 708 . . . . . . . 8  |-  ( ( E. s  e.  E  f  =  ( s `  F )  /\  t  e.  { O } )  <-> 
( E. s  e.  E  f  =  ( s `  F )  /\  t  =  O ) )
34 r19.41v 2928 . . . . . . . 8  |-  ( E. s  e.  E  ( f  =  ( s `
 F )  /\  t  =  O )  <->  ( E. s  e.  E  f  =  ( s `  F )  /\  t  =  O ) )
3533, 34bitr4i 260 . . . . . . 7  |-  ( ( E. s  e.  E  f  =  ( s `  F )  /\  t  e.  { O } )  <->  E. s  e.  E  ( f  =  ( s `  F )  /\  t  =  O ) )
36 df-3an 1009 . . . . . . 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 296 . . . . . 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 261 . . . . 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 265 . . . 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 4989 . . 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 2505 . 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 2475 . . . . 5  |-  ( g  =  <. f ,  t
>.  ->  ( g  = 
<. ( s `  F
) ,  O >.  <->  <. f ,  t >.  =  <. ( s `  F ) ,  O >. )
)
43 vex 3034 . . . . . 6  |-  f  e. 
_V
44 vex 3034 . . . . . 6  |-  t  e. 
_V
4543, 44opth 4676 . . . . 5  |-  ( <.
f ,  t >.  =  <. ( s `  F ) ,  O >.  <-> 
( f  =  ( s `  F )  /\  t  =  O ) )
4642, 45syl6bb 269 . . . 4  |-  ( g  =  <. f ,  t
>.  ->  ( g  = 
<. ( s `  F
) ,  O >.  <->  (
f  =  ( s `
 F )  /\  t  =  O )
) )
4746rexbidv 2892 . . 3  |-  ( g  =  <. f ,  t
>.  ->  ( E. s  e.  E  g  =  <. ( s `  F
) ,  O >.  <->  E. s  e.  E  (
f  =  ( s `
 F )  /\  t  =  O )
) )
4847rabxp 4876 . 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 2523 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   {crab 2760   {csn 3959   <.cop 3965   class class class wbr 4395   {copab 4453    |-> cmpt 4454    _I cid 4749    X. cxp 4837    |` cres 4841   ` cfv 5589   Basecbs 15199   lecple 15275   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795   TEndoctendo 34390   DIsoAcdia 34667   DIsoBcdib 34777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-undef 7038  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tendo 34393  df-disoa 34668  df-dib 34778
This theorem is referenced by:  dib1dim2  34807
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