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Theorem linevala2 26181
Description: Definition of the line xy. It also defines a degenerate line. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval2.1  |-  P  =  (PPoints `  G )
lineval2.2  |-  L  =  (PLines `  G )
lineval2.3  |-  M  =  ( line `  G
)
lineval2.4  |-  ( ph  ->  G  e. Ig )
Assertion
Ref Expression
linevala2  |-  ( ph  ->  M  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) )
Distinct variable groups:    x, l,
y, G    L, l    x, P, y
Allowed substitution hints:    ph( x, y, l)    P( l)    L( x, y)    M( x, y, l)

Proof of Theorem linevala2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 lineval2.3 . 2  |-  M  =  ( line `  G
)
2 lineval2.4 . . 3  |-  ( ph  ->  G  e. Ig )
3 lineval2.1 . . . . . 6  |-  P  =  (PPoints `  G )
4 fvex 5555 . . . . . 6  |-  (PPoints `  G
)  e.  _V
53, 4eqeltri 2366 . . . . 5  |-  P  e. 
_V
65, 5pm3.2i 441 . . . 4  |-  ( P  e.  _V  /\  P  e.  _V )
7 mpt2exga 6213 . . . 4  |-  ( ( P  e.  _V  /\  P  e.  _V )  ->  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) )  e.  _V )
86, 7mp1i 11 . . 3  |-  ( ph  ->  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) )  e.  _V )
9 fveq2 5541 . . . . . 6  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
109, 3syl6eqr 2346 . . . . 5  |-  ( f  =  G  ->  (PPoints `  f )  =  P )
11 fveq2 5541 . . . . . . . 8  |-  ( f  =  G  ->  (PLines `  f )  =  (PLines `  G ) )
12 lineval2.2 . . . . . . . 8  |-  L  =  (PLines `  G )
1311, 12syl6eqr 2346 . . . . . . 7  |-  ( f  =  G  ->  (PLines `  f )  =  L )
1413riotaeqdv 6321 . . . . . 6  |-  ( f  =  G  ->  ( iota_ l  e.  (PLines `  f ) ( x  e.  l  /\  y  e.  l ) )  =  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) )
1514ifeq1d 3592 . . . . 5  |-  ( f  =  G  ->  if ( x  =/=  y ,  ( iota_ l  e.  (PLines `  f )
( x  e.  l  /\  y  e.  l ) ) ,  {
x } )  =  if ( x  =/=  y ,  ( iota_ l  e.  L ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) )
1610, 10, 15mpt2eq123dv 5926 . . . 4  |-  ( f  =  G  ->  (
x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( iota_ l  e.  (PLines `  f
) ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) )  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) )
17 df-li 26180 . . . 4  |-  line  =  ( f  e. Ig  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( iota_ l  e.  (PLines `  f
) ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) ) )
1816, 17fvmptg 5616 . . 3  |-  ( ( G  e. Ig  /\  (
x  e.  P , 
y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) )  e.  _V )  ->  ( line `  G
)  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) )
192, 8, 18syl2anc 642 . 2  |-  ( ph  ->  ( line `  G
)  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) )
201, 19syl5eq 2340 1  |-  ( ph  ->  M  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   ifcif 3578   {csn 3653   ` cfv 5271    e. cmpt2 5876   iota_crio 6313  PPointscpoints 26159  PLinescplines 26161  Igcig 26163   linecline 26179
This theorem is referenced by:  lineval222  26182  lineval3a  26186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-li 26180
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