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Theorem linevala2 26078
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 5539 . . . . . 6  |-  (PPoints `  G
)  e.  _V
53, 4eqeltri 2353 . . . . 5  |-  P  e. 
_V
65, 5pm3.2i 441 . . . 4  |-  ( P  e.  _V  /\  P  e.  _V )
7 mpt2exga 6197 . . . 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 5525 . . . . . 6  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
109, 3syl6eqr 2333 . . . . 5  |-  ( f  =  G  ->  (PPoints `  f )  =  P )
11 fveq2 5525 . . . . . . . 8  |-  ( f  =  G  ->  (PLines `  f )  =  (PLines `  G ) )
12 lineval2.2 . . . . . . . 8  |-  L  =  (PLines `  G )
1311, 12syl6eqr 2333 . . . . . . 7  |-  ( f  =  G  ->  (PLines `  f )  =  L )
1413riotaeqdv 6305 . . . . . 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 3579 . . . . 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 5910 . . . 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 26077 . . . 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 5600 . . 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 2327 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ifcif 3565   {csn 3640   ` cfv 5255    e. cmpt2 5860   iota_crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  lineval222  26079  lineval3a  26083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-li 26077
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