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Theorem lineval222 26182
Description: The line passing through two distinct points  A and  B. (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 )
lineval2.5  |-  ( ph  ->  A  e.  P )
lineval2.6  |-  ( ph  ->  B  e.  P )
lineval2.7  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
lineval222  |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
Distinct variable groups:    A, l    B, l    G, l    L, l    ph, l
Allowed substitution hints:    P( l)    M( l)

Proof of Theorem lineval222
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lineval2.3 . . . 4  |-  M  =  ( line `  G
)
21a1i 10 . . 3  |-  ( ph  ->  M  =  ( line `  G ) )
32oveqd 5891 . 2  |-  ( ph  ->  ( A M B )  =  ( A ( line `  G
) B ) )
4 lineval2.1 . . . 4  |-  P  =  (PPoints `  G )
5 lineval2.2 . . . 4  |-  L  =  (PLines `  G )
6 eqid 2296 . . . 4  |-  ( line `  G )  =  (
line `  G )
7 lineval2.4 . . . 4  |-  ( ph  ->  G  e. Ig )
84, 5, 6, 7linevala2 26181 . . 3  |-  ( ph  ->  ( line `  G
)  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) )
98oveqd 5891 . 2  |-  ( ph  ->  ( A ( line `  G ) B )  =  ( A ( x  e.  P , 
y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L ( x  e.  l  /\  y  e.  l ) ) ,  { x } ) ) B ) )
10 lineval2.5 . . . 4  |-  ( ph  ->  A  e.  P )
11 lineval2.6 . . . 4  |-  ( ph  ->  B  e.  P )
12 riotaex 6324 . . . . . 6  |-  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  e. 
_V
13 snex 4232 . . . . . 6  |-  { A }  e.  _V
1412, 13ifex 3636 . . . . 5  |-  if ( A  =/=  B , 
( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) ,  { A } )  e.  _V
1514a1i 10 . . . 4  |-  ( ph  ->  if ( A  =/= 
B ,  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ,  { A } )  e.  _V )
16 simpl 443 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
17 simpr 447 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
1816, 17neeq12d 2474 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =/=  y  <->  A  =/=  B ) )
1916eleq1d 2362 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  l  <-> 
A  e.  l ) )
2017eleq1d 2362 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  e.  l  <-> 
B  e.  l ) )
2119, 20anbi12d 691 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  e.  l  /\  y  e.  l )  <->  ( A  e.  l  /\  B  e.  l ) ) )
2221riotabidv 6322 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) )  =  (
iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) )
23 sneq 3664 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
2423adantr 451 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  { x }  =  { A } )
2518, 22, 24ifbieq12d 3600 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( x  =/=  y ,  ( iota_ l  e.  L ( x  e.  l  /\  y  e.  l ) ) ,  { x } )  =  if ( A  =/=  B ,  (
iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) ,  { A } ) )
26 eqid 2296 . . . . 5  |-  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( 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 } ) )
2725, 26ovmpt2ga 5993 . . . 4  |-  ( ( A  e.  P  /\  B  e.  P  /\  if ( A  =/=  B ,  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ,  { A } )  e.  _V )  -> 
( A ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) B )  =  if ( A  =/=  B ,  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ,  { A } ) )
2810, 11, 15, 27syl3anc 1182 . . 3  |-  ( ph  ->  ( A ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) B )  =  if ( A  =/=  B ,  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ,  { A } ) )
29 lineval2.7 . . . 4  |-  ( ph  ->  A  =/=  B )
30 iftrue 3584 . . . 4  |-  ( A  =/=  B  ->  if ( A  =/=  B ,  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ,  { A } )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
3129, 30syl 15 . . 3  |-  ( ph  ->  if ( A  =/= 
B ,  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ,  { A } )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
3228, 31eqtrd 2328 . 2  |-  ( ph  ->  ( A ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  ( iota_ l  e.  L
( x  e.  l  /\  y  e.  l ) ) ,  {
x } ) ) B )  =  (
iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) )
333, 9, 323eqtrd 2332 1  |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
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  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313  PPointscpoints 26159  PLinescplines 26161  Igcig 26163   linecline 26179
This theorem is referenced by:  lineval42  26183  lineval12  26184  lineval22  26185  lineval4a  26190
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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-li 26180
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