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Theorem lineval222 26079
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 5875 . 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 2283 . . . 4  |-  ( line `  G )  =  (
line `  G )
7 lineval2.4 . . . 4  |-  ( ph  ->  G  e. Ig )
84, 5, 6, 7linevala2 26078 . . 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 5875 . 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 6308 . . . . . 6  |-  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  e. 
_V
13 snex 4216 . . . . . 6  |-  { A }  e.  _V
1412, 13ifex 3623 . . . . 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 2461 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =/=  y  <->  A  =/=  B ) )
1916eleq1d 2349 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  l  <-> 
A  e.  l ) )
2017eleq1d 2349 . . . . . . . 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 6306 . . . . . 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 3651 . . . . . . 7  |-  ( x  =  A  ->  { x }  =  { A } )
2423adantr 451 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  { x }  =  { A } )
2518, 22, 24ifbieq12d 3587 . . . . 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 2283 . . . . 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 5977 . . . 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 3571 . . . 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 2315 . 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 2319 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ifcif 3565   {csn 3640   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  lineval42  26080  lineval12  26081  lineval22  26082  lineval4a  26087
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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-li 26077
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