Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lineval42 Unicode version

Theorem lineval42 26183
Description: Any line to which  A and  B are incident is the line  ( A M B ). (For my private use only. Don't use.) (Contributed by FL, 1-Apr-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 )
lineval42.5  |-  ( ph  ->  A  e.  P )
lineval42.6  |-  ( ph  ->  B  e.  P )
lineval42.7  |-  ( ph  ->  A  =/=  B )
lineval42.8  |-  ( ph  ->  A  e.  N )
lineval42.9  |-  ( ph  ->  B  e.  N )
lineval42.a  |-  ( ph  ->  N  e.  L )
Assertion
Ref Expression
lineval42  |-  ( ph  ->  N  =  ( A M B ) )

Proof of Theorem lineval42
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 lineval2.1 . . 3  |-  P  =  (PPoints `  G )
2 lineval2.2 . . 3  |-  L  =  (PLines `  G )
3 lineval2.3 . . 3  |-  M  =  ( line `  G
)
4 lineval2.4 . . 3  |-  ( ph  ->  G  e. Ig )
5 lineval42.5 . . 3  |-  ( ph  ->  A  e.  P )
6 lineval42.6 . . 3  |-  ( ph  ->  B  e.  P )
7 lineval42.7 . . 3  |-  ( ph  ->  A  =/=  B )
81, 2, 3, 4, 5, 6, 7lineval222 26182 . 2  |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
9 lineval42.8 . . 3  |-  ( ph  ->  A  e.  N )
10 lineval42.9 . . 3  |-  ( ph  ->  B  e.  N )
11 lineval42.a . . . 4  |-  ( ph  ->  N  e.  L )
121, 2, 4, 5, 6, 7isig2a2 26169 . . . 4  |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )
13 eleq2 2357 . . . . . 6  |-  ( l  =  N  ->  ( A  e.  l  <->  A  e.  N ) )
14 eleq2 2357 . . . . . 6  |-  ( l  =  N  ->  ( B  e.  l  <->  B  e.  N ) )
1513, 14anbi12d 691 . . . . 5  |-  ( l  =  N  ->  (
( A  e.  l  /\  B  e.  l )  <->  ( A  e.  N  /\  B  e.  N ) ) )
1615riota2 6343 . . . 4  |-  ( ( N  e.  L  /\  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )  ->  (
( A  e.  N  /\  B  e.  N
)  <->  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  =  N ) )
1711, 12, 16syl2anc 642 . . 3  |-  ( ph  ->  ( ( A  e.  N  /\  B  e.  N )  <->  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  =  N ) )
189, 10, 17mpbi2and 887 . 2  |-  ( ph  ->  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )  =  N )
198, 18eqtr2d 2329 1  |-  ( ph  ->  N  =  ( A M B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E!wreu 2558   ` cfv 5271  (class class class)co 5874   iota_crio 6313  PPointscpoints 26159  PLinescplines 26161  Igcig 26163   linecline 26179
This theorem is referenced by:  lineval5a  26191  lineval6a  26192  isconcl5a  26204  isconcl5ab  26205  isibg1a6  26228
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-ig2 26164  df-li 26180
  Copyright terms: Public domain W3C validator