Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lineval6a Unicode version

Theorem lineval6a 26192
Description: If  C is a point of AB, AB = AC (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
lineval4a.1  |-  P  =  (PPoints `  G )
lineval4a.3  |-  M  =  ( line `  G
)
lineval4a.4  |-  ( ph  ->  G  e. Ig )
lineval4a.5  |-  ( ph  ->  A  e.  P )
lineval4a.6  |-  ( ph  ->  B  e.  P )
lineval6a.7  |-  ( ph  ->  C  e.  ( A M B ) )
lineval6a.8  |-  ( ph  ->  A  =/=  C )
Assertion
Ref Expression
lineval6a  |-  ( ph  ->  ( A M B )  =  ( A M C ) )

Proof of Theorem lineval6a
StepHypRef Expression
1 lineval4a.1 . 2  |-  P  =  (PPoints `  G )
2 eqid 2296 . 2  |-  (PLines `  G )  =  (PLines `  G )
3 lineval4a.3 . 2  |-  M  =  ( line `  G
)
4 lineval4a.4 . 2  |-  ( ph  ->  G  e. Ig )
5 lineval4a.5 . 2  |-  ( ph  ->  A  e.  P )
6 lineval4a.6 . . . 4  |-  ( ph  ->  B  e.  P )
71, 3, 4, 5, 6lineval12a 26187 . . 3  |-  ( ph  ->  ( A M B )  C_  P )
8 lineval6a.7 . . 3  |-  ( ph  ->  C  e.  ( A M B ) )
97, 8sseldd 3194 . 2  |-  ( ph  ->  C  e.  P )
10 lineval6a.8 . 2  |-  ( ph  ->  A  =/=  C )
111, 3, 4, 5, 6lineval2a 26188 . 2  |-  ( ph  ->  A  e.  ( A M B ) )
12 oveq2 5882 . . . . . . . . 9  |-  ( B  =  A  ->  ( A M B )  =  ( A M A ) )
1312eqcoms 2299 . . . . . . . 8  |-  ( A  =  B  ->  ( A M B )  =  ( A M A ) )
1413eleq2d 2363 . . . . . . 7  |-  ( A  =  B  ->  ( C  e.  ( A M B )  <->  C  e.  ( A M A ) ) )
151, 3, 4, 5lineval3a 26186 . . . . . . . . . 10  |-  ( ph  ->  ( A M A )  =  { A } )
1615eleq2d 2363 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  ( A M A )  <-> 
C  e.  { A } ) )
17 elex 2809 . . . . . . . . . . 11  |-  ( C  e.  ( A M B )  ->  C  e.  _V )
18 elsncg 3675 . . . . . . . . . . . 12  |-  ( C  e.  _V  ->  ( C  e.  { A } 
<->  C  =  A ) )
1918biimpd 198 . . . . . . . . . . 11  |-  ( C  e.  _V  ->  ( C  e.  { A }  ->  C  =  A ) )
208, 17, 193syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( C  e.  { A }  ->  C  =  A ) )
21 df-ne 2461 . . . . . . . . . . . . 13  |-  ( A  =/=  C  <->  -.  A  =  C )
22 pm2.24 101 . . . . . . . . . . . . 13  |-  ( A  =  C  ->  ( -.  A  =  C  ->  A  =/=  B ) )
2321, 22syl5bi 208 . . . . . . . . . . . 12  |-  ( A  =  C  ->  ( A  =/=  C  ->  A  =/=  B ) )
2423eqcoms 2299 . . . . . . . . . . 11  |-  ( C  =  A  ->  ( A  =/=  C  ->  A  =/=  B ) )
2524com12 27 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( C  =  A  ->  A  =/=  B ) )
2610, 20, 25sylsyld 52 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  { A }  ->  A  =/= 
B ) )
2716, 26sylbid 206 . . . . . . . 8  |-  ( ph  ->  ( C  e.  ( A M A )  ->  A  =/=  B
) )
2827com12 27 . . . . . . 7  |-  ( C  e.  ( A M A )  ->  ( ph  ->  A  =/=  B
) )
2914, 28syl6bi 219 . . . . . 6  |-  ( A  =  B  ->  ( C  e.  ( A M B )  ->  ( ph  ->  A  =/=  B
) ) )
3029com3l 75 . . . . 5  |-  ( C  e.  ( A M B )  ->  ( ph  ->  ( A  =  B  ->  A  =/=  B ) ) )
318, 30mpcom 32 . . . 4  |-  ( ph  ->  ( A  =  B  ->  A  =/=  B
) )
32 idd 21 . . . 4  |-  ( ph  ->  ( A  =/=  B  ->  A  =/=  B ) )
3331, 32pm2.61dne 2536 . . 3  |-  ( ph  ->  A  =/=  B )
341, 2, 3, 4, 5, 6, 33lineval12 26184 . 2  |-  ( ph  ->  ( A M B )  e.  (PLines `  G ) )
351, 2, 3, 4, 5, 9, 10, 11, 8, 34lineval42 26183 1  |-  ( ph  ->  ( A M B )  =  ( A M C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   {csn 3653   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163   linecline 26179
This theorem is referenced by:  isibg1a8  26230
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