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Theorem lineval6a 26089
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 2283 . 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 26084 . . 3  |-  ( ph  ->  ( A M B )  C_  P )
8 lineval6a.7 . . 3  |-  ( ph  ->  C  e.  ( A M B ) )
97, 8sseldd 3181 . 2  |-  ( ph  ->  C  e.  P )
10 lineval6a.8 . 2  |-  ( ph  ->  A  =/=  C )
111, 3, 4, 5, 6lineval2a 26085 . 2  |-  ( ph  ->  A  e.  ( A M B ) )
12 oveq2 5866 . . . . . . . . 9  |-  ( B  =  A  ->  ( A M B )  =  ( A M A ) )
1312eqcoms 2286 . . . . . . . 8  |-  ( A  =  B  ->  ( A M B )  =  ( A M A ) )
1413eleq2d 2350 . . . . . . 7  |-  ( A  =  B  ->  ( C  e.  ( A M B )  <->  C  e.  ( A M A ) ) )
151, 3, 4, 5lineval3a 26083 . . . . . . . . . 10  |-  ( ph  ->  ( A M A )  =  { A } )
1615eleq2d 2350 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  ( A M A )  <-> 
C  e.  { A } ) )
17 elex 2796 . . . . . . . . . . 11  |-  ( C  e.  ( A M B )  ->  C  e.  _V )
18 elsncg 3662 . . . . . . . . . . . 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 2448 . . . . . . . . . . . . 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 2286 . . . . . . . . . . 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 2523 . . 3  |-  ( ph  ->  A  =/=  B )
341, 2, 3, 4, 5, 6, 33lineval12 26081 . 2  |-  ( ph  ->  ( A M B )  e.  (PLines `  G ) )
351, 2, 3, 4, 5, 9, 10, 11, 8, 34lineval42 26080 1  |-  ( ph  ->  ( A M B )  =  ( A M C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   {csn 3640   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  isibg1a8  26127
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-ig2 26061  df-li 26077
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