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Theorem tglineneq 23765
Description: Given three non-colinear points, build two different lines. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
tglineintmo.p  |-  P  =  ( Base `  G
)
tglineintmo.i  |-  I  =  (Itv `  G )
tglineintmo.l  |-  L  =  (LineG `  G )
tglineintmo.g  |-  ( ph  ->  G  e. TarskiG )
tglineinteq.a  |-  ( ph  ->  A  e.  P )
tglineinteq.b  |-  ( ph  ->  B  e.  P )
tglineinteq.c  |-  ( ph  ->  C  e.  P )
tglineinteq.d  |-  ( ph  ->  D  e.  P )
tglineinteq.e  |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
Assertion
Ref Expression
tglineneq  |-  ( ph  ->  ( A L B )  =/=  ( C L D ) )

Proof of Theorem tglineneq
StepHypRef Expression
1 tglineintmo.p . . . . 5  |-  P  =  ( Base `  G
)
2 tglineintmo.i . . . . 5  |-  I  =  (Itv `  G )
3 tglineintmo.l . . . . 5  |-  L  =  (LineG `  G )
4 tglineintmo.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
5 tglineinteq.a . . . . 5  |-  ( ph  ->  A  e.  P )
6 tglineinteq.b . . . . 5  |-  ( ph  ->  B  e.  P )
7 tglineinteq.c . . . . . 6  |-  ( ph  ->  C  e.  P )
8 tglineinteq.e . . . . . 6  |-  ( ph  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
91, 2, 3, 4, 5, 6, 7, 8ncolne1 23747 . . . . 5  |-  ( ph  ->  A  =/=  B )
10 eqid 2467 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
111, 10, 2, 4, 5, 6tgbtwntriv1 23638 . . . . 5  |-  ( ph  ->  A  e.  ( A I B ) )
121, 2, 3, 4, 5, 6, 5, 9, 11btwnlng1 23741 . . . 4  |-  ( ph  ->  A  e.  ( A L B ) )
1312adantr 465 . . 3  |-  ( (
ph  /\  C  =  D )  ->  A  e.  ( A L B ) )
14 simplr 754 . . . 4  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  C  =  D )
154adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  G  e. TarskiG )
167adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  C  e.  P )
17 tglineinteq.d . . . . . . . 8  |-  ( ph  ->  D  e.  P )
1817adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  D  e.  P )
19 simpr 461 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  A  e.  ( C L D ) )
201, 3, 2, 15, 16, 18, 19tglngne 23693 . . . . . 6  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  C  =/=  D )
2120adantlr 714 . . . . 5  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  C  =/=  D )
2221neneqd 2669 . . . 4  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  -.  C  =  D )
2314, 22pm2.65da 576 . . 3  |-  ( (
ph  /\  C  =  D )  ->  -.  A  e.  ( C L D ) )
24 nelne1 2796 . . 3  |-  ( ( A  e.  ( A L B )  /\  -.  A  e.  ( C L D ) )  ->  ( A L B )  =/=  ( C L D ) )
2513, 23, 24syl2anc 661 . 2  |-  ( (
ph  /\  C  =  D )  ->  ( A L B )  =/=  ( C L D ) )
264ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  G  e. TarskiG )
276ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  B  e.  P )
287ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  P )
295ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  e.  P )
30 pm2.46 398 . . . . . . . . 9  |-  ( -.  ( A  e.  ( B L C )  \/  B  =  C )  ->  -.  B  =  C )
318, 30syl 16 . . . . . . . 8  |-  ( ph  ->  -.  B  =  C )
3231neqned 2670 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
3332ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  B  =/=  C )
3417ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  D  e.  P )
35 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  =/=  D )
361, 10, 2, 4, 7, 17tgbtwntriv1 23638 . . . . . . . . 9  |-  ( ph  ->  C  e.  ( C I D ) )
3736ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( C I D ) )
381, 2, 3, 26, 28, 34, 28, 35, 37btwnlng1 23741 . . . . . . 7  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( C L D ) )
39 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  ( A L B )  =  ( C L D ) )
4038, 39eleqtrrd 2558 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( A L B ) )
411, 3, 2, 26, 29, 27, 40tglngne 23693 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  =/=  B )
421, 2, 3, 26, 27, 28, 29, 33, 40, 41lnrot1 23745 . . . . 5  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  e.  ( B L C ) )
4342orcd 392 . . . 4  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  ( A  e.  ( B L C )  \/  B  =  C ) )
448ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
4543, 44pm2.65da 576 . . 3  |-  ( (
ph  /\  C  =/=  D )  ->  -.  ( A L B )  =  ( C L D ) )
4645neqned 2670 . 2  |-  ( (
ph  /\  C  =/=  D )  ->  ( A L B )  =/=  ( C L D ) )
4725, 46pm2.61dane 2785 1  |-  ( ph  ->  ( A L B )  =/=  ( C L D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5588  (class class class)co 6284   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606
This theorem is referenced by:  tglineinteq  23766  perpneq  23827
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