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Theorem tglineneq 24768
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 24749 . . . . 5  |-  ( ph  ->  A  =/=  B )
101, 2, 3, 4, 5, 6, 9tglinerflx1 24757 . . . 4  |-  ( ph  ->  A  e.  ( A L B ) )
1110adantr 472 . . 3  |-  ( (
ph  /\  C  =  D )  ->  A  e.  ( A L B ) )
12 simplr 770 . . . 4  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  C  =  D )
134adantr 472 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  G  e. TarskiG )
147adantr 472 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  C  e.  P )
15 tglineinteq.d . . . . . . . 8  |-  ( ph  ->  D  e.  P )
1615adantr 472 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  D  e.  P )
17 simpr 468 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  A  e.  ( C L D ) )
181, 3, 2, 13, 14, 16, 17tglngne 24674 . . . . . 6  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  C  =/=  D )
1918adantlr 729 . . . . 5  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  C  =/=  D )
2019neneqd 2648 . . . 4  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  -.  C  =  D )
2112, 20pm2.65da 586 . . 3  |-  ( (
ph  /\  C  =  D )  ->  -.  A  e.  ( C L D ) )
22 nelne1 2739 . . 3  |-  ( ( A  e.  ( A L B )  /\  -.  A  e.  ( C L D ) )  ->  ( A L B )  =/=  ( C L D ) )
2311, 21, 22syl2anc 673 . 2  |-  ( (
ph  /\  C  =  D )  ->  ( A L B )  =/=  ( C L D ) )
244ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  G  e. TarskiG )
256ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  B  e.  P )
267ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  P )
275ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  e.  P )
28 pm2.46 405 . . . . . . . . 9  |-  ( -.  ( A  e.  ( B L C )  \/  B  =  C )  ->  -.  B  =  C )
298, 28syl 17 . . . . . . . 8  |-  ( ph  ->  -.  B  =  C )
3029neqned 2650 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
3130ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  B  =/=  C )
3215ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  D  e.  P )
33 simplr 770 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  =/=  D )
341, 2, 3, 24, 26, 32, 33tglinerflx1 24757 . . . . . . 7  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( C L D ) )
35 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  ( A L B )  =  ( C L D ) )
3634, 35eleqtrrd 2552 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( A L B ) )
371, 3, 2, 24, 27, 25, 36tglngne 24674 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  =/=  B )
381, 2, 3, 24, 25, 26, 27, 31, 36, 37lnrot1 24747 . . . . 5  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  e.  ( B L C ) )
3938orcd 399 . . . 4  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  ( A  e.  ( B L C )  \/  B  =  C ) )
408ad2antrr 740 . . . 4  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
4139, 40pm2.65da 586 . . 3  |-  ( (
ph  /\  C  =/=  D )  ->  -.  ( A L B )  =  ( C L D ) )
4241neqned 2650 . 2  |-  ( (
ph  /\  C  =/=  D )  ->  ( A L B )  =/=  ( C L D ) )
4323, 42pm2.61dane 2730 1  |-  ( ph  ->  ( A L B )  =/=  ( C L D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ` cfv 5589  (class class class)co 6308   Basecbs 15199  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580
This theorem is referenced by:  tglineinteq  24769  perpneq  24838
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