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Theorem tglineneq 24548
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 24529 . . . . 5  |-  ( ph  ->  A  =/=  B )
101, 2, 3, 4, 5, 6, 9tglinerflx1 24537 . . . 4  |-  ( ph  ->  A  e.  ( A L B ) )
1110adantr 466 . . 3  |-  ( (
ph  /\  C  =  D )  ->  A  e.  ( A L B ) )
12 simplr 760 . . . 4  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  C  =  D )
134adantr 466 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  G  e. TarskiG )
147adantr 466 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  C  e.  P )
15 tglineinteq.d . . . . . . . 8  |-  ( ph  ->  D  e.  P )
1615adantr 466 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  D  e.  P )
17 simpr 462 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  A  e.  ( C L D ) )
181, 3, 2, 13, 14, 16, 17tglngne 24455 . . . . . 6  |-  ( (
ph  /\  A  e.  ( C L D ) )  ->  C  =/=  D )
1918adantlr 719 . . . . 5  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  C  =/=  D )
2019neneqd 2632 . . . 4  |-  ( ( ( ph  /\  C  =  D )  /\  A  e.  ( C L D ) )  ->  -.  C  =  D )
2112, 20pm2.65da 578 . . 3  |-  ( (
ph  /\  C  =  D )  ->  -.  A  e.  ( C L D ) )
22 nelne1 2760 . . 3  |-  ( ( A  e.  ( A L B )  /\  -.  A  e.  ( C L D ) )  ->  ( A L B )  =/=  ( C L D ) )
2311, 21, 22syl2anc 665 . 2  |-  ( (
ph  /\  C  =  D )  ->  ( A L B )  =/=  ( C L D ) )
244ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  G  e. TarskiG )
256ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  B  e.  P )
267ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  P )
275ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  e.  P )
28 pm2.46 399 . . . . . . . . 9  |-  ( -.  ( A  e.  ( B L C )  \/  B  =  C )  ->  -.  B  =  C )
298, 28syl 17 . . . . . . . 8  |-  ( ph  ->  -.  B  =  C )
3029neqned 2634 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
3130ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  B  =/=  C )
3215ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  D  e.  P )
33 simplr 760 . . . . . . . 8  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  =/=  D )
341, 2, 3, 24, 26, 32, 33tglinerflx1 24537 . . . . . . 7  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( C L D ) )
35 simpr 462 . . . . . . 7  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  ( A L B )  =  ( C L D ) )
3634, 35eleqtrrd 2520 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  C  e.  ( A L B ) )
371, 3, 2, 24, 27, 25, 36tglngne 24455 . . . . . 6  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  =/=  B )
381, 2, 3, 24, 25, 26, 27, 31, 36, 37lnrot1 24527 . . . . 5  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  A  e.  ( B L C ) )
3938orcd 393 . . . 4  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  ( A  e.  ( B L C )  \/  B  =  C ) )
408ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  C  =/=  D )  /\  ( A L B )  =  ( C L D ) )  ->  -.  ( A  e.  ( B L C )  \/  B  =  C ) )
4139, 40pm2.65da 578 . . 3  |-  ( (
ph  /\  C  =/=  D )  ->  -.  ( A L B )  =  ( C L D ) )
4241neqned 2634 . 2  |-  ( (
ph  /\  C  =/=  D )  ->  ( A L B )  =/=  ( C L D ) )
4323, 42pm2.61dane 2749 1  |-  ( ph  ->  ( A L B )  =/=  ( C L D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   ` cfv 5601  (class class class)co 6305   Basecbs 15084  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-trkgc 24359  df-trkgb 24360  df-trkgcb 24361  df-trkg 24364
This theorem is referenced by:  tglineinteq  24549  perpneq  24616
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