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Theorem oppne3 24864
Description: Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-2020.)
Hypotheses
Ref Expression
hpg.p  |-  P  =  ( Base `  G
)
hpg.d  |-  .-  =  ( dist `  G )
hpg.i  |-  I  =  (Itv `  G )
hpg.o  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
opphl.l  |-  L  =  (LineG `  G )
opphl.d  |-  ( ph  ->  D  e.  ran  L
)
opphl.g  |-  ( ph  ->  G  e. TarskiG )
oppcom.a  |-  ( ph  ->  A  e.  P )
oppcom.b  |-  ( ph  ->  B  e.  P )
oppcom.o  |-  ( ph  ->  A O B )
Assertion
Ref Expression
oppne3  |-  ( ph  ->  A  =/=  B )
Distinct variable groups:    D, a,
b    I, a, b    P, a, b    t, A    t, B    t, D    t, G    t, L    t, I    t, O    t, P    ph, t    t,  .-    t, a, b
Allowed substitution hints:    ph( a, b)    A( a, b)    B( a, b)    G( a, b)    L( a, b)    .- ( a, b)    O( a, b)

Proof of Theorem oppne3
StepHypRef Expression
1 hpg.p . . . . . 6  |-  P  =  ( Base `  G
)
2 eqid 2471 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
3 hpg.i . . . . . 6  |-  I  =  (Itv `  G )
4 opphl.g . . . . . . 7  |-  ( ph  ->  G  e. TarskiG )
54ad3antrrr 744 . . . . . 6  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  G  e. TarskiG )
6 oppcom.a . . . . . . 7  |-  ( ph  ->  A  e.  P )
76ad3antrrr 744 . . . . . 6  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  A  e.  P )
8 opphl.l . . . . . . 7  |-  L  =  (LineG `  G )
9 opphl.d . . . . . . . 8  |-  ( ph  ->  D  e.  ran  L
)
109ad3antrrr 744 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  D  e.  ran  L )
11 simplr 770 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  t  e.  D )
121, 8, 3, 5, 10, 11tglnpt 24673 . . . . . 6  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  t  e.  P )
13 simpr 468 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  t  e.  ( A I B ) )
14 simpllr 777 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  A  =  B )
1514oveq2d 6324 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  ( A I A )  =  ( A I B ) )
1613, 15eleqtrrd 2552 . . . . . 6  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  t  e.  ( A I A ) )
171, 2, 3, 5, 7, 12, 16axtgbtwnid 24593 . . . . 5  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  A  =  t )
1817, 11eqeltrd 2549 . . . 4  |-  ( ( ( ( ph  /\  A  =  B )  /\  t  e.  D
)  /\  t  e.  ( A I B ) )  ->  A  e.  D )
19 oppcom.o . . . . . . 7  |-  ( ph  ->  A O B )
20 hpg.d . . . . . . . 8  |-  .-  =  ( dist `  G )
21 hpg.o . . . . . . . 8  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
22 oppcom.b . . . . . . . 8  |-  ( ph  ->  B  e.  P )
231, 20, 3, 21, 6, 22islnopp 24860 . . . . . . 7  |-  ( ph  ->  ( A O B  <-> 
( ( -.  A  e.  D  /\  -.  B  e.  D )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
2419, 23mpbid 215 . . . . . 6  |-  ( ph  ->  ( ( -.  A  e.  D  /\  -.  B  e.  D )  /\  E. t  e.  D  t  e.  ( A I B ) ) )
2524simprd 470 . . . . 5  |-  ( ph  ->  E. t  e.  D  t  e.  ( A I B ) )
2625adantr 472 . . . 4  |-  ( (
ph  /\  A  =  B )  ->  E. t  e.  D  t  e.  ( A I B ) )
2718, 26r19.29a 2918 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  e.  D )
281, 20, 3, 21, 8, 9, 4, 6, 22, 19oppne1 24862 . . . 4  |-  ( ph  ->  -.  A  e.  D
)
2928adantr 472 . . 3  |-  ( (
ph  /\  A  =  B )  ->  -.  A  e.  D )
3027, 29pm2.65da 586 . 2  |-  ( ph  ->  -.  A  =  B )
3130neqned 2650 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    \ cdif 3387   class class class wbr 4395   {copab 4453   ran crn 4840   ` cfv 5589  (class class class)co 6308   Basecbs 15199   distcds 15277  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-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
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-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-br 4396  df-opab 4455  df-cnv 4847  df-dm 4849  df-rn 4850  df-iota 5553  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-trkgb 24576  df-trkg 24580
This theorem is referenced by:  colopp  24890  colhp  24891  trgcopyeulem  24926
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