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Theorem islnopp 24317
Description: The property for two points  A and  B to lie on the opposite sides of a set  D Definition 9.1 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
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 ) ) }
islnopp.a  |-  ( ph  ->  A  e.  P )
islnopp.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
islnopp  |-  ( ph  ->  ( A O B  <-> 
( ( -.  A  e.  D  /\  -.  B  e.  D )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
Distinct variable groups:    A, a,
b, t    B, a,
b, t    D, a,
b, t    G, a,
b, t    I, a,
b, t    t, O    P, a, b, t    ph, a,
b, t    .- , a, b, t
Allowed substitution hints:    O( a, b)

Proof of Theorem islnopp
StepHypRef Expression
1 islnopp.a . . 3  |-  ( ph  ->  A  e.  P )
2 islnopp.b . . 3  |-  ( ph  ->  B  e.  P )
3 eleq1 2526 . . . . . 6  |-  ( a  =  A  ->  (
a  e.  ( P 
\  D )  <->  A  e.  ( P  \  D ) ) )
43anbi1d 702 . . . . 5  |-  ( a  =  A  ->  (
( a  e.  ( P  \  D )  /\  b  e.  ( P  \  D ) )  <->  ( A  e.  ( P  \  D
)  /\  b  e.  ( P  \  D ) ) ) )
5 id 22 . . . . . . . 8  |-  ( a  =  A  ->  a  =  A )
65oveq1d 6285 . . . . . . 7  |-  ( a  =  A  ->  (
a I b )  =  ( A I b ) )
76eleq2d 2524 . . . . . 6  |-  ( a  =  A  ->  (
t  e.  ( a I b )  <->  t  e.  ( A I b ) ) )
87rexbidv 2965 . . . . 5  |-  ( a  =  A  ->  ( E. t  e.  D  t  e.  ( a
I b )  <->  E. t  e.  D  t  e.  ( A I b ) ) )
94, 8anbi12d 708 . . . 4  |-  ( a  =  A  ->  (
( ( a  e.  ( P  \  D
)  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) )  <->  ( ( A  e.  ( P  \  D )  /\  b  e.  ( P  \  D
) )  /\  E. t  e.  D  t  e.  ( A I b ) ) ) )
10 eleq1 2526 . . . . . 6  |-  ( b  =  B  ->  (
b  e.  ( P 
\  D )  <->  B  e.  ( P  \  D ) ) )
1110anbi2d 701 . . . . 5  |-  ( b  =  B  ->  (
( A  e.  ( P  \  D )  /\  b  e.  ( P  \  D ) )  <->  ( A  e.  ( P  \  D
)  /\  B  e.  ( P  \  D ) ) ) )
12 oveq2 6278 . . . . . . 7  |-  ( b  =  B  ->  ( A I b )  =  ( A I B ) )
1312eleq2d 2524 . . . . . 6  |-  ( b  =  B  ->  (
t  e.  ( A I b )  <->  t  e.  ( A I B ) ) )
1413rexbidv 2965 . . . . 5  |-  ( b  =  B  ->  ( E. t  e.  D  t  e.  ( A I b )  <->  E. t  e.  D  t  e.  ( A I B ) ) )
1511, 14anbi12d 708 . . . 4  |-  ( b  =  B  ->  (
( ( A  e.  ( P  \  D
)  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( A I b ) )  <->  ( ( A  e.  ( P  \  D )  /\  B  e.  ( P  \  D
) )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
16 hpg.o . . . 4  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
179, 15, 16brabg 4755 . . 3  |-  ( ( A  e.  P  /\  B  e.  P )  ->  ( A O B  <-> 
( ( A  e.  ( P  \  D
)  /\  B  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
181, 2, 17syl2anc 659 . 2  |-  ( ph  ->  ( A O B  <-> 
( ( A  e.  ( P  \  D
)  /\  B  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
191biantrurd 506 . . . . 5  |-  ( ph  ->  ( -.  A  e.  D  <->  ( A  e.  P  /\  -.  A  e.  D ) ) )
20 eldif 3471 . . . . 5  |-  ( A  e.  ( P  \  D )  <->  ( A  e.  P  /\  -.  A  e.  D ) )
2119, 20syl6bbr 263 . . . 4  |-  ( ph  ->  ( -.  A  e.  D  <->  A  e.  ( P  \  D ) ) )
222biantrurd 506 . . . . 5  |-  ( ph  ->  ( -.  B  e.  D  <->  ( B  e.  P  /\  -.  B  e.  D ) ) )
23 eldif 3471 . . . . 5  |-  ( B  e.  ( P  \  D )  <->  ( B  e.  P  /\  -.  B  e.  D ) )
2422, 23syl6bbr 263 . . . 4  |-  ( ph  ->  ( -.  B  e.  D  <->  B  e.  ( P  \  D ) ) )
2521, 24anbi12d 708 . . 3  |-  ( ph  ->  ( ( -.  A  e.  D  /\  -.  B  e.  D )  <->  ( A  e.  ( P  \  D
)  /\  B  e.  ( P  \  D ) ) ) )
2625anbi1d 702 . 2  |-  ( ph  ->  ( ( ( -.  A  e.  D  /\  -.  B  e.  D
)  /\  E. t  e.  D  t  e.  ( A I B ) )  <->  ( ( A  e.  ( P  \  D )  /\  B  e.  ( P  \  D
) )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
2718, 26bitr4d 256 1  |-  ( ph  ->  ( A O B  <-> 
( ( -.  A  e.  D  /\  -.  B  e.  D )  /\  E. t  e.  D  t  e.  ( A I B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805    \ cdif 3458   class class class wbr 4439   {copab 4496   ` cfv 5570  (class class class)co 6270   Basecbs 14719   distcds 14796  Itvcitv 24033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  oppne1  24318  oppne2  24319  oppcom  24320  oppnid  24322  opphllem1  24323  opphllem2  24324  opphllem3  24325  opphllem4  24326  opphllem5  24327  opphllem6  24328  lnopp2hpgb  24336  hpgerlem  24338
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