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Theorem abhp 26173
Description: The half planes delimited by  M. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
abhp.1  |-  ( ph  ->  G  e. Ibg )
abhp.2  |-  ( ph  ->  M  e.  L )
abhp.3  |-  P  =  (PPoints `  G )
abhp.4  |-  .~  =  ( (ss `  G ) `
 M )
abhp.5  |-  L  =  (PLines `  G )
Assertion
Ref Expression
abhp  |-  ( ph  ->  ( H  e.  ( (Halfplane `  G ) `  M )  <->  E. x  e.  ( P  \  M
) ( H  =  [ x ]  .~  \/  H  =  (
( P  \  M
)  \  [ x ]  .~  ) ) ) )
Distinct variable groups:    x,  .~    x, H    x, M    x, P    ph, x
Allowed substitution hints:    G( x)    L( x)

Proof of Theorem abhp
StepHypRef Expression
1 abhp.1 . . . 4  |-  ( ph  ->  G  e. Ibg )
2 abhp.2 . . . 4  |-  ( ph  ->  M  e.  L )
3 abhp.3 . . . 4  |-  P  =  (PPoints `  G )
4 abhp.4 . . . 4  |-  .~  =  ( (ss `  G ) `
 M )
5 abhp.5 . . . 4  |-  L  =  (PLines `  G )
61, 2, 3, 4, 5aishp 26172 . . 3  |-  ( ph  ->  ( (Halfplane `  G
) `  M )  =  ( ( P 
\  M ) /.  .~  ) )
76eleq2d 2350 . 2  |-  ( ph  ->  ( H  e.  ( (Halfplane `  G ) `  M )  <->  H  e.  ( ( P  \  M ) /.  .~  ) ) )
81isibg1a 26111 . . . . 5  |-  ( ph  ->  G  e. Ig )
93, 5, 8, 2gltpntl 26072 . . . 4  |-  ( ph  ->  E. x  e.  P  x  e/  M )
10 df-nel 2449 . . . . . . . . . . . . 13  |-  ( x  e/  M  <->  -.  x  e.  M )
1110biimpi 186 . . . . . . . . . . . 12  |-  ( x  e/  M  ->  -.  x  e.  M )
1211anim2i 552 . . . . . . . . . . 11  |-  ( ( x  e.  P  /\  x  e/  M )  -> 
( x  e.  P  /\  -.  x  e.  M
) )
13 eldif 3162 . . . . . . . . . . 11  |-  ( x  e.  ( P  \  M )  <->  ( x  e.  P  /\  -.  x  e.  M ) )
1412, 13sylibr 203 . . . . . . . . . 10  |-  ( ( x  e.  P  /\  x  e/  M )  ->  x  e.  ( P  \  M ) )
1514adantl 452 . . . . . . . . 9  |-  ( ( H  e.  ( ( P  \  M ) /.  .~  )  /\  ( x  e.  P  /\  x  e/  M ) )  ->  x  e.  ( P  \  M ) )
16 simpl 443 . . . . . . . . 9  |-  ( ( H  e.  ( ( P  \  M ) /.  .~  )  /\  ( x  e.  P  /\  x  e/  M ) )  ->  H  e.  ( ( P  \  M ) /.  .~  ) )
1715, 16jca 518 . . . . . . . 8  |-  ( ( H  e.  ( ( P  \  M ) /.  .~  )  /\  ( x  e.  P  /\  x  e/  M ) )  ->  ( x  e.  ( P  \  M
)  /\  H  e.  ( ( P  \  M ) /.  .~  ) ) )
1817ex 423 . . . . . . 7  |-  ( H  e.  ( ( P 
\  M ) /.  .~  )  ->  ( (
x  e.  P  /\  x  e/  M )  -> 
( x  e.  ( P  \  M )  /\  H  e.  ( ( P  \  M
) /.  .~  )
) ) )
1918reximdv2 2652 . . . . . 6  |-  ( H  e.  ( ( P 
\  M ) /.  .~  )  ->  ( E. x  e.  P  x  e/  M  ->  E. x  e.  ( P  \  M
) H  e.  ( ( P  \  M
) /.  .~  )
) )
2019com12 27 . . . . 5  |-  ( E. x  e.  P  x  e/  M  ->  ( H  e.  ( ( P  \  M ) /.  .~  )  ->  E. x  e.  ( P  \  M
) H  e.  ( ( P  \  M
) /.  .~  )
) )
211adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  G  e. Ibg )
222adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  M  e.  L )
23 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  x  e.  ( P  \  M ) )
243, 5, 4, 21, 22, 23pdiveql 26168 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( H  e.  ( ( P  \  M ) /.  .~  ) 
<->  ( H  =  [
x ]  .~  \/  H  =  ( ( P  \  M )  \  [ x ]  .~  ) ) ) )
2524biimpd 198 . . . . . 6  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( H  e.  ( ( P  \  M ) /.  .~  )  ->  ( H  =  [ x ]  .~  \/  H  =  (
( P  \  M
)  \  [ x ]  .~  ) ) ) )
2625reximdva 2655 . . . . 5  |-  ( ph  ->  ( E. x  e.  ( P  \  M
) H  e.  ( ( P  \  M
) /.  .~  )  ->  E. x  e.  ( P  \  M ) ( H  =  [
x ]  .~  \/  H  =  ( ( P  \  M )  \  [ x ]  .~  ) ) ) )
2720, 26syl9 66 . . . 4  |-  ( E. x  e.  P  x  e/  M  ->  ( ph  ->  ( H  e.  ( ( P  \  M ) /.  .~  )  ->  E. x  e.  ( P  \  M ) ( H  =  [
x ]  .~  \/  H  =  ( ( P  \  M )  \  [ x ]  .~  ) ) ) ) )
289, 27mpcom 32 . . 3  |-  ( ph  ->  ( H  e.  ( ( P  \  M
) /.  .~  )  ->  E. x  e.  ( P  \  M ) ( H  =  [
x ]  .~  \/  H  =  ( ( P  \  M )  \  [ x ]  .~  ) ) ) )
2924biimprd 214 . . . 4  |-  ( (
ph  /\  x  e.  ( P  \  M ) )  ->  ( ( H  =  [ x ]  .~  \/  H  =  ( ( P  \  M )  \  [
x ]  .~  )
)  ->  H  e.  ( ( P  \  M ) /.  .~  ) ) )
3029rexlimdva 2667 . . 3  |-  ( ph  ->  ( E. x  e.  ( P  \  M
) ( H  =  [ x ]  .~  \/  H  =  (
( P  \  M
)  \  [ x ]  .~  ) )  ->  H  e.  ( ( P  \  M ) /.  .~  ) ) )
3128, 30impbid 183 . 2  |-  ( ph  ->  ( H  e.  ( ( P  \  M
) /.  .~  )  <->  E. x  e.  ( P 
\  M ) ( H  =  [ x ]  .~  \/  H  =  ( ( P  \  M )  \  [
x ]  .~  )
) ) )
327, 31bitrd 244 1  |-  ( ph  ->  ( H  e.  ( (Halfplane `  G ) `  M )  <->  E. x  e.  ( P  \  M
) ( H  =  [ x ]  .~  \/  H  =  (
( P  \  M
)  \  [ x ]  .~  ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    e/ wnel 2447   E.wrex 2544    \ cdif 3149   ` cfv 5255   [cec 6658   /.cqs 6659  PPointscpoints 26056  PLinescplines 26058  Ibgcibg 26107  sscsas 26162  Halfplanechalfp 26170
This theorem is referenced by:  bhp2a  26176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-ec 6662  df-qs 6666  df-ig2 26061  df-li 26077  df-col 26091  df-ibg2 26109  df-seg2 26131  df-sside 26163  df-halfplane 26171
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