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Theorem aishp 26172
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
aishp  |-  ( ph  ->  ( (Halfplane `  G
) `  M )  =  ( ( P 
\  M ) /.  .~  ) )

Proof of Theorem aishp
Dummy variables  g 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-halfplane 26171 . . . . 5  |- Halfplane  =  ( g  e. Ibg  |->  ( l  e.  (PLines `  g
)  |->  ( ( (PPoints `  g )  \  l
) /. ( (ss
`  g ) `  l ) ) ) )
21a1i 10 . . . 4  |-  ( ph  -> Halfplane  =  ( g  e. Ibg  |->  ( l  e.  (PLines `  g )  |->  ( ( (PPoints `  g )  \  l ) /. ( (ss `  g ) `
 l ) ) ) ) )
3 fveq2 5525 . . . . . 6  |-  ( g  =  G  ->  (PLines `  g )  =  (PLines `  G ) )
4 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  (PPoints `  g )  =  (PPoints `  G ) )
54difeq1d 3293 . . . . . . . 8  |-  ( g  =  G  ->  (
(PPoints `  g )  \ 
l )  =  ( (PPoints `  G )  \  l ) )
6 qseq1 6709 . . . . . . . 8  |-  ( ( (PPoints `  g )  \  l )  =  ( (PPoints `  G
)  \  l )  ->  ( ( (PPoints `  g
)  \  l ) /. ( (ss `  g
) `  l )
)  =  ( ( (PPoints `  G )  \  l ) /. ( (ss `  g ) `
 l ) ) )
75, 6syl 15 . . . . . . 7  |-  ( g  =  G  ->  (
( (PPoints `  g
)  \  l ) /. ( (ss `  g
) `  l )
)  =  ( ( (PPoints `  G )  \  l ) /. ( (ss `  g ) `
 l ) ) )
8 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  (ss `  g )  =  (ss
`  G ) )
98fveq1d 5527 . . . . . . . 8  |-  ( g  =  G  ->  (
(ss `  g ) `  l )  =  ( (ss `  G ) `
 l ) )
10 qseq2 6710 . . . . . . . 8  |-  ( ( (ss `  g ) `
 l )  =  ( (ss `  G
) `  l )  ->  ( ( (PPoints `  G
)  \  l ) /. ( (ss `  g
) `  l )
)  =  ( ( (PPoints `  G )  \  l ) /. ( (ss `  G ) `
 l ) ) )
119, 10syl 15 . . . . . . 7  |-  ( g  =  G  ->  (
( (PPoints `  G
)  \  l ) /. ( (ss `  g
) `  l )
)  =  ( ( (PPoints `  G )  \  l ) /. ( (ss `  G ) `
 l ) ) )
127, 11eqtrd 2315 . . . . . 6  |-  ( g  =  G  ->  (
( (PPoints `  g
)  \  l ) /. ( (ss `  g
) `  l )
)  =  ( ( (PPoints `  G )  \  l ) /. ( (ss `  G ) `
 l ) ) )
133, 12mpteq12dv 4098 . . . . 5  |-  ( g  =  G  ->  (
l  e.  (PLines `  g )  |->  ( ( (PPoints `  g )  \  l ) /. ( (ss `  g ) `
 l ) ) )  =  ( l  e.  (PLines `  G
)  |->  ( ( (PPoints `  G )  \  l
) /. ( (ss
`  G ) `  l ) ) ) )
1413adantl 452 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  (
l  e.  (PLines `  g )  |->  ( ( (PPoints `  g )  \  l ) /. ( (ss `  g ) `
 l ) ) )  =  ( l  e.  (PLines `  G
)  |->  ( ( (PPoints `  G )  \  l
) /. ( (ss
`  G ) `  l ) ) ) )
15 abhp.1 . . . 4  |-  ( ph  ->  G  e. Ibg )
16 fvex 5539 . . . . 5  |-  (PLines `  G )  e.  _V
17 mptexg 5745 . . . . 5  |-  ( (PLines `  G )  e.  _V  ->  ( l  e.  (PLines `  G )  |->  ( ( (PPoints `  G )  \  l ) /. ( (ss `  G ) `
 l ) ) )  e.  _V )
1816, 17mp1i 11 . . . 4  |-  ( ph  ->  ( l  e.  (PLines `  G )  |->  ( ( (PPoints `  G )  \  l ) /. ( (ss `  G ) `
 l ) ) )  e.  _V )
192, 14, 15, 18fvmptd 5606 . . 3  |-  ( ph  ->  (Halfplane `  G )  =  ( l  e.  (PLines `  G )  |->  ( ( (PPoints `  G
)  \  l ) /. ( (ss `  G
) `  l )
) ) )
20 difeq2 3288 . . . . . 6  |-  ( l  =  M  ->  (
(PPoints `  G )  \ 
l )  =  ( (PPoints `  G )  \  M ) )
21 qseq1 6709 . . . . . 6  |-  ( ( (PPoints `  G )  \  l )  =  ( (PPoints `  G
)  \  M )  ->  ( ( (PPoints `  G
)  \  l ) /. ( (ss `  G
) `  l )
)  =  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  l ) ) )
2220, 21syl 15 . . . . 5  |-  ( l  =  M  ->  (
( (PPoints `  G
)  \  l ) /. ( (ss `  G
) `  l )
)  =  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  l ) ) )
23 fveq2 5525 . . . . . 6  |-  ( l  =  M  ->  (
(ss `  G ) `  l )  =  ( (ss `  G ) `
 M ) )
24 qseq2 6710 . . . . . 6  |-  ( ( (ss `  G ) `
 l )  =  ( (ss `  G
) `  M )  ->  ( ( (PPoints `  G
)  \  M ) /. ( (ss `  G
) `  l )
)  =  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  M ) ) )
2523, 24syl 15 . . . . 5  |-  ( l  =  M  ->  (
( (PPoints `  G
)  \  M ) /. ( (ss `  G
) `  l )
)  =  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  M ) ) )
2622, 25eqtrd 2315 . . . 4  |-  ( l  =  M  ->  (
( (PPoints `  G
)  \  l ) /. ( (ss `  G
) `  l )
)  =  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  M ) ) )
2726adantl 452 . . 3  |-  ( (
ph  /\  l  =  M )  ->  (
( (PPoints `  G
)  \  l ) /. ( (ss `  G
) `  l )
)  =  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  M ) ) )
28 abhp.2 . . . 4  |-  ( ph  ->  M  e.  L )
29 abhp.5 . . . 4  |-  L  =  (PLines `  G )
3028, 29syl6eleq 2373 . . 3  |-  ( ph  ->  M  e.  (PLines `  G ) )
31 fvex 5539 . . . . . 6  |-  (PPoints `  G
)  e.  _V
32 difexg 4162 . . . . . 6  |-  ( (PPoints `  G )  e.  _V  ->  ( (PPoints `  G
)  \  M )  e.  _V )
3331, 32ax-mp 8 . . . . 5  |-  ( (PPoints `  G )  \  M
)  e.  _V
3433qsex 6718 . . . 4  |-  ( ( (PPoints `  G )  \  M ) /. (
(ss `  G ) `  M ) )  e. 
_V
3534a1i 10 . . 3  |-  ( ph  ->  ( ( (PPoints `  G
)  \  M ) /. ( (ss `  G
) `  M )
)  e.  _V )
3619, 27, 30, 35fvmptd 5606 . 2  |-  ( ph  ->  ( (Halfplane `  G
) `  M )  =  ( ( (PPoints `  G )  \  M
) /. ( (ss
`  G ) `  M ) ) )
37 abhp.3 . . . . 5  |-  P  =  (PPoints `  G )
3837eqcomi 2287 . . . 4  |-  (PPoints `  G
)  =  P
3938difeq1i 3290 . . 3  |-  ( (PPoints `  G )  \  M
)  =  ( P 
\  M )
40 qseq1 6709 . . 3  |-  ( ( (PPoints `  G )  \  M )  =  ( P  \  M )  ->  ( ( (PPoints `  G )  \  M
) /. ( (ss
`  G ) `  M ) )  =  ( ( P  \  M ) /. (
(ss `  G ) `  M ) ) )
4139, 40mp1i 11 . 2  |-  ( ph  ->  ( ( (PPoints `  G
)  \  M ) /. ( (ss `  G
) `  M )
)  =  ( ( P  \  M ) /. ( (ss `  G ) `  M
) ) )
42 abhp.4 . . . 4  |-  .~  =  ( (ss `  G ) `
 M )
4342eqcomi 2287 . . 3  |-  ( (ss
`  G ) `  M )  =  .~
44 qseq2 6710 . . 3  |-  ( ( (ss `  G ) `
 M )  =  .~  ->  ( ( P  \  M ) /. ( (ss `  G ) `
 M ) )  =  ( ( P 
\  M ) /.  .~  ) )
4543, 44mp1i 11 . 2  |-  ( ph  ->  ( ( P  \  M ) /. (
(ss `  G ) `  M ) )  =  ( ( P  \  M ) /.  .~  ) )
4636, 41, 453eqtrd 2319 1  |-  ( ph  ->  ( (Halfplane `  G
) `  M )  =  ( ( P 
\  M ) /.  .~  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    e. cmpt 4077   ` cfv 5255   /.cqs 6659  PPointscpoints 26056  PLinescplines 26058  Ibgcibg 26107  sscsas 26162  Halfplanechalfp 26170
This theorem is referenced by:  abhp  26173  abhp1  26174
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-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-sn 3646  df-pr 3647  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-ec 6662  df-qs 6666  df-halfplane 26171
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