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Theorem mirfv 24015
Description: Value of the point inversion function  M. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-2019.)
Hypotheses
Ref Expression
mirval.p  |-  P  =  ( Base `  G
)
mirval.d  |-  .-  =  ( dist `  G )
mirval.i  |-  I  =  (Itv `  G )
mirval.l  |-  L  =  (LineG `  G )
mirval.s  |-  S  =  (pInvG `  G )
mirval.g  |-  ( ph  ->  G  e. TarskiG )
mirval.a  |-  ( ph  ->  A  e.  P )
mirfv.m  |-  M  =  ( S `  A
)
mirfv.b  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
mirfv  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
Distinct variable groups:    z, A    z, B    z, G    z, M    z, I    z, P    ph, z    z,  .-
Allowed substitution hints:    S( z)    L( z)

Proof of Theorem mirfv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mirfv.m . . 3  |-  M  =  ( S `  A
)
2 mirval.p . . . 4  |-  P  =  ( Base `  G
)
3 mirval.d . . . 4  |-  .-  =  ( dist `  G )
4 mirval.i . . . 4  |-  I  =  (Itv `  G )
5 mirval.l . . . 4  |-  L  =  (LineG `  G )
6 mirval.s . . . 4  |-  S  =  (pInvG `  G )
7 mirval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
8 mirval.a . . . 4  |-  ( ph  ->  A  e.  P )
92, 3, 4, 5, 6, 7, 8mirval 24014 . . 3  |-  ( ph  ->  ( S `  A
)  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
101, 9syl5eq 2496 . 2  |-  ( ph  ->  M  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
11 simplr 755 . . . . . 6  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  y  =  B )
1211oveq2d 6297 . . . . 5  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  ( A  .-  y )  =  ( A  .-  B
) )
1312eqeq2d 2457 . . . 4  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
( A  .-  z
)  =  ( A 
.-  y )  <->  ( A  .-  z )  =  ( A  .-  B ) ) )
1411oveq2d 6297 . . . . 5  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
z I y )  =  ( z I B ) )
1514eleq2d 2513 . . . 4  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  ( A  e.  ( z
I y )  <->  A  e.  ( z I B ) ) )
1613, 15anbi12d 710 . . 3  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) )  <->  ( ( A 
.-  z )  =  ( A  .-  B
)  /\  A  e.  ( z I B ) ) ) )
1716riotabidva 6259 . 2  |-  ( (
ph  /\  y  =  B )  ->  ( iota_ z  e.  P  ( ( A  .-  z
)  =  ( A 
.-  y )  /\  A  e.  ( z
I y ) ) )  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
18 mirfv.b . 2  |-  ( ph  ->  B  e.  P )
19 riotaex 6246 . . 3  |-  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  e. 
_V
2019a1i 11 . 2  |-  ( ph  ->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) )  e.  _V )
2110, 17, 18, 20fvmptd 5946 1  |-  ( ph  ->  ( M `  B
)  =  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  B )  /\  A  e.  ( z I B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095    |-> cmpt 4495   ` cfv 5578   iota_crio 6241  (class class class)co 6281   Basecbs 14614   distcds 14688  TarskiGcstrkg 23803  Itvcitv 23810  LineGclng 23811  pInvGcmir 24011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-mir 24012
This theorem is referenced by:  mircgr  24016  mirbtwn  24017  ismir  24018  mirf  24019  mireq  24024
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