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Theorem mirfv 24157
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 24156 . . 3  |-  ( ph  ->  ( S `  A
)  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
101, 9syl5eq 2435 . 2  |-  ( ph  ->  M  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
11 simplr 753 . . . . . 6  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  y  =  B )
1211oveq2d 6212 . . . . 5  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  ( A  .-  y )  =  ( A  .-  B
) )
1312eqeq2d 2396 . . . 4  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
( A  .-  z
)  =  ( A 
.-  y )  <->  ( A  .-  z )  =  ( A  .-  B ) ) )
1411oveq2d 6212 . . . . 5  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
z I y )  =  ( z I B ) )
1514eleq2d 2452 . . . 4  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  ( A  e.  ( z
I y )  <->  A  e.  ( z I B ) ) )
1613, 15anbi12d 708 . . 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 6174 . 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 6162 . . 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 5862 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 367    = wceq 1399    e. wcel 1826   _Vcvv 3034    |-> cmpt 4425   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   distcds 14711  TarskiGcstrkg 23942  Itvcitv 23949  LineGclng 23950  pInvGcmir 24153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-mir 24154
This theorem is referenced by:  mircgr  24158  mirbtwn  24159  ismir  24160  mirf  24161  mireq  24166
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