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Theorem mirfv 23060
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 23059 . . 3  |-  ( ph  ->  ( S `  A
)  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
101, 9syl5eq 2487 . 2  |-  ( ph  ->  M  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
11 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  y  =  B )
1211oveq2d 6107 . . . . 5  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  ( A  .-  y )  =  ( A  .-  B
) )
1312eqeq2d 2454 . . . 4  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
( A  .-  z
)  =  ( A 
.-  y )  <->  ( A  .-  z )  =  ( A  .-  B ) ) )
1411oveq2d 6107 . . . . 5  |-  ( ( ( ph  /\  y  =  B )  /\  z  e.  P )  ->  (
z I y )  =  ( z I B ) )
1514eleq2d 2510 . . . 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 6069 . 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 6056 . . 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 5779 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 1369    e. wcel 1756   _Vcvv 2972    e. cmpt 4350   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897  LineGclng 22898  pInvGcmir 23055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-mir 23056
This theorem is referenced by:  mircgr  23061  mirbtwn  23062  ismir  23063  mirf  23064  mireq  23069
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