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Theorem mirf 23913
Description: Point inversion as function. (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
)
Assertion
Ref Expression
mirf  |-  ( ph  ->  M : P --> P )

Proof of Theorem mirf
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6246 . . 3  |-  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) )  e. 
_V
21a1i 11 . 2  |-  ( (
ph  /\  y  e.  P )  ->  ( iota_ z  e.  P  ( ( A  .-  z
)  =  ( A 
.-  y )  /\  A  e.  ( z
I y ) ) )  e.  _V )
3 mirfv.m . . 3  |-  M  =  ( S `  A
)
4 mirval.p . . . 4  |-  P  =  ( Base `  G
)
5 mirval.d . . . 4  |-  .-  =  ( dist `  G )
6 mirval.i . . . 4  |-  I  =  (Itv `  G )
7 mirval.l . . . 4  |-  L  =  (LineG `  G )
8 mirval.s . . . 4  |-  S  =  (pInvG `  G )
9 mirval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
10 mirval.a . . . 4  |-  ( ph  ->  A  e.  P )
114, 5, 6, 7, 8, 9, 10mirval 23908 . . 3  |-  ( ph  ->  ( S `  A
)  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
123, 11syl5eq 2496 . 2  |-  ( ph  ->  M  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
139adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  G  e. TarskiG )
1410adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  A  e.  P )
15 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  x  e.  P )
164, 5, 6, 7, 8, 13, 14, 3, 15mirfv 23909 . . 3  |-  ( (
ph  /\  x  e.  P )  ->  ( M `  x )  =  ( iota_ z  e.  P  ( ( A 
.-  z )  =  ( A  .-  x
)  /\  A  e.  ( z I x ) ) ) )
174, 5, 6, 13, 15, 14mirreu3 23907 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  E! z  e.  P  (
( A  .-  z
)  =  ( A 
.-  x )  /\  A  e.  ( z
I x ) ) )
18 riotacl 6257 . . . 4  |-  ( E! z  e.  P  ( ( A  .-  z
)  =  ( A 
.-  x )  /\  A  e.  ( z
I x ) )  ->  ( iota_ z  e.  P  ( ( A 
.-  z )  =  ( A  .-  x
)  /\  A  e.  ( z I x ) ) )  e.  P )
1917, 18syl 16 . . 3  |-  ( (
ph  /\  x  e.  P )  ->  ( iota_ z  e.  P  ( ( A  .-  z
)  =  ( A 
.-  x )  /\  A  e.  ( z
I x ) ) )  e.  P )
2016, 19eqeltrd 2531 . 2  |-  ( (
ph  /\  x  e.  P )  ->  ( M `  x )  e.  P )
212, 12, 20fmpt2d 6046 1  |-  ( ph  ->  M : P --> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   E!wreu 2795   _Vcvv 3095    |-> cmpt 4495   -->wf 5574   ` cfv 5578   iota_crio 6241  (class class class)co 6281   Basecbs 14509   distcds 14583  TarskiGcstrkg 23697  Itvcitv 23704  LineGclng 23705  pInvGcmir 23905
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-rmo 2801  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-pw 3999  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-trkgc 23716  df-trkgb 23717  df-trkgcb 23718  df-trkg 23722  df-mir 23906
This theorem is referenced by:  mircl  23914  mirf1o  23921  mirbtwni  23923  mirbtwnb  23924  mirauto  23933  miduniq2  23936  krippenlem  23939
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