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Theorem mirf 23208
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 6168 . . 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 23203 . . 3  |-  ( ph  ->  ( S `  A
)  =  ( y  e.  P  |->  ( iota_ z  e.  P  ( ( A  .-  z )  =  ( A  .-  y )  /\  A  e.  ( z I y ) ) ) ) )
123, 11syl5eq 2507 . 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 23204 . . 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 23202 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  E! z  e.  P  (
( A  .-  z
)  =  ( A 
.-  x )  /\  A  e.  ( z
I x ) ) )
18 riotacl 6179 . . . 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 2542 . 2  |-  ( (
ph  /\  x  e.  P )  ->  ( M `  x )  e.  P )
212, 12, 20fmpt2d 5985 1  |-  ( ph  ->  M : P --> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E!wreu 2801   _Vcvv 3078    |-> cmpt 4461   -->wf 5525   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14295   distcds 14369  TarskiGcstrkg 23025  Itvcitv 23032  LineGclng 23033  pInvGcmir 23200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-trkgc 23044  df-trkgb 23045  df-trkgcb 23046  df-trkg 23050  df-mir 23201
This theorem is referenced by:  mircl  23209  mirf1o  23216  mirbtwni  23218  mirbtwnb  23219  mirauto  23222  miduniq2  23225  krippenlem  23228
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