MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mirf Structured version   Unicode version

Theorem mirf 24242
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 6236 . . 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 24237 . . 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 463 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  G  e. TarskiG )
1410adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  A  e.  P )
15 simpr 459 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  x  e.  P )
164, 5, 6, 7, 8, 13, 14, 3, 15mirfv 24238 . . 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 24236 . . . 4  |-  ( (
ph  /\  x  e.  P )  ->  E! z  e.  P  (
( A  .-  z
)  =  ( A 
.-  x )  /\  A  e.  ( z
I x ) ) )
18 riotacl 6246 . . . 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 6037 1  |-  ( ph  ->  M : P --> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806   _Vcvv 3106    |-> cmpt 4497   -->wf 5566   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  pInvGcmir 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-mir 24235
This theorem is referenced by:  mircl  24243  mirf1o  24250  mirbtwni  24252  mirbtwnb  24253  mirauto  24262  miduniq2  24265  krippenlem  24268
  Copyright terms: Public domain W3C validator