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Theorem mirauto 23201
Description: Point inversion preserves point inversion. (Contributed by Thierry Arnoux, 30-Jul-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 )
mirauto.m  |-  M  =  ( S `  T
)
mirauto.x  |-  X  =  ( M `  A
)
mirauto.y  |-  Y  =  ( M `  B
)
mirauto.z  |-  Z  =  ( M `  C
)
mirauto.0  |-  ( ph  ->  T  e.  P )
mirauto.1  |-  ( ph  ->  A  e.  P )
mirauto.2  |-  ( ph  ->  B  e.  P )
mirauto.3  |-  ( ph  ->  C  e.  P )
mirauto.4  |-  ( ph  ->  ( ( S `  A ) `  B
)  =  C )
Assertion
Ref Expression
mirauto  |-  ( ph  ->  ( ( S `  X ) `  Y
)  =  Z )

Proof of Theorem mirauto
StepHypRef Expression
1 mirval.p . . 3  |-  P  =  ( Base `  G
)
2 mirval.d . . 3  |-  .-  =  ( dist `  G )
3 mirval.i . . 3  |-  I  =  (Itv `  G )
4 mirval.l . . 3  |-  L  =  (LineG `  G )
5 mirval.s . . 3  |-  S  =  (pInvG `  G )
6 mirval.g . . 3  |-  ( ph  ->  G  e. TarskiG )
7 mirauto.x . . . 4  |-  X  =  ( M `  A
)
8 mirauto.0 . . . . . 6  |-  ( ph  ->  T  e.  P )
9 mirauto.m . . . . . 6  |-  M  =  ( S `  T
)
101, 2, 3, 4, 5, 6, 8, 9mirf 23187 . . . . 5  |-  ( ph  ->  M : P --> P )
11 mirauto.1 . . . . 5  |-  ( ph  ->  A  e.  P )
1210, 11ffvelrnd 5940 . . . 4  |-  ( ph  ->  ( M `  A
)  e.  P )
137, 12syl5eqel 2541 . . 3  |-  ( ph  ->  X  e.  P )
14 eqid 2451 . . 3  |-  ( S `
 X )  =  ( S `  X
)
15 mirauto.y . . . 4  |-  Y  =  ( M `  B
)
16 mirauto.2 . . . . 5  |-  ( ph  ->  B  e.  P )
1710, 16ffvelrnd 5940 . . . 4  |-  ( ph  ->  ( M `  B
)  e.  P )
1815, 17syl5eqel 2541 . . 3  |-  ( ph  ->  Y  e.  P )
19 mirauto.z . . . 4  |-  Z  =  ( M `  C
)
20 mirauto.3 . . . . 5  |-  ( ph  ->  C  e.  P )
2110, 20ffvelrnd 5940 . . . 4  |-  ( ph  ->  ( M `  C
)  e.  P )
2219, 21syl5eqel 2541 . . 3  |-  ( ph  ->  Z  e.  P )
23 mirauto.4 . . . . . 6  |-  ( ph  ->  ( ( S `  A ) `  B
)  =  C )
2423, 20eqeltrd 2537 . . . . 5  |-  ( ph  ->  ( ( S `  A ) `  B
)  e.  P )
25 eqid 2451 . . . . . 6  |-  ( S `
 A )  =  ( S `  A
)
261, 2, 3, 4, 5, 6, 11, 25, 16mircgr 23184 . . . . 5  |-  ( ph  ->  ( A  .-  (
( S `  A
) `  B )
)  =  ( A 
.-  B ) )
271, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26mircgrs 23199 . . . 4  |-  ( ph  ->  ( ( M `  A )  .-  ( M `  ( ( S `  A ) `  B ) ) )  =  ( ( M `
 A )  .-  ( M `  B ) ) )
287a1i 11 . . . . 5  |-  ( ph  ->  X  =  ( M `
 A ) )
2923fveq2d 5790 . . . . . 6  |-  ( ph  ->  ( M `  (
( S `  A
) `  B )
)  =  ( M `
 C ) )
3029, 19syl6reqr 2510 . . . . 5  |-  ( ph  ->  Z  =  ( M `
 ( ( S `
 A ) `  B ) ) )
3128, 30oveq12d 6205 . . . 4  |-  ( ph  ->  ( X  .-  Z
)  =  ( ( M `  A ) 
.-  ( M `  ( ( S `  A ) `  B
) ) ) )
327, 15oveq12i 6199 . . . . 5  |-  ( X 
.-  Y )  =  ( ( M `  A )  .-  ( M `  B )
)
3332a1i 11 . . . 4  |-  ( ph  ->  ( X  .-  Y
)  =  ( ( M `  A ) 
.-  ( M `  B ) ) )
3427, 31, 333eqtr4d 2501 . . 3  |-  ( ph  ->  ( X  .-  Z
)  =  ( X 
.-  Y ) )
351, 2, 3, 4, 5, 6, 11, 25, 16mirbtwn 23185 . . . . . 6  |-  ( ph  ->  A  e.  ( ( ( S `  A
) `  B )
I B ) )
3623oveq1d 6202 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  B ) I B )  =  ( C I B ) )
3735, 36eleqtrd 2539 . . . . 5  |-  ( ph  ->  A  e.  ( C I B ) )
381, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37mirbtwni 23197 . . . 4  |-  ( ph  ->  ( M `  A
)  e.  ( ( M `  C ) I ( M `  B ) ) )
3919, 15oveq12i 6199 . . . 4  |-  ( Z I Y )  =  ( ( M `  C ) I ( M `  B ) )
4038, 7, 393eltr4g 2555 . . 3  |-  ( ph  ->  X  e.  ( Z I Y ) )
411, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40ismir 23186 . 2  |-  ( ph  ->  Z  =  ( ( S `  X ) `
 Y ) )
4241eqcomd 2458 1  |-  ( ph  ->  ( ( S `  X ) `  Y
)  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` cfv 5513  (class class class)co 6187   Basecbs 14273   distcds 14346  TarskiGcstrkg 23002  Itvcitv 23009  LineGclng 23010  pInvGcmir 23178
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-pm 7314  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-card 8207  df-cda 8435  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-hash 12202  df-word 12328  df-concat 12330  df-s1 12331  df-s2 12574  df-s3 12575  df-trkgc 23021  df-trkgb 23022  df-trkgcb 23023  df-trkg 23027  df-cgrg 23080  df-mir 23179
This theorem is referenced by:  miduniq2  23204  krippenlem  23207
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