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Theorem mirne 24791
Description: Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020.)
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
)
mirinv.b  |-  ( ph  ->  B  e.  P )
mirne.1  |-  ( ph  ->  B  =/=  A )
Assertion
Ref Expression
mirne  |-  ( ph  ->  ( M `  B
)  =/=  A )

Proof of Theorem mirne
StepHypRef Expression
1 simpr 468 . . . . 5  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  ( M `  B )  =  A )
21fveq2d 5883 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  ( M `  ( M `  B
) )  =  ( M `  A ) )
3 mirval.p . . . . . 6  |-  P  =  ( Base `  G
)
4 mirval.d . . . . . 6  |-  .-  =  ( dist `  G )
5 mirval.i . . . . . 6  |-  I  =  (Itv `  G )
6 mirval.l . . . . . 6  |-  L  =  (LineG `  G )
7 mirval.s . . . . . 6  |-  S  =  (pInvG `  G )
8 mirval.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
9 mirval.a . . . . . 6  |-  ( ph  ->  A  e.  P )
10 mirfv.m . . . . . 6  |-  M  =  ( S `  A
)
11 mirinv.b . . . . . 6  |-  ( ph  ->  B  e.  P )
123, 4, 5, 6, 7, 8, 9, 10, 11mirmir 24786 . . . . 5  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
1312adantr 472 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  ( M `  ( M `  B
) )  =  B )
14 eqid 2471 . . . . . 6  |-  A  =  A
153, 4, 5, 6, 7, 8, 9, 10, 9mirinv 24790 . . . . . 6  |-  ( ph  ->  ( ( M `  A )  =  A  <-> 
A  =  A ) )
1614, 15mpbiri 241 . . . . 5  |-  ( ph  ->  ( M `  A
)  =  A )
1716adantr 472 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  ( M `  A )  =  A )
182, 13, 173eqtr3d 2513 . . 3  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  B  =  A )
19 mirne.1 . . . . 5  |-  ( ph  ->  B  =/=  A )
2019adantr 472 . . . 4  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  B  =/=  A )
2120neneqd 2648 . . 3  |-  ( (
ph  /\  ( M `  B )  =  A )  ->  -.  B  =  A )
2218, 21pm2.65da 586 . 2  |-  ( ph  ->  -.  ( M `  B )  =  A )
2322neqned 2650 1  |-  ( ph  ->  ( M `  B
)  =/=  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ` cfv 5589   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564  pInvGcmir 24776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580  df-mir 24777
This theorem is referenced by:  sacgr  24951
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