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Theorem lmimid 24848
Description: If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
ismid.p  |-  P  =  ( Base `  G
)
ismid.d  |-  .-  =  ( dist `  G )
ismid.i  |-  I  =  (Itv `  G )
ismid.g  |-  ( ph  ->  G  e. TarskiG )
ismid.1  |-  ( ph  ->  GDimTarskiG 2 )
lmif.m  |-  M  =  ( (lInvG `  G
) `  D )
lmif.l  |-  L  =  (LineG `  G )
lmif.d  |-  ( ph  ->  D  e.  ran  L
)
lmicl.1  |-  ( ph  ->  A  e.  P )
lmimid.s  |-  S  =  ( (pInvG `  G
) `  B )
lmimid.r  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
lmimid.a  |-  ( ph  ->  A  e.  D )
lmimid.b  |-  ( ph  ->  B  e.  D )
lmimid.c  |-  ( ph  ->  C  e.  P )
lmimid.d  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
lmimid  |-  ( ph  ->  ( M `  C
)  =  ( S `
 C ) )

Proof of Theorem lmimid
StepHypRef Expression
1 lmimid.s . . . . . . 7  |-  S  =  ( (pInvG `  G
) `  B )
21a1i 11 . . . . . 6  |-  ( ph  ->  S  =  ( (pInvG `  G ) `  B
) )
32fveq1d 5872 . . . . 5  |-  ( ph  ->  ( S `  C
)  =  ( ( (pInvG `  G ) `  B ) `  C
) )
4 ismid.p . . . . . 6  |-  P  =  ( Base `  G
)
5 ismid.d . . . . . 6  |-  .-  =  ( dist `  G )
6 ismid.i . . . . . 6  |-  I  =  (Itv `  G )
7 ismid.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
8 ismid.1 . . . . . 6  |-  ( ph  ->  GDimTarskiG 2 )
9 lmimid.c . . . . . 6  |-  ( ph  ->  C  e.  P )
10 lmif.l . . . . . . 7  |-  L  =  (LineG `  G )
11 eqid 2453 . . . . . . 7  |-  (pInvG `  G )  =  (pInvG `  G )
12 lmif.d . . . . . . . 8  |-  ( ph  ->  D  e.  ran  L
)
13 lmimid.b . . . . . . . 8  |-  ( ph  ->  B  e.  D )
144, 10, 6, 7, 12, 13tglnpt 24606 . . . . . . 7  |-  ( ph  ->  B  e.  P )
154, 5, 6, 10, 11, 7, 14, 1, 9mircl 24718 . . . . . 6  |-  ( ph  ->  ( S `  C
)  e.  P )
164, 5, 6, 7, 8, 9, 15, 11, 14ismidb 24832 . . . . 5  |-  ( ph  ->  ( ( S `  C )  =  ( ( (pInvG `  G
) `  B ) `  C )  <->  ( C
(midG `  G )
( S `  C
) )  =  B ) )
173, 16mpbid 214 . . . 4  |-  ( ph  ->  ( C (midG `  G ) ( S `
 C ) )  =  B )
1817, 13eqeltrd 2531 . . 3  |-  ( ph  ->  ( C (midG `  G ) ( S `
 C ) )  e.  D )
19 df-ne 2626 . . . . . 6  |-  ( C  =/=  ( S `  C )  <->  -.  C  =  ( S `  C ) )
207adantr 467 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  G  e. TarskiG )
2112adantr 467 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  D  e.  ran  L )
229adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  e.  P )
2315adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  ( S `  C )  e.  P
)
24 simpr 463 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  =/=  ( S `  C ) )
254, 6, 10, 20, 22, 23, 24tgelrnln 24687 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  ( C L ( S `  C ) )  e. 
ran  L )
2613adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  D )
2714adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  P )
284, 5, 6, 7, 8, 9, 15midbtwn 24833 . . . . . . . . . . . 12  |-  ( ph  ->  ( C (midG `  G ) ( S `
 C ) )  e.  ( C I ( S `  C
) ) )
2917, 28eqeltrrd 2532 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ( C I ( S `  C ) ) )
3029adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  ( C I ( S `
 C ) ) )
314, 6, 10, 20, 22, 23, 27, 24, 30btwnlng1 24676 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  ( C L ( S `
 C ) ) )
3226, 31elind 3620 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  ( D  i^i  ( C L ( S `  C ) ) ) )
33 lmimid.a . . . . . . . . 9  |-  ( ph  ->  A  e.  D )
3433adantr 467 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  A  e.  D )
354, 6, 10, 20, 22, 23, 24tglinerflx1 24690 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  e.  ( C L ( S `
 C ) ) )
36 lmimid.d . . . . . . . . 9  |-  ( ph  ->  A  =/=  B )
3736adantr 467 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  A  =/=  B )
384, 5, 6, 10, 11, 7, 14, 1, 9mirinv 24723 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( S `  C )  =  C  <-> 
B  =  C ) )
39 eqcom 2460 . . . . . . . . . . . . . 14  |-  ( B  =  C  <->  C  =  B )
4038, 39syl6bb 265 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( S `  C )  =  C  <-> 
C  =  B ) )
4140biimpar 488 . . . . . . . . . . . 12  |-  ( (
ph  /\  C  =  B )  ->  ( S `  C )  =  C )
4241eqcomd 2459 . . . . . . . . . . 11  |-  ( (
ph  /\  C  =  B )  ->  C  =  ( S `  C ) )
4342ex 436 . . . . . . . . . 10  |-  ( ph  ->  ( C  =  B  ->  C  =  ( S `  C ) ) )
4443necon3d 2647 . . . . . . . . 9  |-  ( ph  ->  ( C  =/=  ( S `  C )  ->  C  =/=  B ) )
4544imp 431 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  =/=  B )
46 lmimid.r . . . . . . . . 9  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
4746adantr 467 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  <" A B C ">  e.  (∟G `  G ) )
484, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47ragperp 24774 . . . . . . 7  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  D (⟂G `  G ) ( C L ( S `  C ) ) )
4948ex 436 . . . . . 6  |-  ( ph  ->  ( C  =/=  ( S `  C )  ->  D (⟂G `  G
) ( C L ( S `  C
) ) ) )
5019, 49syl5bir 222 . . . . 5  |-  ( ph  ->  ( -.  C  =  ( S `  C
)  ->  D (⟂G `  G ) ( C L ( S `  C ) ) ) )
5150orrd 380 . . . 4  |-  ( ph  ->  ( C  =  ( S `  C )  \/  D (⟂G `  G
) ( C L ( S `  C
) ) ) )
5251orcomd 390 . . 3  |-  ( ph  ->  ( D (⟂G `  G
) ( C L ( S `  C
) )  \/  C  =  ( S `  C ) ) )
53 lmif.m . . . 4  |-  M  =  ( (lInvG `  G
) `  D )
544, 5, 6, 7, 8, 53, 10, 12, 9, 15islmib 24841 . . 3  |-  ( ph  ->  ( ( S `  C )  =  ( M `  C )  <-> 
( ( C (midG `  G ) ( S `
 C ) )  e.  D  /\  ( D (⟂G `  G )
( C L ( S `  C ) )  \/  C  =  ( S `  C
) ) ) ) )
5518, 52, 54mpbir2and 934 . 2  |-  ( ph  ->  ( S `  C
)  =  ( M `
 C ) )
5655eqcomd 2459 1  |-  ( ph  ->  ( M `  C
)  =  ( S `
 C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1446    e. wcel 1889    =/= wne 2624   class class class wbr 4405   ran crn 4838   ` cfv 5585  (class class class)co 6295   2c2 10666   <"cs3 12945   Basecbs 15133   distcds 15211  TarskiGcstrkg 24490  DimTarskiGcstrkgld 24494  Itvcitv 24496  LineGclng 24497  pInvGcmir 24709  ∟Gcrag 24750  ⟂Gcperpg 24752  midGcmid 24826  lInvGclmi 24827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-concat 12673  df-s1 12674  df-s2 12951  df-s3 12952  df-trkgc 24508  df-trkgb 24509  df-trkgcb 24510  df-trkgld 24512  df-trkg 24513  df-cgrg 24568  df-leg 24640  df-mir 24710  df-rag 24751  df-perpg 24753  df-mid 24828  df-lmi 24829
This theorem is referenced by:  hypcgrlem1  24853
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