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Theorem lmimid 24376
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 5874 . . . . 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 2457 . . . . . . 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 24153 . . . . . . 7  |-  ( ph  ->  B  e.  P )
154, 5, 6, 10, 11, 7, 14, 1, 9mircl 24259 . . . . . 6  |-  ( ph  ->  ( S `  C
)  e.  P )
164, 5, 6, 7, 8, 9, 15, 11, 14ismidb 24361 . . . . 5  |-  ( ph  ->  ( ( S `  C )  =  ( ( (pInvG `  G
) `  B ) `  C )  <->  ( C
(midG `  G )
( S `  C
) )  =  B ) )
173, 16mpbid 210 . . . 4  |-  ( ph  ->  ( C (midG `  G ) ( S `
 C ) )  =  B )
1817, 13eqeltrd 2545 . . 3  |-  ( ph  ->  ( C (midG `  G ) ( S `
 C ) )  e.  D )
19 df-ne 2654 . . . . . 6  |-  ( C  =/=  ( S `  C )  <->  -.  C  =  ( S `  C ) )
207adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  G  e. TarskiG )
2112adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  D  e.  ran  L )
229adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  e.  P )
2315adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  ( S `  C )  e.  P
)
24 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  =/=  ( S `  C ) )
254, 6, 10, 20, 22, 23, 24tgelrnln 24227 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  ( C L ( S `  C ) )  e. 
ran  L )
2613adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  D )
2714adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  P )
284, 5, 6, 7, 8, 9, 15midbtwn 24362 . . . . . . . . . . . 12  |-  ( ph  ->  ( C (midG `  G ) ( S `
 C ) )  e.  ( C I ( S `  C
) ) )
2917, 28eqeltrrd 2546 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ( C I ( S `  C ) ) )
3029adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  ( C I ( S `
 C ) ) )
314, 6, 10, 20, 22, 23, 27, 24, 30btwnlng1 24216 . . . . . . . . 9  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  ( C L ( S `
 C ) ) )
3226, 31elind 3684 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  B  e.  ( D  i^i  ( C L ( S `  C ) ) ) )
33 lmimid.a . . . . . . . . 9  |-  ( ph  ->  A  e.  D )
3433adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  A  e.  D )
354, 6, 10, 20, 22, 23, 24tglinerflx1 24230 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  e.  ( C L ( S `
 C ) ) )
36 lmimid.d . . . . . . . . 9  |-  ( ph  ->  A  =/=  B )
3736adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  A  =/=  B )
384, 5, 6, 10, 11, 7, 14, 1, 9mirinv 24264 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( S `  C )  =  C  <-> 
B  =  C ) )
39 eqcom 2466 . . . . . . . . . . . . . 14  |-  ( B  =  C  <->  C  =  B )
4038, 39syl6bb 261 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( S `  C )  =  C  <-> 
C  =  B ) )
4140biimpar 485 . . . . . . . . . . . 12  |-  ( (
ph  /\  C  =  B )  ->  ( S `  C )  =  C )
4241eqcomd 2465 . . . . . . . . . . 11  |-  ( (
ph  /\  C  =  B )  ->  C  =  ( S `  C ) )
4342ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( C  =  B  ->  C  =  ( S `  C ) ) )
4443necon3d 2681 . . . . . . . . 9  |-  ( ph  ->  ( C  =/=  ( S `  C )  ->  C  =/=  B ) )
4544imp 429 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  C  =/=  B )
46 lmimid.r . . . . . . . . 9  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
4746adantr 465 . . . . . . . 8  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  <" A B C ">  e.  (∟G `  G ) )
484, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47ragperp 24311 . . . . . . 7  |-  ( (
ph  /\  C  =/=  ( S `  C ) )  ->  D (⟂G `  G ) ( C L ( S `  C ) ) )
4948ex 434 . . . . . 6  |-  ( ph  ->  ( C  =/=  ( S `  C )  ->  D (⟂G `  G
) ( C L ( S `  C
) ) ) )
5019, 49syl5bir 218 . . . . 5  |-  ( ph  ->  ( -.  C  =  ( S `  C
)  ->  D (⟂G `  G ) ( C L ( S `  C ) ) ) )
5150orrd 378 . . . 4  |-  ( ph  ->  ( C  =  ( S `  C )  \/  D (⟂G `  G
) ( C L ( S `  C
) ) ) )
5251orcomd 388 . . 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 24370 . . 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 922 . 2  |-  ( ph  ->  ( S `  C
)  =  ( M `
 C ) )
5655eqcomd 2465 1  |-  ( ph  ->  ( M `  C
)  =  ( S `
 C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   2c2 10606   <"cs3 12819   Basecbs 14735   distcds 14812  TarskiGcstrkg 24042  DimTarskiGcstrkgld 24046  Itvcitv 24049  LineGclng 24050  pInvGcmir 24250  ∟Gcrag 24287  ⟂Gcperpg 24289  midGcmid 24355  lInvGclmi 24356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 24061  df-trkgb 24062  df-trkgcb 24063  df-trkgld 24065  df-trkg 24067  df-cgrg 24120  df-leg 24187  df-mir 24251  df-rag 24288  df-perpg 24290  df-mid 24357  df-lmi 24358
This theorem is referenced by:  hypcgrlem1  24381
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