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Theorem ragmir 24281
Description: Right angle property is preserved by point inversion. Theorem 8.4 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 25-Aug-2019.)
Hypotheses
Ref Expression
israg.p  |-  P  =  ( Base `  G
)
israg.d  |-  .-  =  ( dist `  G )
israg.i  |-  I  =  (Itv `  G )
israg.l  |-  L  =  (LineG `  G )
israg.s  |-  S  =  (pInvG `  G )
israg.g  |-  ( ph  ->  G  e. TarskiG )
israg.a  |-  ( ph  ->  A  e.  P )
israg.b  |-  ( ph  ->  B  e.  P )
israg.c  |-  ( ph  ->  C  e.  P )
ragmir.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
Assertion
Ref Expression
ragmir  |-  ( ph  ->  <" A B ( ( S `  B ) `  C
) ">  e.  (∟G `  G ) )

Proof of Theorem ragmir
StepHypRef Expression
1 israg.p . . . . 5  |-  P  =  ( Base `  G
)
2 israg.d . . . . 5  |-  .-  =  ( dist `  G )
3 israg.i . . . . 5  |-  I  =  (Itv `  G )
4 israg.l . . . . 5  |-  L  =  (LineG `  G )
5 israg.s . . . . 5  |-  S  =  (pInvG `  G )
6 israg.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
7 israg.b . . . . 5  |-  ( ph  ->  B  e.  P )
8 eqid 2454 . . . . 5  |-  ( S `
 B )  =  ( S `  B
)
9 israg.c . . . . 5  |-  ( ph  ->  C  e.  P )
101, 2, 3, 4, 5, 6, 7, 8, 9mirmir 24247 . . . 4  |-  ( ph  ->  ( ( S `  B ) `  (
( S `  B
) `  C )
)  =  C )
1110oveq2d 6286 . . 3  |-  ( ph  ->  ( A  .-  (
( S `  B
) `  ( ( S `  B ) `  C ) ) )  =  ( A  .-  C ) )
12 ragmir.1 . . . 4  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
13 israg.a . . . . 5  |-  ( ph  ->  A  e.  P )
141, 2, 3, 4, 5, 6, 13, 7, 9israg 24278 . . . 4  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
1512, 14mpbid 210 . . 3  |-  ( ph  ->  ( A  .-  C
)  =  ( A 
.-  ( ( S `
 B ) `  C ) ) )
1611, 15eqtr2d 2496 . 2  |-  ( ph  ->  ( A  .-  (
( S `  B
) `  C )
)  =  ( A 
.-  ( ( S `
 B ) `  ( ( S `  B ) `  C
) ) ) )
171, 2, 3, 4, 5, 6, 7, 8, 9mircl 24246 . . 3  |-  ( ph  ->  ( ( S `  B ) `  C
)  e.  P )
181, 2, 3, 4, 5, 6, 13, 7, 17israg 24278 . 2  |-  ( ph  ->  ( <" A B ( ( S `
 B ) `  C ) ">  e.  (∟G `  G )  <->  ( A  .-  ( ( S `  B ) `
 C ) )  =  ( A  .-  ( ( S `  B ) `  (
( S `  B
) `  C )
) ) ) )
1916, 18mpbird 232 1  |-  ( ph  ->  <" A B ( ( S `  B ) `  C
) ">  e.  (∟G `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   <"cs3 12801   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  Itvcitv 24033  LineGclng 24034  pInvGcmir 24237  ∟Gcrag 24274
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-8 1825  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-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  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-nel 2652  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-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532  df-s2 12807  df-s3 12808  df-trkgc 24045  df-trkgb 24046  df-trkgcb 24047  df-trkg 24051  df-mir 24238  df-rag 24275
This theorem is referenced by: (None)
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