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Theorem ragflat2 24644
Description: Deduce equality from two right angles. Theorem 8.6 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 3-Sep-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 )
ragflat2.d  |-  ( ph  ->  D  e.  P )
ragflat2.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
ragflat2.2  |-  ( ph  ->  <" D B C ">  e.  (∟G `  G ) )
ragflat2.3  |-  ( ph  ->  C  e.  ( A I D ) )
Assertion
Ref Expression
ragflat2  |-  ( ph  ->  B  =  C )

Proof of Theorem ragflat2
StepHypRef Expression
1 israg.p . . . 4  |-  P  =  ( Base `  G
)
2 israg.l . . . 4  |-  L  =  (LineG `  G )
3 israg.i . . . 4  |-  I  =  (Itv `  G )
4 israg.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
5 israg.a . . . 4  |-  ( ph  ->  A  e.  P )
6 ragflat2.d . . . 4  |-  ( ph  ->  D  e.  P )
7 israg.c . . . 4  |-  ( ph  ->  C  e.  P )
8 eqid 2420 . . . 4  |-  (cgrG `  G )  =  (cgrG `  G )
9 israg.d . . . . 5  |-  .-  =  ( dist `  G )
10 israg.s . . . . 5  |-  S  =  (pInvG `  G )
11 israg.b . . . . 5  |-  ( ph  ->  B  e.  P )
12 eqid 2420 . . . . 5  |-  ( S `
 B )  =  ( S `  B
)
131, 9, 3, 2, 10, 4, 11, 12, 7mircl 24605 . . . 4  |-  ( ph  ->  ( ( S `  B ) `  C
)  e.  P )
14 ragflat2.3 . . . 4  |-  ( ph  ->  C  e.  ( A I D ) )
15 ragflat2.1 . . . . 5  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
161, 9, 3, 2, 10, 4, 5, 11, 7israg 24638 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
1715, 16mpbid 213 . . . 4  |-  ( ph  ->  ( A  .-  C
)  =  ( A 
.-  ( ( S `
 B ) `  C ) ) )
18 ragflat2.2 . . . . 5  |-  ( ph  ->  <" D B C ">  e.  (∟G `  G ) )
191, 9, 3, 2, 10, 4, 6, 11, 7israg 24638 . . . . 5  |-  ( ph  ->  ( <" D B C ">  e.  (∟G `  G )  <->  ( D  .-  C )  =  ( D  .-  ( ( S `  B ) `
 C ) ) ) )
2018, 19mpbid 213 . . . 4  |-  ( ph  ->  ( D  .-  C
)  =  ( D 
.-  ( ( S `
 B ) `  C ) ) )
211, 2, 3, 4, 5, 6, 7, 8, 13, 5, 9, 14, 17, 20tgidinside 24515 . . 3  |-  ( ph  ->  C  =  ( ( S `  B ) `
 C ) )
2221eqcomd 2428 . 2  |-  ( ph  ->  ( ( S `  B ) `  C
)  =  C )
231, 9, 3, 2, 10, 4, 11, 12, 7mirinv 24610 . 2  |-  ( ph  ->  ( ( ( S `
 B ) `  C )  =  C  <-> 
B  =  C ) )
2422, 23mpbid 213 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296   <"cs3 12912   Basecbs 15081   distcds 15159  TarskiGcstrkg 24380  Itvcitv 24386  LineGclng 24387  cgrGccgrg 24457  pInvGcmir 24596  ∟Gcrag 24634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-fzo 11903  df-hash 12502  df-word 12640  df-concat 12642  df-s1 12643  df-s2 12918  df-s3 12919  df-trkgc 24398  df-trkgb 24399  df-trkgcb 24400  df-trkg 24403  df-cgrg 24458  df-mir 24597  df-rag 24635
This theorem is referenced by:  ragflat  24645  opphllem5  24689  opphllem6  24690
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