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Theorem ragflat2 23885
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 2467 . . . 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 2467 . . . . 5  |-  ( S `
 B )  =  ( S `  B
)
131, 9, 3, 2, 10, 4, 11, 12, 7mircl 23852 . . . 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 23879 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
1715, 16mpbid 210 . . . 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 23879 . . . . 5  |-  ( ph  ->  ( <" D B C ">  e.  (∟G `  G )  <->  ( D  .-  C )  =  ( D  .-  ( ( S `  B ) `
 C ) ) ) )
2018, 19mpbid 210 . . . 4  |-  ( ph  ->  ( D  .-  C
)  =  ( D 
.-  ( ( S `
 B ) `  C ) ) )
211, 2, 3, 4, 5, 6, 7, 8, 13, 5, 9, 14, 17, 20tgidinside 23782 . . 3  |-  ( ph  ->  C  =  ( ( S `  B ) `
 C ) )
2221eqcomd 2475 . 2  |-  ( ph  ->  ( ( S `  B ) `  C
)  =  C )
231, 9, 3, 2, 10, 4, 11, 12, 7mirinv 23857 . 2  |-  ( ph  ->  ( ( ( S `
 B ) `  C )  =  C  <-> 
B  =  C ) )
2422, 23mpbid 210 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   <"cs3 12773   Basecbs 14493   distcds 14567  TarskiGcstrkg 23650  Itvcitv 23657  LineGclng 23658  cgrGccgrg 23727  pInvGcmir 23843  ∟Gcrag 23875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-concat 12511  df-s1 12512  df-s2 12779  df-s3 12780  df-trkgc 23669  df-trkgb 23670  df-trkgcb 23671  df-trkg 23675  df-cgrg 23728  df-mir 23844  df-rag 23876
This theorem is referenced by:  ragflat  23886
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