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Theorem ragflat 23782
Description: Deduce equality from two right angles. Theorem 8.7 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 )
ragflat.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
ragflat.2  |-  ( ph  ->  <" A C B ">  e.  (∟G `  G ) )
Assertion
Ref Expression
ragflat  |-  ( ph  ->  B  =  C )

Proof of Theorem ragflat
StepHypRef Expression
1 simpr 461 . 2  |-  ( (
ph  /\  B  =  C )  ->  B  =  C )
2 israg.p . . 3  |-  P  =  ( Base `  G
)
3 israg.d . . 3  |-  .-  =  ( dist `  G )
4 israg.i . . 3  |-  I  =  (Itv `  G )
5 israg.l . . 3  |-  L  =  (LineG `  G )
6 israg.s . . 3  |-  S  =  (pInvG `  G )
7 israg.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
87adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  G  e. TarskiG )
9 israg.a . . . 4  |-  ( ph  ->  A  e.  P )
109adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  A  e.  P )
11 israg.b . . . 4  |-  ( ph  ->  B  e.  P )
1211adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  P )
13 israg.c . . . 4  |-  ( ph  ->  C  e.  P )
1413adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  P )
15 eqid 2460 . . . 4  |-  ( S `
 C )  =  ( S `  C
)
162, 3, 4, 5, 6, 8, 14, 15, 10mircl 23748 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  ( ( S `  C ) `  A )  e.  P
)
17 ragflat.1 . . . 4  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
1817adantr 465 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  <" A B C ">  e.  (∟G `  G ) )
192, 3, 4, 5, 6, 8, 14, 15, 10mircgr 23744 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  ( ( S `  C ) `  A
) )  =  ( C  .-  A ) )
202, 3, 4, 8, 14, 16, 14, 10, 19tgcgrcomlr 23592 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  C
) `  A )  .-  C )  =  ( A  .-  C ) )
212, 3, 4, 5, 6, 8, 10, 12, 14israg 23775 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( <" A B C ">  e.  (∟G `  G
)  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
2218, 21mpbid 210 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) )
23 eqid 2460 . . . . . . 7  |-  ( S `
 B )  =  ( S `  B
)
242, 3, 4, 5, 6, 8, 12, 23, 14mircl 23748 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( ( S `  B ) `  C )  e.  P
)
25 ragflat.2 . . . . . . . . . 10  |-  ( ph  ->  <" A C B ">  e.  (∟G `  G ) )
2625adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  B  =/=  C )  ->  <" A C B ">  e.  (∟G `  G ) )
272, 3, 4, 5, 6, 8, 10, 14, 12, 26ragcom 23776 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  C )  ->  <" B C A ">  e.  (∟G `  G ) )
28 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  C )  ->  B  =/=  C )
292, 3, 4, 5, 6, 8, 12, 23, 14mirbtwn 23745 . . . . . . . . . 10  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( ( ( S `
 B ) `  C ) I C ) )
302, 3, 4, 8, 24, 12, 14, 29tgbtwncom 23600 . . . . . . . . 9  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( C I ( ( S `  B ) `
 C ) ) )
312, 5, 4, 8, 14, 24, 12, 30btwncolg1 23663 . . . . . . . 8  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  e.  ( C L ( ( S `  B
) `  C )
)  \/  C  =  ( ( S `  B ) `  C
) ) )
322, 3, 4, 5, 6, 8, 12, 14, 10, 24, 27, 28, 31ragcol 23777 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  <" (
( S `  B
) `  C ) C A ">  e.  (∟G `  G ) )
332, 3, 4, 5, 6, 8, 24, 14, 10israg 23775 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( <" ( ( S `  B ) `  C
) C A ">  e.  (∟G `  G
)  <->  ( ( ( S `  B ) `
 C )  .-  A )  =  ( ( ( S `  B ) `  C
)  .-  ( ( S `  C ) `  A ) ) ) )
3432, 33mpbid 210 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  B
) `  C )  .-  A )  =  ( ( ( S `  B ) `  C
)  .-  ( ( S `  C ) `  A ) ) )
352, 3, 4, 8, 24, 10, 24, 16, 34tgcgrcomlr 23592 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  ( ( S `  B ) `  C
) )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) )
3620, 22, 353eqtrd 2505 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  C
) `  A )  .-  C )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) )
372, 3, 4, 5, 6, 8, 16, 12, 14israg 23775 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( <" ( ( S `  C ) `  A
) B C ">  e.  (∟G `  G
)  <->  ( ( ( S `  C ) `
 A )  .-  C )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) ) )
3836, 37mpbird 232 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  <" (
( S `  C
) `  A ) B C ">  e.  (∟G `  G ) )
392, 3, 4, 5, 6, 8, 14, 15, 10mirbtwn 23745 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( ( ( S `
 C ) `  A ) I A ) )
402, 3, 4, 8, 16, 14, 10, 39tgbtwncom 23600 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( A I ( ( S `  C ) `
 A ) ) )
412, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40ragflat2 23781 . 2  |-  ( (
ph  /\  B  =/=  C )  ->  B  =  C )
421, 41pm2.61dane 2778 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   ` cfv 5579  (class class class)co 6275   <"cs3 12757   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554  pInvGcmir 23739  ∟Gcrag 23771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-s2 12763  df-s3 12764  df-trkgc 23565  df-trkgb 23566  df-trkgcb 23567  df-trkg 23571  df-cgrg 23624  df-mir 23740  df-rag 23772
This theorem is referenced by:  ragtriva  23783  footex  23796  foot  23797
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