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Theorem ragflat 23953
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 2443 . . . 4  |-  ( S `
 C )  =  ( S `  C
)
162, 3, 4, 5, 6, 8, 14, 15, 10mircl 23914 . . 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 23910 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  ( ( S `  C ) `  A
) )  =  ( C  .-  A ) )
202, 3, 4, 8, 14, 16, 14, 10, 19tgcgrcomlr 23743 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  C
) `  A )  .-  C )  =  ( A  .-  C ) )
212, 3, 4, 5, 6, 8, 10, 12, 14israg 23946 . . . . . 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 2443 . . . . . . 7  |-  ( S `
 B )  =  ( S `  B
)
242, 3, 4, 5, 6, 8, 12, 23, 14mircl 23914 . . . . . 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 23947 . . . . . . . 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 23911 . . . . . . . . . 10  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( ( ( S `
 B ) `  C ) I C ) )
302, 3, 4, 8, 24, 12, 14, 29tgbtwncom 23751 . . . . . . . . 9  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( C I ( ( S `  B ) `
 C ) ) )
312, 5, 4, 8, 14, 24, 12, 30btwncolg1 23814 . . . . . . . 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 23948 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  <" (
( S `  B
) `  C ) C A ">  e.  (∟G `  G ) )
332, 3, 4, 5, 6, 8, 24, 14, 10israg 23946 . . . . . . 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 23743 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  ( ( S `  B ) `  C
) )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) )
3620, 22, 353eqtrd 2488 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  C
) `  A )  .-  C )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) )
372, 3, 4, 5, 6, 8, 16, 12, 14israg 23946 . . . 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 23911 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( ( ( S `
 C ) `  A ) I A ) )
402, 3, 4, 8, 16, 14, 10, 39tgbtwncom 23751 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( A I ( ( S `  C ) `
 A ) ) )
412, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40ragflat2 23952 . 2  |-  ( (
ph  /\  B  =/=  C )  ->  B  =  C )
421, 41pm2.61dane 2761 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   <"cs3 12786   Basecbs 14509   distcds 14583  TarskiGcstrkg 23697  Itvcitv 23704  LineGclng 23705  pInvGcmir 23905  ∟Gcrag 23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-s2 12792  df-s3 12793  df-trkgc 23716  df-trkgb 23717  df-trkgcb 23718  df-trkg 23722  df-cgrg 23775  df-mir 23906  df-rag 23943
This theorem is referenced by:  ragtriva  23954  footex  23967  foot  23968
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