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Theorem ragflat 23226
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 2451 . . . 4  |-  ( S `
 C )  =  ( S `  C
)
162, 3, 4, 5, 6, 8, 14, 15, 10mircl 23193 . . 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 23189 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  ( ( S `  C ) `  A
) )  =  ( C  .-  A ) )
202, 3, 4, 8, 14, 16, 14, 10, 19tgcgrcomlr 23053 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  C
) `  A )  .-  C )  =  ( A  .-  C ) )
212, 3, 4, 5, 6, 8, 10, 12, 14israg 23219 . . . . . 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 2451 . . . . . . 7  |-  ( S `
 B )  =  ( S `  B
)
242, 3, 4, 5, 6, 8, 12, 23, 14mircl 23193 . . . . . 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 23220 . . . . . . . 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 23190 . . . . . . . . . 10  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( ( ( S `
 B ) `  C ) I C ) )
302, 3, 4, 8, 24, 12, 14, 29tgbtwncom 23061 . . . . . . . . 9  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( C I ( ( S `  B ) `
 C ) ) )
312, 5, 4, 8, 14, 24, 12, 30btwncolg1 23110 . . . . . . . 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 23221 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  <" (
( S `  B
) `  C ) C A ">  e.  (∟G `  G ) )
332, 3, 4, 5, 6, 8, 24, 14, 10israg 23219 . . . . . . 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 23053 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  ( ( S `  B ) `  C
) )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) )
3620, 22, 353eqtrd 2496 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  C
) `  A )  .-  C )  =  ( ( ( S `  C ) `  A
)  .-  ( ( S `  B ) `  C ) ) )
372, 3, 4, 5, 6, 8, 16, 12, 14israg 23219 . . . 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 23190 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( ( ( S `
 C ) `  A ) I A ) )
402, 3, 4, 8, 16, 14, 10, 39tgbtwncom 23061 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  ( A I ( ( S `  C ) `
 A ) ) )
412, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 38, 40ragflat2 23225 . 2  |-  ( (
ph  /\  B  =/=  C )  ->  B  =  C )
421, 41pm2.61dane 2766 1  |-  ( ph  ->  B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   ` cfv 5518  (class class class)co 6192   <"cs3 12573   Basecbs 14278   distcds 14351  TarskiGcstrkg 23007  Itvcitv 23014  LineGclng 23015  pInvGcmir 23183  ∟Gcrag 23215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-concat 12335  df-s1 12336  df-s2 12579  df-s3 12580  df-trkgc 23026  df-trkgb 23027  df-trkgcb 23028  df-trkg 23032  df-cgrg 23085  df-mir 23184  df-rag 23216
This theorem is referenced by:  ragtriva  23227  footex  23239  foot  23240
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