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Theorem ragcgr 23907
Description: Right angle and colinearity. Theorem 8.10 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-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 )
ragcgr.c  |-  .~  =  (cgrG `  G )
ragcgr.d  |-  ( ph  ->  D  e.  P )
ragcgr.e  |-  ( ph  ->  E  e.  P )
ragcgr.f  |-  ( ph  ->  F  e.  P )
ragcgr.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
ragcgr.2  |-  ( ph  ->  <" A B C ">  .~  <" D E F "> )
Assertion
Ref Expression
ragcgr  |-  ( ph  ->  <" D E F ">  e.  (∟G `  G ) )

Proof of Theorem ragcgr
StepHypRef Expression
1 eqidd 2468 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  D  =  D )
2 israg.p . . . . 5  |-  P  =  ( Base `  G
)
3 israg.d . . . . 5  |-  .-  =  ( dist `  G )
4 israg.i . . . . 5  |-  I  =  (Itv `  G )
5 israg.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
65adantr 465 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  G  e. TarskiG )
7 israg.b . . . . . 6  |-  ( ph  ->  B  e.  P )
87adantr 465 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  B  e.  P )
9 israg.c . . . . . 6  |-  ( ph  ->  C  e.  P )
109adantr 465 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  C  e.  P )
11 ragcgr.e . . . . . 6  |-  ( ph  ->  E  e.  P )
1211adantr 465 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  E  e.  P )
13 ragcgr.f . . . . . 6  |-  ( ph  ->  F  e.  P )
1413adantr 465 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  F  e.  P )
15 ragcgr.c . . . . . 6  |-  .~  =  (cgrG `  G )
16 israg.a . . . . . . 7  |-  ( ph  ->  A  e.  P )
1716adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  C )  ->  A  e.  P )
18 ragcgr.d . . . . . . 7  |-  ( ph  ->  D  e.  P )
1918adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  C )  ->  D  e.  P )
20 ragcgr.2 . . . . . . 7  |-  ( ph  ->  <" A B C ">  .~  <" D E F "> )
2120adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =  C )  ->  <" A B C ">  .~  <" D E F "> )
222, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21cgr3simp2 23755 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  ( B  .-  C )  =  ( E  .-  F
) )
23 simpr 461 . . . . 5  |-  ( (
ph  /\  B  =  C )  ->  B  =  C )
242, 3, 4, 6, 8, 10, 12, 14, 22, 23tgcgreq 23716 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  E  =  F )
25 eqidd 2468 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  F  =  F )
261, 24, 25s3eqd 12803 . . 3  |-  ( (
ph  /\  B  =  C )  ->  <" D E F ">  =  <" D F F "> )
27 israg.l . . . 4  |-  L  =  (LineG `  G )
28 israg.s . . . 4  |-  S  =  (pInvG `  G )
292, 3, 4, 27, 28, 6, 19, 14, 12ragtrivb 23902 . . 3  |-  ( (
ph  /\  B  =  C )  ->  <" D F F ">  e.  (∟G `  G ) )
3026, 29eqeltrd 2555 . 2  |-  ( (
ph  /\  B  =  C )  ->  <" D E F ">  e.  (∟G `  G ) )
31 ragcgr.1 . . . . . 6  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
3231adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  <" A B C ">  e.  (∟G `  G ) )
335adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  G  e. TarskiG )
3416adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  A  e.  P )
357adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  P )
369adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  C  e.  P )
372, 3, 4, 27, 28, 33, 34, 35, 36israg 23897 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( <" A B C ">  e.  (∟G `  G
)  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
3832, 37mpbid 210 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) )
3913adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  F  e.  P )
4018adantr 465 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  D  e.  P )
4111adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  P )
4220adantr 465 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  <" A B C ">  .~  <" D E F "> )
432, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp3 23756 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  A )  =  ( F  .-  D ) )
442, 3, 4, 33, 36, 34, 39, 40, 43tgcgrcomlr 23714 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  C )  =  ( D  .-  F ) )
45 eqid 2467 . . . . . 6  |-  ( S `
 B )  =  ( S `  B
)
462, 3, 4, 27, 28, 33, 35, 45, 36mircl 23870 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( ( S `  B ) `  C )  e.  P
)
47 eqid 2467 . . . . . 6  |-  ( S `
 E )  =  ( S `  E
)
482, 3, 4, 27, 28, 33, 41, 47, 39mircl 23870 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( ( S `  E ) `  F )  e.  P
)
49 simpr 461 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  B  =/=  C )
5049necomd 2738 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  C  =/=  B )
512, 3, 4, 27, 28, 33, 35, 45, 36mirbtwn 23867 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( ( ( S `
 B ) `  C ) I C ) )
522, 3, 4, 33, 46, 35, 36, 51tgbtwncom 23722 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( C I ( ( S `  B ) `
 C ) ) )
532, 3, 4, 27, 28, 33, 41, 47, 39mirbtwn 23867 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  ( ( ( S `
 E ) `  F ) I F ) )
542, 3, 4, 33, 48, 41, 39, 53tgbtwncom 23722 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  ( F I ( ( S `  E ) `
 F ) ) )
552, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp2 23755 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  C )  =  ( E  .-  F ) )
562, 3, 4, 33, 35, 36, 41, 39, 55tgcgrcomlr 23714 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  B )  =  ( F  .-  E ) )
572, 3, 4, 27, 28, 33, 35, 45, 36mircgr 23866 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  ( ( S `  B ) `  C
) )  =  ( B  .-  C ) )
582, 3, 4, 27, 28, 33, 41, 47, 39mircgr 23866 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( E  .-  ( ( S `  E ) `  F
) )  =  ( E  .-  F ) )
5955, 57, 583eqtr4d 2518 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  ( ( S `  B ) `  C
) )  =  ( E  .-  ( ( S `  E ) `
 F ) ) )
602, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp1 23754 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  B )  =  ( D  .-  E ) )
612, 3, 4, 33, 34, 35, 40, 41, 60tgcgrcomlr 23714 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  A )  =  ( E  .-  D ) )
622, 3, 4, 33, 36, 35, 46, 39, 41, 48, 34, 40, 50, 52, 54, 56, 59, 43, 61axtg5seg 23705 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  B
) `  C )  .-  A )  =  ( ( ( S `  E ) `  F
)  .-  D )
)
632, 3, 4, 33, 46, 34, 48, 40, 62tgcgrcomlr 23714 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  ( ( S `  B ) `  C
) )  =  ( D  .-  ( ( S `  E ) `
 F ) ) )
6438, 44, 633eqtr3d 2516 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  ( D  .-  F )  =  ( D  .-  ( ( S `  E ) `
 F ) ) )
652, 3, 4, 27, 28, 33, 40, 41, 39israg 23897 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  ( <" D E F ">  e.  (∟G `  G
)  <->  ( D  .-  F )  =  ( D  .-  ( ( S `  E ) `
 F ) ) ) )
6664, 65mpbird 232 . 2  |-  ( (
ph  /\  B  =/=  C )  ->  <" D E F ">  e.  (∟G `  G ) )
6730, 66pm2.61dane 2785 1  |-  ( ph  ->  <" D E F ">  e.  (∟G `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   <"cs3 12782   Basecbs 14502   distcds 14576  TarskiGcstrkg 23668  Itvcitv 23675  LineGclng 23676  cgrGccgrg 23745  pInvGcmir 23861  ∟Gcrag 23893
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-hash 12384  df-word 12518  df-concat 12520  df-s1 12521  df-s2 12788  df-s3 12789  df-trkgc 23687  df-trkgb 23688  df-trkgcb 23689  df-trkg 23693  df-cgrg 23746  df-mir 23862  df-rag 23894
This theorem is referenced by:  motrag  23908  footex  23918
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