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Theorem ragcgr 23101
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 2444 . . . 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 22973 . . . . 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 22936 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  E  =  F )
25 eqidd 2444 . . . 4  |-  ( (
ph  /\  B  =  C )  ->  F  =  F )
261, 24, 25s3eqd 12490 . . 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 23096 . . 3  |-  ( (
ph  /\  B  =  C )  ->  <" D F F ">  e.  (∟G `  G ) )
3026, 29eqeltrd 2517 . 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 23091 . . . . 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 22974 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  A )  =  ( F  .-  D ) )
442, 3, 4, 33, 36, 34, 39, 40, 43tgcgrcomlr 22934 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  C )  =  ( D  .-  F ) )
45 eqid 2443 . . . . . 6  |-  ( S `
 B )  =  ( S `  B
)
462, 3, 4, 27, 28, 33, 35, 45, 36mircl 23065 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( ( S `  B ) `  C )  e.  P
)
47 eqid 2443 . . . . . 6  |-  ( S `
 E )  =  ( S `  E
)
482, 3, 4, 27, 28, 33, 41, 47, 39mircl 23065 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( ( S `  E ) `  F )  e.  P
)
49 simpr 461 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  B  =/=  C )
5049necomd 2695 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  C  =/=  B )
512, 3, 4, 27, 28, 33, 35, 45, 36mirbtwn 23062 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( ( ( S `
 B ) `  C ) I C ) )
522, 3, 4, 33, 46, 35, 36, 51tgbtwncom 22942 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  B  e.  ( C I ( ( S `  B ) `
 C ) ) )
532, 3, 4, 27, 28, 33, 41, 47, 39mirbtwn 23062 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  ( ( ( S `
 E ) `  F ) I F ) )
542, 3, 4, 33, 48, 41, 39, 53tgbtwncom 22942 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  E  e.  ( F I ( ( S `  E ) `
 F ) ) )
552, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp2 22973 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  C )  =  ( E  .-  F ) )
562, 3, 4, 33, 35, 36, 41, 39, 55tgcgrcomlr 22934 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( C  .-  B )  =  ( F  .-  E ) )
572, 3, 4, 27, 28, 33, 35, 45, 36mircgr 23061 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  ( ( S `  B ) `  C
) )  =  ( B  .-  C ) )
582, 3, 4, 27, 28, 33, 41, 47, 39mircgr 23061 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( E  .-  ( ( S `  E ) `  F
) )  =  ( E  .-  F ) )
5955, 57, 583eqtr4d 2485 . . . . . 6  |-  ( (
ph  /\  B  =/=  C )  ->  ( B  .-  ( ( S `  B ) `  C
) )  =  ( E  .-  ( ( S `  E ) `
 F ) ) )
602, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp1 22972 . . . . . . 7  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  B )  =  ( D  .-  E ) )
612, 3, 4, 33, 34, 35, 40, 41, 60tgcgrcomlr 22934 . . . . . 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 22926 . . . . 5  |-  ( (
ph  /\  B  =/=  C )  ->  ( (
( S `  B
) `  C )  .-  A )  =  ( ( ( S `  E ) `  F
)  .-  D )
)
632, 3, 4, 33, 46, 34, 48, 40, 62tgcgrcomlr 22934 . . . 4  |-  ( (
ph  /\  B  =/=  C )  ->  ( A  .-  ( ( S `  B ) `  C
) )  =  ( D  .-  ( ( S `  E ) `
 F ) ) )
6438, 44, 633eqtr3d 2483 . . 3  |-  ( (
ph  /\  B  =/=  C )  ->  ( D  .-  F )  =  ( D  .-  ( ( S `  E ) `
 F ) ) )
652, 3, 4, 27, 28, 33, 40, 41, 39israg 23091 . . 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 2689 1  |-  ( ph  ->  <" D E F ">  e.  (∟G `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   <"cs3 12469   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897  LineGclng 22898  cgrGccgrg 22963  pInvGcmir 23055  ∟Gcrag 23087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-s2 12475  df-s3 12476  df-trkgc 22909  df-trkgb 22910  df-trkgcb 22911  df-trkg 22916  df-cgrg 22964  df-mir 23056  df-rag 23088
This theorem is referenced by:  footex  23109
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