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Theorem ragperp 24484
Description: Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)
Hypotheses
Ref Expression
isperp.p  |-  P  =  ( Base `  G
)
isperp.d  |-  .-  =  ( dist `  G )
isperp.i  |-  I  =  (Itv `  G )
isperp.l  |-  L  =  (LineG `  G )
isperp.g  |-  ( ph  ->  G  e. TarskiG )
isperp.a  |-  ( ph  ->  A  e.  ran  L
)
ragperp.b  |-  ( ph  ->  B  e.  ran  L
)
ragperp.x  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
ragperp.u  |-  ( ph  ->  U  e.  A )
ragperp.v  |-  ( ph  ->  V  e.  B )
ragperp.1  |-  ( ph  ->  U  =/=  X )
ragperp.2  |-  ( ph  ->  V  =/=  X )
ragperp.r  |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
Assertion
Ref Expression
ragperp  |-  ( ph  ->  A (⟂G `  G
) B )

Proof of Theorem ragperp
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isperp.p . . . 4  |-  P  =  ( Base `  G
)
2 isperp.d . . . 4  |-  .-  =  ( dist `  G )
3 isperp.i . . . 4  |-  I  =  (Itv `  G )
4 isperp.l . . . 4  |-  L  =  (LineG `  G )
5 eqid 2404 . . . 4  |-  (pInvG `  G )  =  (pInvG `  G )
6 isperp.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76adantr 465 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  G  e. TarskiG )
8 ragperp.b . . . . . 6  |-  ( ph  ->  B  e.  ran  L
)
98adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  B  e.  ran  L )
10 simprr 760 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  B )
111, 4, 3, 7, 9, 10tglnpt 24321 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  P )
12 isperp.a . . . . . 6  |-  ( ph  ->  A  e.  ran  L
)
1312adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  A  e.  ran  L )
14 inss1 3661 . . . . . . 7  |-  ( A  i^i  B )  C_  A
15 ragperp.x . . . . . . 7  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
1614, 15sseldi 3442 . . . . . 6  |-  ( ph  ->  X  e.  A )
1716adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  A )
181, 4, 3, 7, 13, 17tglnpt 24321 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  P )
19 simprl 758 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  A )
201, 4, 3, 7, 13, 19tglnpt 24321 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  P )
21 ragperp.v . . . . . . 7  |-  ( ph  ->  V  e.  B )
2221adantr 465 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  B )
231, 4, 3, 7, 9, 22tglnpt 24321 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  P )
24 ragperp.u . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
2524adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  A )
261, 4, 3, 7, 13, 25tglnpt 24321 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  P )
27 ragperp.r . . . . . . . 8  |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" U X V ">  e.  (∟G `  G ) )
29 ragperp.1 . . . . . . . 8  |-  ( ph  ->  U  =/=  X )
3029adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  =/=  X )
3124ad2antrr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  A )
326ad2antrr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  G  e. TarskiG )
3318adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  P )
3420adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  P )
35 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  -.  X  =  u )
3635neqned 2608 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  =/=  u )
3712ad2antrr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  e.  ran  L )
3816ad2antrr 726 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  A )
3919adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  A )
401, 3, 4, 32, 33, 34, 36, 36, 37, 38, 39tglinethru 24405 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  =  ( X L u ) )
4131, 40eleqtrd 2494 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  ( X L u ) )
4241ex 434 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  u  ->  U  e.  ( X L u ) ) )
4342orrd 378 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  u  \/  U  e.  ( X L u ) ) )
4443orcomd 388 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( U  e.  ( X L u )  \/  X  =  u ) )
451, 2, 3, 4, 5, 7, 26, 18, 23, 20, 28, 30, 44ragcol 24466 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X V ">  e.  (∟G `  G ) )
461, 2, 3, 4, 5, 7, 20, 18, 23, 45ragcom 24465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" V X u ">  e.  (∟G `  G ) )
47 ragperp.2 . . . . . 6  |-  ( ph  ->  V  =/=  X )
4847adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  =/=  X )
4921ad2antrr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  B )
506ad2antrr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  G  e. TarskiG )
5118adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  P )
5211adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  P )
53 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  -.  X  =  v
)
5453neqned 2608 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  =/=  v )
558ad2antrr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  e.  ran  L )
56 inss2 3662 . . . . . . . . . . . 12  |-  ( A  i^i  B )  C_  B
5756, 15sseldi 3442 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
5857ad2antrr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  B )
5910adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  B )
601, 3, 4, 50, 51, 52, 54, 54, 55, 58, 59tglinethru 24405 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  =  ( X L v ) )
6149, 60eleqtrd 2494 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  ( X L v ) )
6261ex 434 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  v  ->  V  e.  ( X L v ) ) )
6362orrd 378 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  v  \/  V  e.  ( X L v ) ) )
6463orcomd 388 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( V  e.  ( X L v )  \/  X  =  v ) )
651, 2, 3, 4, 5, 7, 23, 18, 20, 11, 46, 48, 64ragcol 24466 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" v X u ">  e.  (∟G `  G ) )
661, 2, 3, 4, 5, 7, 11, 18, 20, 65ragcom 24465 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X v ">  e.  (∟G `  G ) )
6766ralrimivva 2827 . 2  |-  ( ph  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G ) )
681, 2, 3, 4, 6, 12, 8, 15isperp2 24482 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
6967, 68mpbird 234 1  |-  ( ph  ->  A (⟂G `  G
) B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756    i^i cin 3415   class class class wbr 4397   ran crn 4826   ` cfv 5571  (class class class)co 6280   <"cs3 12865   Basecbs 14843   distcds 14920  TarskiGcstrkg 24208  Itvcitv 24214  LineGclng 24215  pInvGcmir 24422  ∟Gcrag 24460  ⟂Gcperpg 24462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-concat 12595  df-s1 12596  df-s2 12871  df-s3 12872  df-trkgc 24226  df-trkgb 24227  df-trkgcb 24228  df-trkg 24231  df-cgrg 24286  df-mir 24423  df-rag 24461  df-perpg 24463
This theorem is referenced by:  footex  24485  colperpexlem3  24496  mideulem2  24498  lmimid  24555  hypcgrlem1  24560  hypcgrlem2  24561  trgcopyeulem  24566
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