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Theorem ragperp 24841
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 2471 . . . 4  |-  (pInvG `  G )  =  (pInvG `  G )
6 isperp.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76adantr 472 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  G  e. TarskiG )
8 ragperp.b . . . . . 6  |-  ( ph  ->  B  e.  ran  L
)
98adantr 472 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  B  e.  ran  L )
10 simprr 774 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  B )
111, 4, 3, 7, 9, 10tglnpt 24673 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  P )
12 isperp.a . . . . . 6  |-  ( ph  ->  A  e.  ran  L
)
1312adantr 472 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  A  e.  ran  L )
14 inss1 3643 . . . . . . 7  |-  ( A  i^i  B )  C_  A
15 ragperp.x . . . . . . 7  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
1614, 15sseldi 3416 . . . . . 6  |-  ( ph  ->  X  e.  A )
1716adantr 472 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  A )
181, 4, 3, 7, 13, 17tglnpt 24673 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  P )
19 simprl 772 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  A )
201, 4, 3, 7, 13, 19tglnpt 24673 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  P )
21 ragperp.v . . . . . . 7  |-  ( ph  ->  V  e.  B )
2221adantr 472 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  B )
231, 4, 3, 7, 9, 22tglnpt 24673 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  P )
24 ragperp.u . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
2524adantr 472 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  A )
261, 4, 3, 7, 13, 25tglnpt 24673 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  P )
27 ragperp.r . . . . . . . 8  |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
2827adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" U X V ">  e.  (∟G `  G ) )
29 ragperp.1 . . . . . . . 8  |-  ( ph  ->  U  =/=  X )
3029adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  =/=  X )
3124ad2antrr 740 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  A )
326ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  G  e. TarskiG )
3318adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  P )
3420adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  P )
35 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  -.  X  =  u )
3635neqned 2650 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  =/=  u )
3712ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  e.  ran  L )
3816ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  A )
3919adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  A )
401, 3, 4, 32, 33, 34, 36, 36, 37, 38, 39tglinethru 24760 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  =  ( X L u ) )
4131, 40eleqtrd 2551 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  ( X L u ) )
4241ex 441 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  u  ->  U  e.  ( X L u ) ) )
4342orrd 385 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  u  \/  U  e.  ( X L u ) ) )
4443orcomd 395 . . . . . . 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 24823 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X V ">  e.  (∟G `  G ) )
461, 2, 3, 4, 5, 7, 20, 18, 23, 45ragcom 24822 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" V X u ">  e.  (∟G `  G ) )
47 ragperp.2 . . . . . 6  |-  ( ph  ->  V  =/=  X )
4847adantr 472 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  =/=  X )
4921ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  B )
506ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  G  e. TarskiG )
5118adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  P )
5211adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  P )
53 simpr 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  -.  X  =  v
)
5453neqned 2650 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  =/=  v )
558ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  e.  ran  L )
56 inss2 3644 . . . . . . . . . . . 12  |-  ( A  i^i  B )  C_  B
5756, 15sseldi 3416 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
5857ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  B )
5910adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  B )
601, 3, 4, 50, 51, 52, 54, 54, 55, 58, 59tglinethru 24760 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  =  ( X L v ) )
6149, 60eleqtrd 2551 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  ( X L v ) )
6261ex 441 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  v  ->  V  e.  ( X L v ) ) )
6362orrd 385 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  v  \/  V  e.  ( X L v ) ) )
6463orcomd 395 . . . . 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 24823 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" v X u ">  e.  (∟G `  G ) )
661, 2, 3, 4, 5, 7, 11, 18, 20, 65ragcom 24822 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X v ">  e.  (∟G `  G ) )
6766ralrimivva 2814 . 2  |-  ( ph  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G ) )
681, 2, 3, 4, 6, 12, 8, 15isperp2 24839 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
6967, 68mpbird 240 1  |-  ( ph  ->  A (⟂G `  G
) B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756    i^i cin 3389   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308   <"cs3 12997   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564  pInvGcmir 24776  ∟Gcrag 24817  ⟂Gcperpg 24819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580  df-cgrg 24635  df-mir 24777  df-rag 24818  df-perpg 24820
This theorem is referenced by:  footex  24842  colperpexlem3  24853  mideulem2  24855  lmimid  24915  hypcgrlem1  24920  hypcgrlem2  24921  trgcopyeulem  24926
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