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Theorem ragperp 24755
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 2450 . . . 4  |-  (pInvG `  G )  =  (pInvG `  G )
6 isperp.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76adantr 467 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  G  e. TarskiG )
8 ragperp.b . . . . . 6  |-  ( ph  ->  B  e.  ran  L
)
98adantr 467 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  B  e.  ran  L )
10 simprr 765 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  B )
111, 4, 3, 7, 9, 10tglnpt 24587 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  P )
12 isperp.a . . . . . 6  |-  ( ph  ->  A  e.  ran  L
)
1312adantr 467 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  A  e.  ran  L )
14 inss1 3651 . . . . . . 7  |-  ( A  i^i  B )  C_  A
15 ragperp.x . . . . . . 7  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
1614, 15sseldi 3429 . . . . . 6  |-  ( ph  ->  X  e.  A )
1716adantr 467 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  A )
181, 4, 3, 7, 13, 17tglnpt 24587 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  P )
19 simprl 763 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  A )
201, 4, 3, 7, 13, 19tglnpt 24587 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  P )
21 ragperp.v . . . . . . 7  |-  ( ph  ->  V  e.  B )
2221adantr 467 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  B )
231, 4, 3, 7, 9, 22tglnpt 24587 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  P )
24 ragperp.u . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
2524adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  A )
261, 4, 3, 7, 13, 25tglnpt 24587 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  P )
27 ragperp.r . . . . . . . 8  |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
2827adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" U X V ">  e.  (∟G `  G ) )
29 ragperp.1 . . . . . . . 8  |-  ( ph  ->  U  =/=  X )
3029adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  =/=  X )
3124ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  A )
326ad2antrr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  G  e. TarskiG )
3318adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  P )
3420adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  P )
35 simpr 463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  -.  X  =  u )
3635neqned 2630 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  =/=  u )
3712ad2antrr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  e.  ran  L )
3816ad2antrr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  A )
3919adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  A )
401, 3, 4, 32, 33, 34, 36, 36, 37, 38, 39tglinethru 24674 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  =  ( X L u ) )
4131, 40eleqtrd 2530 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  ( X L u ) )
4241ex 436 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  u  ->  U  e.  ( X L u ) ) )
4342orrd 380 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  u  \/  U  e.  ( X L u ) ) )
4443orcomd 390 . . . . . . 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 24737 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X V ">  e.  (∟G `  G ) )
461, 2, 3, 4, 5, 7, 20, 18, 23, 45ragcom 24736 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" V X u ">  e.  (∟G `  G ) )
47 ragperp.2 . . . . . 6  |-  ( ph  ->  V  =/=  X )
4847adantr 467 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  =/=  X )
4921ad2antrr 731 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  B )
506ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  G  e. TarskiG )
5118adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  P )
5211adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  P )
53 simpr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  -.  X  =  v
)
5453neqned 2630 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  =/=  v )
558ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  e.  ran  L )
56 inss2 3652 . . . . . . . . . . . 12  |-  ( A  i^i  B )  C_  B
5756, 15sseldi 3429 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
5857ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  B )
5910adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  B )
601, 3, 4, 50, 51, 52, 54, 54, 55, 58, 59tglinethru 24674 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  =  ( X L v ) )
6149, 60eleqtrd 2530 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  ( X L v ) )
6261ex 436 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  v  ->  V  e.  ( X L v ) ) )
6362orrd 380 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  v  \/  V  e.  ( X L v ) ) )
6463orcomd 390 . . . . 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 24737 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" v X u ">  e.  (∟G `  G ) )
661, 2, 3, 4, 5, 7, 11, 18, 20, 65ragcom 24736 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X v ">  e.  (∟G `  G ) )
6766ralrimivva 2808 . 2  |-  ( ph  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G ) )
681, 2, 3, 4, 6, 12, 8, 15isperp2 24753 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
6967, 68mpbird 236 1  |-  ( ph  ->  A (⟂G `  G
) B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736    i^i cin 3402   class class class wbr 4401   ran crn 4834   ` cfv 5581  (class class class)co 6288   <"cs3 12933   Basecbs 15114   distcds 15192  TarskiGcstrkg 24471  Itvcitv 24477  LineGclng 24478  pInvGcmir 24690  ∟Gcrag 24731  ⟂Gcperpg 24733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-word 12661  df-concat 12663  df-s1 12664  df-s2 12939  df-s3 12940  df-trkgc 24489  df-trkgb 24490  df-trkgcb 24491  df-trkg 24494  df-cgrg 24549  df-mir 24691  df-rag 24732  df-perpg 24734
This theorem is referenced by:  footex  24756  colperpexlem3  24767  mideulem2  24769  lmimid  24829  hypcgrlem1  24834  hypcgrlem2  24835  trgcopyeulem  24840
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