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Theorem isperp2 23128
Description: Property for 2 lines A, B, intersecting at a point X to be perpendicular. Item (i) of definition 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 16-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
)
isperp2.b  |-  ( ph  ->  B  e.  ran  L
)
isperp2.x  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
Assertion
Ref Expression
isperp2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
Distinct variable groups:    v, u, A    u, B, v    u, G, v    ph, u, v   
u, X, v
Allowed substitution hints:    P( v, u)    I( v, u)    L( v, u)   
.- ( v, u)

Proof of Theorem isperp2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqidd 2444 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  u  =  u )
2 isperp.p . . . . . . . . . 10  |-  P  =  ( Base `  G
)
3 isperp.i . . . . . . . . . 10  |-  I  =  (Itv `  G )
4 isperp.l . . . . . . . . . 10  |-  L  =  (LineG `  G )
5 isperp.g . . . . . . . . . . 11  |-  ( ph  ->  G  e. TarskiG )
65ad4antr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  G  e. TarskiG )
7 isperp.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ran  L
)
87ad4antr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  A  e.  ran  L )
9 isperp2.b . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ran  L
)
109ad4antr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  B  e.  ran  L )
11 isperp.d . . . . . . . . . . 11  |-  .-  =  ( dist `  G )
12 simp-4r 766 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  A (⟂G `  G ) B )
132, 11, 3, 4, 6, 8, 10, 12perpneq 23127 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  A  =/=  B )
14 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  x  e.  ( A  i^i  B ) )
15 isperp2.x . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
1615ad4antr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  X  e.  ( A  i^i  B ) )
172, 3, 4, 6, 8, 10, 13, 14, 16tglineineq 23070 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  x  =  X )
18 eqidd 2444 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  v  =  v )
191, 17, 18s3eqd 12511 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  <" u x v ">  =  <" u X v "> )
2019eleq1d 2509 . . . . . . 7  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  ( <" u x v ">  e.  (∟G `  G
)  <->  <" u X v ">  e.  (∟G `  G ) ) )
2120biimpd 207 . . . . . 6  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  ( <" u x v ">  e.  (∟G `  G
)  ->  <" u X v ">  e.  (∟G `  G )
) )
2221ralimdva 2815 . . . . 5  |-  ( ( ( ( ph  /\  A (⟂G `  G ) B )  /\  x  e.  ( A  i^i  B
) )  /\  u  e.  A )  ->  ( A. v  e.  B  <" u x v ">  e.  (∟G `  G )  ->  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
2322ralimdva 2815 . . . 4  |-  ( ( ( ph  /\  A
(⟂G `  G ) B )  /\  x  e.  ( A  i^i  B
) )  ->  ( A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G )  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
2423imp 429 . . 3  |-  ( ( ( ( ph  /\  A (⟂G `  G ) B )  /\  x  e.  ( A  i^i  B
) )  /\  A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G
) )  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
)
252, 11, 3, 4, 5, 7, 9isperp 23125 . . . 4  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  E. x  e.  ( A  i^i  B
) A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G ) ) )
2625biimpa 484 . . 3  |-  ( (
ph  /\  A (⟂G `  G ) B )  ->  E. x  e.  ( A  i^i  B ) A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G ) )
2724, 26r19.29a 2883 . 2  |-  ( (
ph  /\  A (⟂G `  G ) B )  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G ) )
28 eqidd 2444 . . . . . . . 8  |-  ( x  =  X  ->  u  =  u )
29 id 22 . . . . . . . 8  |-  ( x  =  X  ->  x  =  X )
30 eqidd 2444 . . . . . . . 8  |-  ( x  =  X  ->  v  =  v )
3128, 29, 30s3eqd 12511 . . . . . . 7  |-  ( x  =  X  ->  <" u x v ">  =  <" u X v "> )
3231eleq1d 2509 . . . . . 6  |-  ( x  =  X  ->  ( <" u x v ">  e.  (∟G `  G )  <->  <" u X v ">  e.  (∟G `  G )
) )
33322ralbidv 2778 . . . . 5  |-  ( x  =  X  ->  ( A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G )  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
3433rspcev 3094 . . . 4  |-  ( ( X  e.  ( A  i^i  B )  /\  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G
) )  ->  E. x  e.  ( A  i^i  B
) A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G ) )
3515, 34sylan 471 . . 3  |-  ( (
ph  /\  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
)  ->  E. x  e.  ( A  i^i  B
) A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G ) )
3625adantr 465 . . 3  |-  ( (
ph  /\  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
)  ->  ( A
(⟂G `  G ) B  <->  E. x  e.  ( A  i^i  B ) A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G
) ) )
3735, 36mpbird 232 . 2  |-  ( (
ph  /\  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
)  ->  A (⟂G `  G ) B )
3827, 37impbida 828 1  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737    i^i cin 3348   class class class wbr 4313   ran crn 4862   ` cfv 5439   <"cs3 12490   Basecbs 14195   distcds 14268  TarskiGcstrkg 22911  Itvcitv 22919  LineGclng 22920  ∟Gcrag 23109  ⟂Gcperpg 23111
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-s2 12496  df-s3 12497  df-trkgc 22931  df-trkgb 22932  df-trkgcb 22933  df-trkg 22938  df-cgrg 22986  df-mir 23078  df-rag 23110  df-perpg 23112
This theorem is referenced by:  isperp2d  23129  ragperp  23130  foot  23132
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