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Theorem isperp2 23947
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 2468 . . . . . . . . 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 23946 . . . . . . . . . 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 23881 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  x  =  X )
18 eqidd 2468 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  A (⟂G `  G
) B )  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  v  e.  B
)  ->  v  =  v )
191, 17, 18s3eqd 12807 . . . . . . . 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 2536 . . . . . . 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 2875 . . . . 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 2875 . . . 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 23944 . . . 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 3008 . 2  |-  ( (
ph  /\  A (⟂G `  G ) B )  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G ) )
28 eqidd 2468 . . . . . . . 8  |-  ( x  =  X  ->  u  =  u )
29 id 22 . . . . . . . 8  |-  ( x  =  X  ->  x  =  X )
30 eqidd 2468 . . . . . . . 8  |-  ( x  =  X  ->  v  =  v )
3128, 29, 30s3eqd 12807 . . . . . . 7  |-  ( x  =  X  ->  <" u x v ">  =  <" u X v "> )
3231eleq1d 2536 . . . . . 6  |-  ( x  =  X  ->  ( <" u x v ">  e.  (∟G `  G )  <->  <" u X v ">  e.  (∟G `  G )
) )
33322ralbidv 2911 . . . . 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 3219 . . . 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 830 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 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    i^i cin 3480   class class class wbr 4453   ran crn 5006   ` cfv 5594   <"cs3 12786   Basecbs 14506   distcds 14580  TarskiGcstrkg 23689  Itvcitv 23696  LineGclng 23697  ∟Gcrag 23925  ⟂Gcperpg 23927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12522  df-concat 12524  df-s1 12525  df-s2 12792  df-s3 12793  df-trkgc 23708  df-trkgb 23709  df-trkgcb 23710  df-trkg 23714  df-cgrg 23767  df-mir 23892  df-rag 23926  df-perpg 23928
This theorem is referenced by:  isperp2d  23948  ragperp  23949  foot  23951
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