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Theorem isperp 24290
Description: Property for 2 lines A, B to be perpendicular. Item (ii) of definition 8.11 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
)
isperp.b  |-  ( ph  ->  B  e.  ran  L
)
Assertion
Ref Expression
isperp  |-  ( 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 ) ) )
Distinct variable groups:    v, u, x, A    u, B, v, x    u, G, v, x    ph, u, v, x
Allowed substitution hints:    P( x, v, u)    I( x, v, u)    L( x, v, u)    .- ( x, v, u)

Proof of Theorem isperp
Dummy variables  a 
b  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4440 . . 3  |-  ( A (⟂G `  G ) B 
<-> 
<. A ,  B >.  e.  (⟂G `  G )
)
2 df-perpg 24274 . . . . . 6  |- ⟂G  =  ( g  e.  _V  |->  {
<. a ,  b >.  |  ( ( a  e.  ran  (LineG `  g )  /\  b  e.  ran  (LineG `  g
) )  /\  E. x  e.  ( a  i^i  b ) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  g )
) } )
32a1i 11 . . . . 5  |-  ( ph  -> ⟂G  =  ( g  e. 
_V  |->  { <. a ,  b >.  |  ( ( a  e.  ran  (LineG `  g )  /\  b  e.  ran  (LineG `  g ) )  /\  E. x  e.  ( a  i^i  b ) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  g
) ) } ) )
4 simpr 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
54fveq2d 5852 . . . . . . . . . . 11  |-  ( (
ph  /\  g  =  G )  ->  (LineG `  g )  =  (LineG `  G ) )
6 isperp.l . . . . . . . . . . 11  |-  L  =  (LineG `  G )
75, 6syl6eqr 2513 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (LineG `  g )  =  L )
87rneqd 5219 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ran  (LineG `  g )  =  ran  L )
98eleq2d 2524 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
a  e.  ran  (LineG `  g )  <->  a  e.  ran  L ) )
108eleq2d 2524 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
b  e.  ran  (LineG `  g )  <->  b  e.  ran  L ) )
119, 10anbi12d 708 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  (
( a  e.  ran  (LineG `  g )  /\  b  e.  ran  (LineG `  g ) )  <->  ( a  e.  ran  L  /\  b  e.  ran  L ) ) )
124fveq2d 5852 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (∟G `  g )  =  (∟G `  G ) )
1312eleq2d 2524 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ( <" u x v ">  e.  (∟G `  g )  <->  <" u x v ">  e.  (∟G `  G )
) )
1413ralbidv 2893 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  ( A. v  e.  b  <" u x v ">  e.  (∟G `  g )  <->  A. v  e.  b  <" u x v ">  e.  (∟G `  G )
) )
1514rexralbidv 2973 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  ( E. 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 ) ) )
1611, 15anbi12d 708 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (
( ( a  e. 
ran  (LineG `  g )  /\  b  e.  ran  (LineG `  g ) )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  g ) )  <-> 
( ( a  e. 
ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b ) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G
) ) ) )
1716opabbidv 4502 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  { <. a ,  b >.  |  ( ( a  e.  ran  (LineG `  g )  /\  b  e.  ran  (LineG `  g ) )  /\  E. x  e.  ( a  i^i  b ) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  g
) ) }  =  { <. a ,  b
>.  |  ( (
a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) } )
18 isperp.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
19 elex 3115 . . . . . 6  |-  ( G  e. TarskiG  ->  G  e.  _V )
2018, 19syl 16 . . . . 5  |-  ( ph  ->  G  e.  _V )
21 fvex 5858 . . . . . . . . 9  |-  (LineG `  G )  e.  _V
226, 21eqeltri 2538 . . . . . . . 8  |-  L  e. 
_V
23 rnexg 6705 . . . . . . . 8  |-  ( L  e.  _V  ->  ran  L  e.  _V )
2422, 23mp1i 12 . . . . . . 7  |-  ( ph  ->  ran  L  e.  _V )
25 xpexg 6575 . . . . . . 7  |-  ( ( ran  L  e.  _V  /\ 
ran  L  e.  _V )  ->  ( ran  L  X.  ran  L )  e. 
_V )
2624, 24, 25syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ran  L  X.  ran  L )  e.  _V )
27 opabssxp 5063 . . . . . . 7  |-  { <. a ,  b >.  |  ( ( a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) }  C_  ( ran  L  X.  ran  L )
2827a1i 11 . . . . . 6  |-  ( ph  ->  { <. a ,  b
>.  |  ( (
a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) }  C_  ( ran  L  X.  ran  L ) )
2926, 28ssexd 4584 . . . . 5  |-  ( ph  ->  { <. a ,  b
>.  |  ( (
a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) }  e.  _V )
303, 17, 20, 29fvmptd 5936 . . . 4  |-  ( ph  ->  (⟂G `  G )  =  { <. a ,  b
>.  |  ( (
a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) } )
3130eleq2d 2524 . . 3  |-  ( ph  ->  ( <. A ,  B >.  e.  (⟂G `  G )  <->  <. A ,  B >.  e. 
{ <. a ,  b
>.  |  ( (
a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) } ) )
321, 31syl5bb 257 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  <. A ,  B >.  e.  { <. a ,  b >.  |  ( ( a  e.  ran  L  /\  b  e.  ran  L )  /\  E. x  e.  ( a  i^i  b
) A. u  e.  a  A. v  e.  b  <" u x v ">  e.  (∟G `  G ) ) } ) )
33 isperp.a . . 3  |-  ( ph  ->  A  e.  ran  L
)
34 isperp.b . . 3  |-  ( ph  ->  B  e.  ran  L
)
35 ineq12 3681 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  i^i  b
)  =  ( A  i^i  B ) )
36 simpll 751 . . . . . 6  |-  ( ( ( a  =  A  /\  b  =  B )  /\  x  e.  ( a  i^i  b
) )  ->  a  =  A )
37 simpllr 758 . . . . . . 7  |-  ( ( ( ( a  =  A  /\  b  =  B )  /\  x  e.  ( a  i^i  b
) )  /\  u  e.  a )  ->  b  =  B )
3837raleqdv 3057 . . . . . 6  |-  ( ( ( ( a  =  A  /\  b  =  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 )
) )
3936, 38raleqbidva 3067 . . . . 5  |-  ( ( ( a  =  A  /\  b  =  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 )
) )
4035, 39rexeqbidva 3068 . . . 4  |-  ( ( a  =  A  /\  b  =  B )  ->  ( E. 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 ) ) )
4140opelopab2a 4751 . . 3  |-  ( ( A  e.  ran  L  /\  B  e.  ran  L )  ->  ( <. A ,  B >.  e.  { <. a ,  b >.  |  ( ( a  e.  ran  L  /\  b  e.  ran  L )  /\  E. 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 ) ) )
4233, 34, 41syl2anc 659 . 2  |-  ( ph  ->  ( <. A ,  B >.  e.  { <. a ,  b >.  |  ( ( a  e.  ran  L  /\  b  e.  ran  L )  /\  E. 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 ) ) )
4332, 42bitrd 253 1  |-  ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   <.cop 4022   class class class wbr 4439   {copab 4496    |-> cmpt 4497    X. cxp 4986   ran crn 4989   ` cfv 5570   <"cs3 12798   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  ∟Gcrag 24271  ⟂Gcperpg 24273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-perpg 24274
This theorem is referenced by:  perpcom  24291  perpneq  24292  isperp2  24293
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