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Theorem isperp 23790
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 4441 . . 3  |-  ( A (⟂G `  G ) B 
<-> 
<. A ,  B >.  e.  (⟂G `  G )
)
2 df-perpg 23774 . . . . . 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 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
54fveq2d 5861 . . . . . . . . . . 11  |-  ( (
ph  /\  g  =  G )  ->  (LineG `  g )  =  (LineG `  G ) )
6 isperp.l . . . . . . . . . . 11  |-  L  =  (LineG `  G )
75, 6syl6eqr 2519 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (LineG `  g )  =  L )
87rneqd 5221 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ran  (LineG `  g )  =  ran  L )
98eleq2d 2530 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
a  e.  ran  (LineG `  g )  <->  a  e.  ran  L ) )
108eleq2d 2530 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
b  e.  ran  (LineG `  g )  <->  b  e.  ran  L ) )
119, 10anbi12d 710 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  (
( a  e.  ran  (LineG `  g )  /\  b  e.  ran  (LineG `  g ) )  <->  ( a  e.  ran  L  /\  b  e.  ran  L ) ) )
124fveq2d 5861 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (∟G `  g )  =  (∟G `  G ) )
1312eleq2d 2530 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ( <" u x v ">  e.  (∟G `  g )  <->  <" u x v ">  e.  (∟G `  G )
) )
1413ralbidv 2896 . . . . . . . 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 2974 . . . . . . 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 710 . . . . . 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 4503 . . . . 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 opabssxp 5065 . . . . . . 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 )
2221a1i 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 ) )
23 fvex 5867 . . . . . . . . 9  |-  (LineG `  G )  e.  _V
246, 23eqeltri 2544 . . . . . . . 8  |-  L  e. 
_V
25 rnexg 6706 . . . . . . . 8  |-  ( L  e.  _V  ->  ran  L  e.  _V )
2624, 25mp1i 12 . . . . . . 7  |-  ( ph  ->  ran  L  e.  _V )
27 xpexg 6702 . . . . . . 7  |-  ( ( ran  L  e.  _V  /\ 
ran  L  e.  _V )  ->  ( ran  L  X.  ran  L )  e. 
_V )
2826, 26, 27syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ran  L  X.  ran  L )  e.  _V )
29 ssexg 4586 . . . . . 6  |-  ( ( { <. 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 )  /\  ( ran  L  X.  ran  L )  e. 
_V )  ->  { <. 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 )
3022, 28, 29syl2anc 661 . . . . 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 )
313, 17, 20, 30fvmptd 5946 . . . 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 ) ) } )
3231eleq2d 2530 . . 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 ) ) } ) )
331, 32syl5bb 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 ) ) } ) )
34 isperp.a . . 3  |-  ( ph  ->  A  e.  ran  L
)
35 isperp.b . . 3  |-  ( ph  ->  B  e.  ran  L
)
36 ineq12 3688 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a  i^i  b
)  =  ( A  i^i  B ) )
37 simpll 753 . . . . . 6  |-  ( ( ( a  =  A  /\  b  =  B )  /\  x  e.  ( a  i^i  b
) )  ->  a  =  A )
38 simpllr 758 . . . . . . 7  |-  ( ( ( ( a  =  A  /\  b  =  B )  /\  x  e.  ( a  i^i  b
) )  /\  u  e.  a )  ->  b  =  B )
3938raleqdv 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 )
) )
4037, 39raleqbidva 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 )
) )
4136, 40rexeqbidva 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 ) ) )
4241opelopab2a 4755 . . 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 ) ) )
4334, 35, 42syl2anc 661 . 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 ) ) )
4433, 43bitrd 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 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   _Vcvv 3106    i^i cin 3468    C_ wss 3469   <.cop 4026   class class class wbr 4440   {copab 4497    |-> cmpt 4498    X. cxp 4990   ran crn 4993   ` cfv 5579   <"cs3 12757   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554  ∟Gcrag 23771  ⟂Gcperpg 23773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fv 5587  df-perpg 23774
This theorem is referenced by:  perpcom  23791  perpneq  23792  isperp2  23793
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