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Theorem isperp 23101
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 4291 . . 3  |-  ( A (⟂G `  G ) B 
<-> 
<. A ,  B >.  e.  (⟂G `  G )
)
2 df-perpg 23088 . . . . . 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 5693 . . . . . . . . . . 11  |-  ( (
ph  /\  g  =  G )  ->  (LineG `  g )  =  (LineG `  G ) )
6 isperp.l . . . . . . . . . . 11  |-  L  =  (LineG `  G )
75, 6syl6eqr 2491 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (LineG `  g )  =  L )
87rneqd 5065 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ran  (LineG `  g )  =  ran  L )
98eleq2d 2508 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
a  e.  ran  (LineG `  g )  <->  a  e.  ran  L ) )
108eleq2d 2508 . . . . . . . 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 5693 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (∟G `  g )  =  (∟G `  G ) )
1312eleq2d 2508 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ( <" u x v ">  e.  (∟G `  g )  <->  <" u x v ">  e.  (∟G `  G )
) )
1413ralbidv 2733 . . . . . . . 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 2757 . . . . . . 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 4353 . . . . 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 2979 . . . . . 6  |-  ( G  e. TarskiG  ->  G  e.  _V )
2018, 19syl 16 . . . . 5  |-  ( ph  ->  G  e.  _V )
21 opabssxp 4909 . . . . . . 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 5699 . . . . . . . . 9  |-  (LineG `  G )  e.  _V
246, 23eqeltri 2511 . . . . . . . 8  |-  L  e. 
_V
25 rnexg 6508 . . . . . . . 8  |-  ( L  e.  _V  ->  ran  L  e.  _V )
2624, 25mp1i 12 . . . . . . 7  |-  ( ph  ->  ran  L  e.  _V )
27 xpexg 6505 . . . . . . 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 4436 . . . . . 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 5777 . . . 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 2508 . . 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 3545 . . . . 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 2921 . . . . . 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 2931 . . . . 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 2932 . . . 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 4602 . . 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 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   _Vcvv 2970    i^i cin 3325    C_ wss 3326   <.cop 3881   class class class wbr 4290   {copab 4347    e. cmpt 4348    X. cxp 4836   ran crn 4839   ` cfv 5416   <"cs3 12467   Basecbs 14172   distcds 14245  TarskiGcstrkg 22887  Itvcitv 22895  LineGclng 22896  ∟Gcrag 23085  ⟂Gcperpg 23087
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fv 5424  df-perpg 23088
This theorem is referenced by:  perpcom  23102  perpneq  23103  isperp2  23104
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