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Theorem perpneq 23127
Description: Two perpendicular lines are different. Theorem 8.14 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 18-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
)
perpcom.1  |-  ( ph  ->  A (⟂G `  G
) B )
Assertion
Ref Expression
perpneq  |-  ( ph  ->  A  =/=  B )

Proof of Theorem perpneq
Dummy variables  u  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isperp.p . . . . . . 7  |-  P  =  ( Base `  G
)
2 isperp.i . . . . . . 7  |-  I  =  (Itv `  G )
3 isperp.l . . . . . . 7  |-  L  =  (LineG `  G )
4 isperp.g . . . . . . . . 9  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  G  e. TarskiG )
65ad5antr 733 . . . . . . 7  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  G  e. TarskiG )
75ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  G  e. TarskiG )
8 isperp.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ran  L
)
98adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  A  e.  ran  L )
109ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  e.  ran  L )
11 inss1 3591 . . . . . . . . . . 11  |-  ( A  i^i  B )  C_  A
12 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  ( A  i^i  B ) )
1311, 12sseldi 3375 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  A )
1413ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  A )
151, 3, 2, 7, 10, 14tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  P )
1615adantl4r 750 . . . . . . 7  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  P )
17 isperp.b . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ran  L
)
1817adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  B  e.  ran  L )
1918ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  e.  ran  L )
20 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  B )
211, 3, 2, 7, 19, 20tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  P )
2221adantl4r 750 . . . . . . 7  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  P )
23 simp-4r 766 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  A )
241, 3, 2, 7, 10, 23tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  P )
2524adantl4r 750 . . . . . . 7  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  P )
26 isperp.d . . . . . . . . 9  |-  .-  =  ( dist `  G )
27 eqid 2443 . . . . . . . . 9  |-  (pInvG `  G )  =  (pInvG `  G )
28 simp-4r 766 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  A )
29 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  B )
30 simp-5r 768 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A. y  e.  A  A. z  e.  B  <" y
x z ">  e.  (∟G `  G )
)
31 id 22 . . . . . . . . . . . . . 14  |-  ( y  =  u  ->  y  =  u )
32 eqidd 2444 . . . . . . . . . . . . . 14  |-  ( y  =  u  ->  x  =  x )
33 eqidd 2444 . . . . . . . . . . . . . 14  |-  ( y  =  u  ->  z  =  z )
3431, 32, 33s3eqd 12511 . . . . . . . . . . . . 13  |-  ( y  =  u  ->  <" y
x z ">  =  <" u x z "> )
3534eleq1d 2509 . . . . . . . . . . . 12  |-  ( y  =  u  ->  ( <" y x z ">  e.  (∟G `  G )  <->  <" u x z ">  e.  (∟G `  G )
) )
36 eqidd 2444 . . . . . . . . . . . . . 14  |-  ( z  =  v  ->  u  =  u )
37 eqidd 2444 . . . . . . . . . . . . . 14  |-  ( z  =  v  ->  x  =  x )
38 id 22 . . . . . . . . . . . . . 14  |-  ( z  =  v  ->  z  =  v )
3936, 37, 38s3eqd 12511 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  <" u x z ">  =  <" u x v "> )
4039eleq1d 2509 . . . . . . . . . . . 12  |-  ( z  =  v  ->  ( <" u x z ">  e.  (∟G `  G )  <->  <" u x v ">  e.  (∟G `  G )
) )
4135, 40rspc2v 3100 . . . . . . . . . . 11  |-  ( ( u  e.  A  /\  v  e.  B )  ->  ( A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G )  ->  <" u x v ">  e.  (∟G `  G ) ) )
4241imp 429 . . . . . . . . . 10  |-  ( ( ( u  e.  A  /\  v  e.  B
)  /\  A. y  e.  A  A. z  e.  B  <" y
x z ">  e.  (∟G `  G )
)  ->  <" u x v ">  e.  (∟G `  G )
)
4328, 29, 30, 42syl21anc 1217 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  <" u x v ">  e.  (∟G `  G )
)
44 simpllr 758 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  =/=  u )
4544necomd 2640 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  =/=  x )
4645adantl4r 750 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  =/=  x )
47 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  =/=  v )
4847necomd 2640 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  =/=  x )
4948adantl4r 750 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  =/=  x )
501, 26, 2, 3, 27, 6, 25, 16, 22, 43, 46, 49ragncol 23124 . . . . . . . 8  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  -.  ( v  e.  ( u L x )  \/  u  =  x ) )
511, 3, 2, 6, 25, 16, 22, 50ncolrot2 23019 . . . . . . 7  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  -.  ( x  e.  (
v L u )  \/  v  =  u ) )
521, 2, 3, 6, 16, 22, 25, 16, 51tglineneq 23071 . . . . . 6  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  (
x L v )  =/=  ( u L x ) )
5352necomd 2640 . . . . 5  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  (
u L x )  =/=  ( x L v ) )
541, 2, 3, 7, 24, 15, 45, 45, 10, 23, 14tglinethru 23064 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  =  ( u L x ) )
5554adantl4r 750 . . . . 5  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  =  ( u L x ) )
56 inss2 3592 . . . . . . . . 9  |-  ( A  i^i  B )  C_  B
5756, 12sseldi 3375 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  B )
5857ad4antr 731 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  B )
591, 2, 3, 7, 15, 21, 47, 47, 19, 58, 20tglinethru 23064 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  =  ( x L v ) )
6059adantl4r 750 . . . . 5  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  =  ( x L v ) )
6153, 55, 603netr4d 2665 . . . 4  |-  ( ( ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  =/=  B )
621, 2, 3, 5, 18, 57tglnpt2 23068 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  E. v  e.  B  x  =/=  v )
6362ad3antrrr 729 . . . 4  |-  ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G ) )  /\  u  e.  A
)  /\  x  =/=  u )  ->  E. v  e.  B  x  =/=  v )
6461, 63r19.29a 2883 . . 3  |-  ( ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G ) )  /\  u  e.  A
)  /\  x  =/=  u )  ->  A  =/=  B )
651, 2, 3, 5, 9, 13tglnpt2 23068 . . . 4  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  E. u  e.  A  x  =/=  u )
6665adantr 465 . . 3  |-  ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  ->  E. u  e.  A  x  =/=  u )
6764, 66r19.29a 2883 . 2  |-  ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  ->  A  =/=  B )
68 perpcom.1 . . 3  |-  ( ph  ->  A (⟂G `  G
) B )
691, 26, 2, 3, 4, 8, 17isperp 23125 . . 3  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  E. x  e.  ( A  i^i  B
) A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G ) ) )
7068, 69mpbid 210 . 2  |-  ( ph  ->  E. x  e.  ( A  i^i  B ) A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G ) )
7167, 70r19.29a 2883 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737    i^i cin 3348   class class class wbr 4313   ran crn 4862   ` cfv 5439  (class class class)co 6112   <"cs3 12490   Basecbs 14195   distcds 14268  TarskiGcstrkg 22911  Itvcitv 22919  LineGclng 22920  pInvGcmir 23077  ∟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:  isperp2  23128
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