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Theorem perpneq 24217
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 )
74ad5antr 733 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  G  e. TarskiG )
8 isperp.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  ran  L
)
98ad5antr 733 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  e.  ran  L )
10 inss1 3714 . . . . . . . . . . 11  |-  ( A  i^i  B )  C_  A
11 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  ( A  i^i  B ) )
1210, 11sseldi 3497 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  A )
1312ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  A )
141, 3, 2, 7, 9, 13tglnpt 24062 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  P )
1514adantl4r 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 )
16 isperp.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  ran  L
)
1716ad5antr 733 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  e.  ran  L )
18 simplr 755 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  B )
191, 3, 2, 7, 17, 18tglnpt 24062 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  P )
2019adantl4r 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 )
21 simp-4r 768 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  A )
221, 3, 2, 7, 9, 21tglnpt 24062 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  P )
2322adantl4r 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 )
24 isperp.d . . . . . . . . 9  |-  .-  =  ( dist `  G )
25 eqid 2457 . . . . . . . . 9  |-  (pInvG `  G )  =  (pInvG `  G )
26 simp-4r 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 )  ->  u  e.  A )
27 simplr 755 . . . . . . . . . 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 )
28 simp-5r 770 . . . . . . . . . 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 )
)
29 id 22 . . . . . . . . . . . . 13  |-  ( y  =  u  ->  y  =  u )
30 eqidd 2458 . . . . . . . . . . . . 13  |-  ( y  =  u  ->  x  =  x )
31 eqidd 2458 . . . . . . . . . . . . 13  |-  ( y  =  u  ->  z  =  z )
3229, 30, 31s3eqd 12840 . . . . . . . . . . . 12  |-  ( y  =  u  ->  <" y
x z ">  =  <" u x z "> )
3332eleq1d 2526 . . . . . . . . . . 11  |-  ( y  =  u  ->  ( <" y x z ">  e.  (∟G `  G )  <->  <" u x z ">  e.  (∟G `  G )
) )
34 eqidd 2458 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  u  =  u )
35 eqidd 2458 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  x  =  x )
36 id 22 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  z  =  v )
3734, 35, 36s3eqd 12840 . . . . . . . . . . . 12  |-  ( z  =  v  ->  <" u x z ">  =  <" u x v "> )
3837eleq1d 2526 . . . . . . . . . . 11  |-  ( z  =  v  ->  ( <" u x z ">  e.  (∟G `  G )  <->  <" u x v ">  e.  (∟G `  G )
) )
3933, 38rspc2va 3220 . . . . . . . . . 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 )
)
4026, 27, 28, 39syl21anc 1227 . . . . . . . . 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 )
)
41 simpllr 760 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  =/=  u )
4241necomd 2728 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  =/=  x )
4342adantl4r 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 )
44 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  =/=  v )
4544necomd 2728 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  =/=  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 )  ->  v  =/=  x )
471, 24, 2, 3, 25, 6, 23, 15, 20, 40, 43, 46ragncol 24212 . . . . . . . 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 ) )
481, 3, 2, 6, 23, 15, 20, 47ncolrot2 24076 . . . . . . 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 ) )
491, 2, 3, 6, 15, 20, 23, 15, 48tglineneq 24150 . . . . . 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 ) )
5049necomd 2728 . . . . 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 ) )
511, 2, 3, 7, 22, 14, 42, 42, 9, 21, 13tglinethru 24142 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  =  ( u L x ) )
5251adantl4r 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 ) )
53 inss2 3715 . . . . . . . . 9  |-  ( A  i^i  B )  C_  B
5453, 11sseldi 3497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  B )
5554ad4antr 731 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  B )
561, 2, 3, 7, 14, 19, 44, 44, 17, 55, 18tglinethru 24142 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  =  ( x L v ) )
5756adantl4r 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 ) )
5850, 52, 573netr4d 2762 . . . 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 )
5916adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  B  e.  ran  L )
601, 2, 3, 5, 59, 54tglnpt2 24147 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  E. v  e.  B  x  =/=  v )
6160ad3antrrr 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 )
6258, 61r19.29a 2999 . . 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 )
638adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  A  e.  ran  L )
641, 2, 3, 5, 63, 12tglnpt2 24147 . . . 4  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  E. u  e.  A  x  =/=  u )
6564adantr 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 )
6662, 65r19.29a 2999 . 2  |-  ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G
) )  ->  A  =/=  B )
67 perpcom.1 . . 3  |-  ( ph  ->  A (⟂G `  G
) B )
681, 24, 2, 3, 4, 8, 16isperp 24215 . . 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 ) ) )
6967, 68mpbid 210 . 2  |-  ( ph  ->  E. x  e.  ( A  i^i  B ) A. y  e.  A  A. z  e.  B  <" y x z ">  e.  (∟G `  G ) )
7066, 69r19.29a 2999 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    i^i cin 3470   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   <"cs3 12819   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959  pInvGcmir 24159  ∟Gcrag 24196  ⟂Gcperpg 24198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-cgrg 24029  df-mir 24160  df-rag 24197  df-perpg 24199
This theorem is referenced by:  isperp2  24218  footne  24223  lmieu  24276
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