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Theorem perpneq 24808
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 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  G  e. TarskiG )
65ad5antr 745 . . . . . . 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 745 . . . . . . . . 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 745 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  e.  ran  L )
10 inss1 3664 . . . . . . . . . . 11  |-  ( A  i^i  B )  C_  A
11 simpr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  ( A  i^i  B ) )
1210, 11sseldi 3442 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  A )
1312ad4antr 743 . . . . . . . . 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 24643 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  e.  P )
1514adantl4r 762 . . . . . . 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 745 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  e.  ran  L )
18 simplr 767 . . . . . . . . 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 24643 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  e.  P )
2019adantl4r 762 . . . . . . 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 782 . . . . . . . . 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 24643 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  e.  P )
2322adantl4r 762 . . . . . . 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 2462 . . . . . . . . 9  |-  (pInvG `  G )  =  (pInvG `  G )
26 simp-4r 782 . . . . . . . . . 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 767 . . . . . . . . . 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 784 . . . . . . . . . 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 2463 . . . . . . . . . . . . 13  |-  ( y  =  u  ->  x  =  x )
31 eqidd 2463 . . . . . . . . . . . . 13  |-  ( y  =  u  ->  z  =  z )
3229, 30, 31s3eqd 12996 . . . . . . . . . . . 12  |-  ( y  =  u  ->  <" y
x z ">  =  <" u x z "> )
3332eleq1d 2524 . . . . . . . . . . 11  |-  ( y  =  u  ->  ( <" y x z ">  e.  (∟G `  G )  <->  <" u x z ">  e.  (∟G `  G )
) )
34 eqidd 2463 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  u  =  u )
35 eqidd 2463 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  x  =  x )
36 id 22 . . . . . . . . . . . . 13  |-  ( z  =  v  ->  z  =  v )
3734, 35, 36s3eqd 12996 . . . . . . . . . . . 12  |-  ( z  =  v  ->  <" u x z ">  =  <" u x v "> )
3837eleq1d 2524 . . . . . . . . . . 11  |-  ( z  =  v  ->  ( <" u x z ">  e.  (∟G `  G )  <->  <" u x v ">  e.  (∟G `  G )
) )
3933, 38rspc2va 3172 . . . . . . . . . 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 1275 . . . . . . . . 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 774 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  =/=  u )
4241necomd 2691 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  u  =/=  x )
4342adantl4r 762 . . . . . . . . 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 467 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  x  =/=  v )
4544necomd 2691 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  v  =/=  x )
4645adantl4r 762 . . . . . . . . 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 24803 . . . . . . . 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 24657 . . . . . . 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 24738 . . . . . 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 2691 . . . . 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 24730 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  A  =  ( u L x ) )
5251adantl4r 762 . . . . 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 3665 . . . . . . . . 9  |-  ( A  i^i  B )  C_  B
5453, 11sseldi 3442 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  x  e.  B )
5554ad4antr 743 . . . . . . 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 24730 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  x  e.  ( A  i^i  B ) )  /\  u  e.  A )  /\  x  =/=  u )  /\  v  e.  B )  /\  x  =/=  v )  ->  B  =  ( x L v ) )
5756adantl4r 762 . . . . 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 2713 . . . 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 471 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  B  e.  ran  L )
601, 2, 3, 5, 59, 54tglnpt2 24735 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  E. v  e.  B  x  =/=  v )
6160ad3antrrr 741 . . . 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 2944 . . 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 471 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  A  e.  ran  L )
641, 2, 3, 5, 63, 12tglnpt2 24735 . . . 4  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  E. u  e.  A  x  =/=  u )
6564adantr 471 . . 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 2944 . 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 24806 . . 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 215 . 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 2944 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750    i^i cin 3415   class class class wbr 4416   ran crn 4854   ` cfv 5601  (class class class)co 6315   <"cs3 12975   Basecbs 15170   distcds 15248  TarskiGcstrkg 24527  Itvcitv 24533  LineGclng 24534  pInvGcmir 24746  ∟Gcrag 24787  ⟂Gcperpg 24789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-fzo 11947  df-hash 12548  df-word 12697  df-concat 12699  df-s1 12700  df-s2 12981  df-s3 12982  df-trkgc 24545  df-trkgb 24546  df-trkgcb 24547  df-trkg 24550  df-cgrg 24605  df-mir 24747  df-rag 24788  df-perpg 24790
This theorem is referenced by:  isperp2  24809  footne  24814  lmieu  24875
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