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Theorem perpcom 23126
Description: The "perpendicular" relation commutes. Theorem 8.12 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
)
perpcom.1  |-  ( ph  ->  A (⟂G `  G
) B )
Assertion
Ref Expression
perpcom  |-  ( ph  ->  B (⟂G `  G
) A )

Proof of Theorem perpcom
Dummy variables  u  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 perpcom.1 . 2  |-  ( ph  ->  A (⟂G `  G
) B )
2 incom 3564 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32a1i 11 . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  ( B  i^i  A ) )
4 ralcom 2902 . . . . 5  |-  ( A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G
)  <->  A. v  e.  B  A. u  e.  A  <" u x v ">  e.  (∟G `  G ) )
5 isperp.p . . . . . . . 8  |-  P  =  ( Base `  G
)
6 isperp.d . . . . . . . 8  |-  .-  =  ( dist `  G )
7 isperp.i . . . . . . . 8  |-  I  =  (Itv `  G )
8 isperp.l . . . . . . . 8  |-  L  =  (LineG `  G )
9 eqid 2443 . . . . . . . 8  |-  (pInvG `  G )  =  (pInvG `  G )
10 isperp.g . . . . . . . . 9  |-  ( ph  ->  G  e. TarskiG )
1110ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  G  e. TarskiG )
12 isperp.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  ran  L
)
1312ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  A  e.  ran  L )
14 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  u  e.  A )
155, 8, 7, 11, 13, 14tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  u  e.  P )
16 inss1 3591 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
17 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  x  e.  ( A  i^i  B
) )
1816, 17sseldi 3375 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  x  e.  A )
195, 8, 7, 11, 13, 18tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  x  e.  P )
20 isperp.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  ran  L
)
2120ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  B  e.  ran  L )
22 simplrl 759 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  v  e.  B )
235, 8, 7, 11, 21, 22tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  v  e.  P )
24 simpr 461 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  <" u x v ">  e.  (∟G `  G )
)
255, 6, 7, 8, 9, 11, 15, 19, 23, 24ragcom 23114 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" u x v ">  e.  (∟G `  G
) )  ->  <" v
x u ">  e.  (∟G `  G )
)
2610ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  G  e. TarskiG )
2720ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  B  e.  ran  L )
28 simplrl 759 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  v  e.  B )
295, 8, 7, 26, 27, 28tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  v  e.  P )
3012ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  A  e.  ran  L )
31 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  x  e.  ( A  i^i  B
) )
3216, 31sseldi 3375 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  x  e.  A )
335, 8, 7, 26, 30, 32tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  x  e.  P )
34 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  u  e.  A )
355, 8, 7, 26, 30, 34tglnpt 23005 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  u  e.  P )
36 simpr 461 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  <" v
x u ">  e.  (∟G `  G )
)
375, 6, 7, 8, 9, 26, 29, 33, 35, 36ragcom 23114 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  <" u x v ">  e.  (∟G `  G )
)
3825, 37impbida 828 . . . . . 6  |-  ( ( ( ph  /\  x  e.  ( A  i^i  B
) )  /\  (
v  e.  B  /\  u  e.  A )
)  ->  ( <" u x v ">  e.  (∟G `  G
)  <->  <" v x u ">  e.  (∟G `  G ) ) )
39382ralbidva 2776 . . . . 5  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  ( A. v  e.  B  A. u  e.  A  <" u x v ">  e.  (∟G `  G
)  <->  A. v  e.  B  A. u  e.  A  <" v x u ">  e.  (∟G `  G ) ) )
404, 39syl5bb 257 . . . 4  |-  ( (
ph  /\  x  e.  ( A  i^i  B ) )  ->  ( A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G
)  <->  A. v  e.  B  A. u  e.  A  <" v x u ">  e.  (∟G `  G ) ) )
413, 40rexeqbidva 2955 . . 3  |-  ( ph  ->  ( E. x  e.  ( A  i^i  B
) A. u  e.  A  A. v  e.  B  <" u x v ">  e.  (∟G `  G )  <->  E. x  e.  ( B  i^i  A
) A. v  e.  B  A. u  e.  A  <" v x u ">  e.  (∟G `  G ) ) )
425, 6, 7, 8, 10, 12, 20isperp 23125 . . 3  |-  ( 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 ) ) )
435, 6, 7, 8, 10, 20, 12isperp 23125 . . 3  |-  ( ph  ->  ( B (⟂G `  G
) A  <->  E. x  e.  ( B  i^i  A
) A. v  e.  B  A. u  e.  A  <" v x u ">  e.  (∟G `  G ) ) )
4441, 42, 433bitr4d 285 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  B (⟂G `  G ) A ) )
451, 44mpbid 210 1  |-  ( ph  ->  B (⟂G `  G
) A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737    i^i cin 3348   class class class wbr 4313   ran crn 4862   ` cfv 5439   <"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-mir 23078  df-rag 23110  df-perpg 23112
This theorem is referenced by: (None)
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