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Theorem perpcom 24480
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 3634 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32a1i 11 . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  ( B  i^i  A ) )
4 ralcom 2970 . . . . 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 2404 . . . . . . . 8  |-  (pInvG `  G )  =  (pInvG `  G )
10 isperp.g . . . . . . . . 9  |-  ( ph  ->  G  e. TarskiG )
1110ad3antrrr 730 . . . . . . . 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 730 . . . . . . . . 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 765 . . . . . . . . 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 24321 . . . . . . . 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 3661 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
17 simpllr 763 . . . . . . . . . 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 3442 . . . . . . . . 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 24321 . . . . . . . 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 730 . . . . . . . . 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 764 . . . . . . . . 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 24321 . . . . . . . 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 24465 . . . . . . 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 730 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  G  e. TarskiG )
2720ad3antrrr 730 . . . . . . . . 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 764 . . . . . . . . 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 24321 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  ( A  i^i  B ) )  /\  ( v  e.  B  /\  u  e.  A
) )  /\  <" v x u ">  e.  (∟G `  G
) )  ->  v  e.  P )
3012ad3antrrr 730 . . . . . . . . 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 763 . . . . . . . . . 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 3442 . . . . . . . . 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 24321 . . . . . . . 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 765 . . . . . . . . 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 24321 . . . . . . . 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 24465 . . . . . . 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 835 . . . . . 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 2848 . . . . 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 259 . . . 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 3023 . . 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 24479 . . 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 24479 . . 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 287 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  B (⟂G `  G ) A ) )
451, 44mpbid 212 1  |-  ( ph  ->  B (⟂G `  G
) A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757    i^i cin 3415   class class class wbr 4397   ran crn 4826   ` cfv 5571   <"cs3 12865   Basecbs 14843   distcds 14920  TarskiGcstrkg 24208  Itvcitv 24214  LineGclng 24215  pInvGcmir 24422  ∟Gcrag 24460  ⟂Gcperpg 24462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-concat 12595  df-s1 12596  df-s2 12871  df-s3 12872  df-trkgc 24226  df-trkgb 24227  df-trkgcb 24228  df-trkg 24231  df-mir 24423  df-rag 24461  df-perpg 24463
This theorem is referenced by:  hlperpnel  24489  colperpexlem3  24496  mideulem2  24498  midex  24501  opphllem5  24514  opphllem6  24515  opphl  24517  lmieu  24545  lnperpex  24564  trgcopy  24565
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