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Theorem perpcom 23826
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 3691 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
32a1i 11 . . . 4  |-  ( ph  ->  ( A  i^i  B
)  =  ( B  i^i  A ) )
4 ralcom 3022 . . . . 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 2467 . . . . . . . 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 23692 . . . . . . . 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 3718 . . . . . . . . . 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 3502 . . . . . . . . 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 23692 . . . . . . . 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 23692 . . . . . . . 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 23811 . . . . . . 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 23692 . . . . . . . 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 3502 . . . . . . . . 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 23692 . . . . . . . 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 23692 . . . . . . . 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 23811 . . . . . . 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 830 . . . . . 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 2906 . . . . 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 3075 . . 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 23825 . . 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 23825 . . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475   class class class wbr 4447   ran crn 5000   ` cfv 5588   <"cs3 12770   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589  pInvGcmir 23774  ∟Gcrag 23806  ⟂Gcperpg 23808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-s2 12776  df-s3 12777  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606  df-mir 23775  df-rag 23807  df-perpg 23809
This theorem is referenced by:  colperpexlem3  23839  mideulem  23841  mideu  23842  lmieu  23855
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