MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lnperpex Structured version   Unicode version

Theorem lnperpex 24832
Description: Existence of a perpendicular to a line  L at a given point  A. Theorem 10.15 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)
Hypotheses
Ref Expression
lmiopp.p  |-  P  =  ( Base `  G
)
lmiopp.m  |-  .-  =  ( dist `  G )
lmiopp.i  |-  I  =  (Itv `  G )
lmiopp.l  |-  L  =  (LineG `  G )
lmiopp.g  |-  ( ph  ->  G  e. TarskiG )
lmiopp.h  |-  ( ph  ->  GDimTarskiG 2 )
lmiopp.d  |-  ( ph  ->  D  e.  ran  L
)
lmiopp.o  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
lnperpex.a  |-  ( ph  ->  A  e.  D )
lnperpex.q  |-  ( ph  ->  Q  e.  P )
lnperpex.1  |-  ( ph  ->  -.  Q  e.  D
)
Assertion
Ref Expression
lnperpex  |-  ( ph  ->  E. p  e.  P  ( D (⟂G `  G
) ( p L A )  /\  p
( (hpG `  G
) `  D ) Q ) )
Distinct variable groups:    .- , a, b, p, t    A, a, b, p, t    D, a, b, p, t    G, a, b, p, t    I,
a, b, p, t    L, a, b, p, t    O, a, b, p, t    P, a, b, p, t    Q, a, b, p, t    ph, a, b, p, t

Proof of Theorem lnperpex
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmiopp.p . . . . . 6  |-  P  =  ( Base `  G
)
2 lmiopp.m . . . . . 6  |-  .-  =  ( dist `  G )
3 lmiopp.i . . . . . 6  |-  I  =  (Itv `  G )
4 lmiopp.l . . . . . 6  |-  L  =  (LineG `  G )
5 lmiopp.g . . . . . . . . 9  |-  ( ph  ->  G  e. TarskiG )
65ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  G  e. TarskiG )
76ad2antrr 730 . . . . . . 7  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  G  e. TarskiG )
87adantr 466 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  G  e. TarskiG )
9 simprl 762 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  p  e.  P )
10 lmiopp.d . . . . . . . . . 10  |-  ( ph  ->  D  e.  ran  L
)
11 lnperpex.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  D )
121, 4, 3, 5, 10, 11tglnpt 24581 . . . . . . . . 9  |-  ( ph  ->  A  e.  P )
1312ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  A  e.  P )
1413ad3antrrr 734 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  A  e.  P )
15 simprrl 772 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( A L p ) (⟂G `  G ) D )
164, 8, 15perpln1 24742 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( A L p )  e. 
ran  L )
171, 3, 4, 8, 14, 9, 16tglnne 24660 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  A  =/=  p )
1817necomd 2695 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  p  =/=  A )
191, 3, 4, 8, 9, 14, 18tgelrnln 24662 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( p L A )  e.  ran  L )
2010ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  D  e.  ran  L )
2120ad2antrr 730 . . . . . . 7  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  D  e.  ran  L )
2221adantr 466 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  D  e.  ran  L )
231, 3, 4, 8, 9, 14, 18tglinecom 24667 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( p L A )  =  ( A L p ) )
2423, 15eqbrtrd 4441 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( p L A ) (⟂G `  G
) D )
251, 2, 3, 4, 8, 19, 22, 24perpcom 24745 . . . . 5  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  D (⟂G `  G ) ( p L A ) )
26 simpr 462 . . . . . . 7  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  Q O c )
2726adantr 466 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  Q O
c )
28 lmiopp.o . . . . . . 7  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
29 lnperpex.q . . . . . . . . . 10  |-  ( ph  ->  Q  e.  P )
3029ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  Q  e.  P )
3130ad2antrr 730 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  Q  e.  P )
3231adantr 466 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  Q  e.  P )
33 simplr 760 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  -> 
c  e.  P )
3433adantr 466 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  c  e.  P )
35 simprrr 773 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  c O p )
361, 2, 3, 28, 4, 22, 8, 34, 9, 35oppcom 24773 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  p O
c )
371, 3, 4, 28, 8, 22, 9, 32, 34, 36lnopp2hpgb 24792 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( Q O c  <->  p (
(hpG `  G ) `  D ) Q ) )
3827, 37mpbid 213 . . . . 5  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  p (
(hpG `  G ) `  D ) Q )
3925, 38jca 534 . . . 4  |-  ( ( ( ( ( (
ph  /\  d  e.  D )  /\  A  =/=  d )  /\  c  e.  P )  /\  Q O c )  /\  ( p  e.  P  /\  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) ) )  ->  ( D
(⟂G `  G ) ( p L A )  /\  p ( (hpG
`  G ) `  D ) Q ) )
40 eqid 2422 . . . . 5  |-  (hlG `  G )  =  (hlG
`  G )
4111ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  A  e.  D )
4241ad2antrr 730 . . . . 5  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  A  e.  D )
431, 2, 3, 28, 4, 21, 7, 31, 33, 26oppne2 24771 . . . . 5  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  -.  c  e.  D
)
44 lmiopp.h . . . . . . 7  |-  ( ph  ->  GDimTarskiG 2 )
4544ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  GDimTarskiG 2
)
4645ad2antrr 730 . . . . 5  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  GDimTarskiG 2 )
471, 2, 3, 28, 4, 21, 7, 40, 42, 33, 43, 46oppperpex 24782 . . . 4  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) D  /\  c O p ) )
4839, 47reximddv 2901 . . 3  |-  ( ( ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d
)  /\  c  e.  P )  /\  Q O c )  ->  E. p  e.  P  ( D (⟂G `  G
) ( p L A )  /\  p
( (hpG `  G
) `  D ) Q ) )
49 lnperpex.1 . . . . 5  |-  ( ph  ->  -.  Q  e.  D
)
501, 3, 4, 5, 10, 29, 28, 49hpgerlem 24794 . . . 4  |-  ( ph  ->  E. c  e.  P  Q O c )
5150ad2antrr 730 . . 3  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  E. c  e.  P  Q O
c )
5248, 51r19.29a 2970 . 2  |-  ( ( ( ph  /\  d  e.  D )  /\  A  =/=  d )  ->  E. p  e.  P  ( D
(⟂G `  G ) ( p L A )  /\  p ( (hpG
`  G ) `  D ) Q ) )
531, 3, 4, 5, 10, 11tglnpt2 24673 . 2  |-  ( ph  ->  E. d  e.  D  A  =/=  d )
5452, 53r19.29a 2970 1  |-  ( ph  ->  E. p  e.  P  ( D (⟂G `  G
) ( p L A )  /\  p
( (hpG `  G
) `  D ) Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776    \ cdif 3433   class class class wbr 4420   {copab 4478   ran crn 4851   ` cfv 5598  (class class class)co 6302   2c2 10660   Basecbs 15109   distcds 15187  TarskiGcstrkg 24465  DimTarskiGcstrkgld 24469  Itvcitv 24471  LineGclng 24472  hlGchlg 24632  ⟂Gcperpg 24727  hpGchpg 24786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-hash 12516  df-word 12657  df-concat 12659  df-s1 12660  df-s2 12935  df-s3 12936  df-trkgc 24483  df-trkgb 24484  df-trkgcb 24485  df-trkgld 24487  df-trkg 24488  df-cgrg 24543  df-leg 24615  df-hlg 24633  df-mir 24685  df-rag 24726  df-perpg 24728  df-hpg 24787
This theorem is referenced by:  trgcopy  24833
  Copyright terms: Public domain W3C validator