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

Theorem foot 24297
Description: From a point  C outside of a line  A, there exists a unique point  x on  A such that  ( C L x ) is perpendicular to  A. That point is called the foot from  C on  A. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-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
)
foot.x  |-  ( ph  ->  C  e.  P )
foot.y  |-  ( ph  ->  -.  C  e.  A
)
Assertion
Ref Expression
foot  |-  ( ph  ->  E! x  e.  A  ( C L x ) (⟂G `  G ) A )
Distinct variable groups:    x, A    x, G    ph, x    x, C    x, I    x,  .-    x, L   
x, P

Proof of Theorem foot
Dummy variables  u  v  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isperp.p . . 3  |-  P  =  ( Base `  G
)
2 isperp.d . . 3  |-  .-  =  ( dist `  G )
3 isperp.i . . 3  |-  I  =  (Itv `  G )
4 isperp.l . . 3  |-  L  =  (LineG `  G )
5 isperp.g . . 3  |-  ( ph  ->  G  e. TarskiG )
6 isperp.a . . 3  |-  ( ph  ->  A  e.  ran  L
)
7 foot.x . . 3  |-  ( ph  ->  C  e.  P )
8 foot.y . . 3  |-  ( ph  ->  -.  C  e.  A
)
91, 2, 3, 4, 5, 6, 7, 8footex 24296 . 2  |-  ( ph  ->  E. x  e.  A  ( C L x ) (⟂G `  G ) A )
10 eqid 2454 . . . . . 6  |-  (pInvG `  G )  =  (pInvG `  G )
115ad2antrr 723 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  G  e. TarskiG )
127ad2antrr 723 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  e.  P )
135adantr 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  G  e. TarskiG )
146adantr 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  A  e.  ran  L )
15 simprl 754 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  A )
161, 4, 3, 13, 14, 15tglnpt 24137 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  P )
1716adantr 463 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  x  e.  P )
18 simprr 755 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  A )
191, 4, 3, 13, 14, 18tglnpt 24137 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  P )
2019adantr 463 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
z  e.  P )
218adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  -.  C  e.  A
)
22 nelne2 2784 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  -.  C  e.  A
)  ->  x  =/=  C )
2315, 21, 22syl2anc 659 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  =/=  C )
2423necomd 2725 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  =/=  x )
2524adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  =/=  x )
261, 3, 4, 11, 12, 17, 25tglinerflx1 24214 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  e.  ( C L x ) )
2718adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
z  e.  A )
28 simprl 754 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
( C L x ) (⟂G `  G
) A )
297adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  e.  P )
301, 3, 4, 13, 29, 16, 24tgelrnln 24211 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( C L x )  e.  ran  L
)
311, 3, 4, 13, 29, 16, 24tglinerflx2 24215 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  ( C L x ) )
3231, 15elind 3674 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  ( ( C L x )  i^i 
A ) )
331, 2, 3, 4, 13, 30, 14, 32isperp2 24293 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( ( C L x ) (⟂G `  G
) A  <->  A. u  e.  ( C L x ) A. v  e.  A  <" u x v ">  e.  (∟G `  G ) ) )
3433adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
( ( C L x ) (⟂G `  G
) A  <->  A. u  e.  ( C L x ) A. v  e.  A  <" u x v ">  e.  (∟G `  G ) ) )
3528, 34mpbid 210 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  A. u  e.  ( C L x ) A. v  e.  A  <" u x v ">  e.  (∟G `  G
) )
36 id 22 . . . . . . . . . 10  |-  ( u  =  C  ->  u  =  C )
37 eqidd 2455 . . . . . . . . . 10  |-  ( u  =  C  ->  x  =  x )
38 eqidd 2455 . . . . . . . . . 10  |-  ( u  =  C  ->  v  =  v )
3936, 37, 38s3eqd 12819 . . . . . . . . 9  |-  ( u  =  C  ->  <" u x v ">  =  <" C x v "> )
4039eleq1d 2523 . . . . . . . 8  |-  ( u  =  C  ->  ( <" u x v ">  e.  (∟G `  G )  <->  <" C x v ">  e.  (∟G `  G )
) )
41 eqidd 2455 . . . . . . . . . 10  |-  ( v  =  z  ->  C  =  C )
42 eqidd 2455 . . . . . . . . . 10  |-  ( v  =  z  ->  x  =  x )
43 id 22 . . . . . . . . . 10  |-  ( v  =  z  ->  v  =  z )
4441, 42, 43s3eqd 12819 . . . . . . . . 9  |-  ( v  =  z  ->  <" C x v ">  =  <" C x z "> )
4544eleq1d 2523 . . . . . . . 8  |-  ( v  =  z  ->  ( <" C x v ">  e.  (∟G `  G )  <->  <" C x z ">  e.  (∟G `  G )
) )
4640, 45rspc2va 3217 . . . . . . 7  |-  ( ( ( C  e.  ( C L x )  /\  z  e.  A
)  /\  A. u  e.  ( C L x ) A. v  e.  A  <" u x v ">  e.  (∟G `  G ) )  ->  <" C x z ">  e.  (∟G `  G ) )
4726, 27, 35, 46syl21anc 1225 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  <" C x z ">  e.  (∟G `  G ) )
48 nelne2 2784 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  -.  C  e.  A
)  ->  z  =/=  C )
4918, 21, 48syl2anc 659 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  =/=  C )
5049necomd 2725 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  =/=  z )
5150adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  =/=  z )
521, 3, 4, 11, 12, 20, 51tglinerflx1 24214 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  e.  ( C L z ) )
5315adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  x  e.  A )
54 simprr 755 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
( C L z ) (⟂G `  G
) A )
551, 3, 4, 13, 29, 19, 50tgelrnln 24211 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( C L z )  e.  ran  L
)
561, 3, 4, 13, 29, 19, 50tglinerflx2 24215 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  ( C L z ) )
5756, 18elind 3674 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  ( ( C L z )  i^i  A ) )
581, 2, 3, 4, 13, 55, 14, 57isperp2 24293 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( ( C L z ) (⟂G `  G
) A  <->  A. u  e.  ( C L z ) A. v  e.  A  <" u z v ">  e.  (∟G `  G ) ) )
5958adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
( ( C L z ) (⟂G `  G
) A  <->  A. u  e.  ( C L z ) A. v  e.  A  <" u z v ">  e.  (∟G `  G ) ) )
6054, 59mpbid 210 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  A. u  e.  ( C L z ) A. v  e.  A  <" u z v ">  e.  (∟G `  G
) )
61 eqidd 2455 . . . . . . . . . 10  |-  ( u  =  C  ->  z  =  z )
6236, 61, 38s3eqd 12819 . . . . . . . . 9  |-  ( u  =  C  ->  <" u
z v ">  =  <" C z v "> )
6362eleq1d 2523 . . . . . . . 8  |-  ( u  =  C  ->  ( <" u z v ">  e.  (∟G `  G )  <->  <" C
z v ">  e.  (∟G `  G )
) )
64 eqidd 2455 . . . . . . . . . 10  |-  ( v  =  x  ->  C  =  C )
65 eqidd 2455 . . . . . . . . . 10  |-  ( v  =  x  ->  z  =  z )
66 id 22 . . . . . . . . . 10  |-  ( v  =  x  ->  v  =  x )
6764, 65, 66s3eqd 12819 . . . . . . . . 9  |-  ( v  =  x  ->  <" C
z v ">  =  <" C z x "> )
6867eleq1d 2523 . . . . . . . 8  |-  ( v  =  x  ->  ( <" C z v ">  e.  (∟G `  G )  <->  <" C
z x ">  e.  (∟G `  G )
) )
6963, 68rspc2va 3217 . . . . . . 7  |-  ( ( ( C  e.  ( C L z )  /\  x  e.  A
)  /\  A. u  e.  ( C L z ) A. v  e.  A  <" u z v ">  e.  (∟G `  G ) )  ->  <" C z x ">  e.  (∟G `  G ) )
7052, 53, 60, 69syl21anc 1225 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  <" C z x ">  e.  (∟G `  G ) )
711, 2, 3, 4, 10, 11, 12, 17, 20, 47, 70ragflat 24282 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  x  =  z )
7271ex 432 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A )  ->  x  =  z ) )
7372ralrimivva 2875 . . 3  |-  ( ph  ->  A. x  e.  A  A. z  e.  A  ( ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A )  ->  x  =  z ) )
74 oveq2 6278 . . . . 5  |-  ( x  =  z  ->  ( C L x )  =  ( C L z ) )
7574breq1d 4449 . . . 4  |-  ( x  =  z  ->  (
( C L x ) (⟂G `  G
) A  <->  ( C L z ) (⟂G `  G ) A ) )
7675rmo4 3289 . . 3  |-  ( E* x  e.  A  ( C L x ) (⟂G `  G ) A 
<-> 
A. x  e.  A  A. z  e.  A  ( ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A )  ->  x  =  z ) )
7773, 76sylibr 212 . 2  |-  ( ph  ->  E* x  e.  A  ( C L x ) (⟂G `  G ) A )
78 reu5 3070 . 2  |-  ( E! x  e.  A  ( C L x ) (⟂G `  G ) A 
<->  ( E. x  e.  A  ( C L x ) (⟂G `  G
) A  /\  E* x  e.  A  ( C L x ) (⟂G `  G ) A ) )
799, 77, 78sylanbrc 662 1  |-  ( ph  ->  E! x  e.  A  ( C L x ) (⟂G `  G ) A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   E!wreu 2806   E*wrmo 2807   class class class wbr 4439   ran crn 4989   ` cfv 5570  (class class class)co 6270   <"cs3 12798   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  pInvGcmir 24234  ∟Gcrag 24271  ⟂Gcperpg 24273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-s2 12804  df-s3 12805  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-cgrg 24104  df-leg 24171  df-mir 24235  df-rag 24272  df-perpg 24274
This theorem is referenced by:  footeq  24299  mideulem2  24309  lmieu  24351
  Copyright terms: Public domain W3C validator