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

Theorem foot 24706
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 24705 . 2  |-  ( ph  ->  E. x  e.  A  ( C L x ) (⟂G `  G ) A )
10 eqid 2428 . . . . . 6  |-  (pInvG `  G )  =  (pInvG `  G )
115ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  G  e. TarskiG )
127ad2antrr 730 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  e.  P )
135adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  G  e. TarskiG )
146adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  A  e.  ran  L )
15 simprl 762 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  A )
161, 4, 3, 13, 14, 15tglnpt 24536 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  P )
1716adantr 466 . . . . . 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 764 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  A )
191, 4, 3, 13, 14, 18tglnpt 24536 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  P )
2019adantr 466 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
z  e.  P )
218adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  -.  C  e.  A
)
22 nelne2 2698 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  -.  C  e.  A
)  ->  x  =/=  C )
2315, 21, 22syl2anc 665 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  =/=  C )
2423necomd 2656 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  =/=  x )
2524adantr 466 . . . . . . . 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 24620 . . . . . . 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 466 . . . . . . 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 762 . . . . . . . 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 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  e.  P )
301, 3, 4, 13, 29, 16, 24tgelrnln 24617 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( C L x )  e.  ran  L
)
311, 3, 4, 13, 29, 16, 24tglinerflx2 24621 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  ( C L x ) )
3231, 15elind 3593 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  ( ( C L x )  i^i 
A ) )
331, 2, 3, 4, 13, 30, 14, 32isperp2 24702 . . . . . . . . 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 466 . . . . . . . 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 213 . . . . . . 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 2429 . . . . . . . . . 10  |-  ( u  =  C  ->  x  =  x )
38 eqidd 2429 . . . . . . . . . 10  |-  ( u  =  C  ->  v  =  v )
3936, 37, 38s3eqd 12905 . . . . . . . . 9  |-  ( u  =  C  ->  <" u x v ">  =  <" C x v "> )
4039eleq1d 2490 . . . . . . . 8  |-  ( u  =  C  ->  ( <" u x v ">  e.  (∟G `  G )  <->  <" C x v ">  e.  (∟G `  G )
) )
41 eqidd 2429 . . . . . . . . . 10  |-  ( v  =  z  ->  C  =  C )
42 eqidd 2429 . . . . . . . . . 10  |-  ( v  =  z  ->  x  =  x )
43 id 22 . . . . . . . . . 10  |-  ( v  =  z  ->  v  =  z )
4441, 42, 43s3eqd 12905 . . . . . . . . 9  |-  ( v  =  z  ->  <" C x v ">  =  <" C x z "> )
4544eleq1d 2490 . . . . . . . 8  |-  ( v  =  z  ->  ( <" C x v ">  e.  (∟G `  G )  <->  <" C x z ">  e.  (∟G `  G )
) )
4640, 45rspc2va 3135 . . . . . . 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 1263 . . . . . 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 2698 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  -.  C  e.  A
)  ->  z  =/=  C )
4918, 21, 48syl2anc 665 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  =/=  C )
5049necomd 2656 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  =/=  z )
5150adantr 466 . . . . . . . 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 24620 . . . . . . 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 466 . . . . . . 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 764 . . . . . . . 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 24617 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( C L z )  e.  ran  L
)
561, 3, 4, 13, 29, 19, 50tglinerflx2 24621 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  ( C L z ) )
5756, 18elind 3593 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  ( ( C L z )  i^i  A ) )
581, 2, 3, 4, 13, 55, 14, 57isperp2 24702 . . . . . . . . 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 466 . . . . . . . 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 213 . . . . . . 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 2429 . . . . . . . . . 10  |-  ( u  =  C  ->  z  =  z )
6236, 61, 38s3eqd 12905 . . . . . . . . 9  |-  ( u  =  C  ->  <" u
z v ">  =  <" C z v "> )
6362eleq1d 2490 . . . . . . . 8  |-  ( u  =  C  ->  ( <" u z v ">  e.  (∟G `  G )  <->  <" C
z v ">  e.  (∟G `  G )
) )
64 eqidd 2429 . . . . . . . . . 10  |-  ( v  =  x  ->  C  =  C )
65 eqidd 2429 . . . . . . . . . 10  |-  ( v  =  x  ->  z  =  z )
66 id 22 . . . . . . . . . 10  |-  ( v  =  x  ->  v  =  x )
6764, 65, 66s3eqd 12905 . . . . . . . . 9  |-  ( v  =  x  ->  <" C
z v ">  =  <" C z x "> )
6867eleq1d 2490 . . . . . . . 8  |-  ( v  =  x  ->  ( <" C z v ">  e.  (∟G `  G )  <->  <" C
z x ">  e.  (∟G `  G )
) )
6963, 68rspc2va 3135 . . . . . . 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 1263 . . . . . 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 24691 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  x  =  z )
7271ex 435 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A )  ->  x  =  z ) )
7372ralrimivva 2786 . . 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 6257 . . . . 5  |-  ( x  =  z  ->  ( C L x )  =  ( C L z ) )
7574breq1d 4376 . . . 4  |-  ( x  =  z  ->  (
( C L x ) (⟂G `  G
) A  <->  ( C L z ) (⟂G `  G ) A ) )
7675rmo4 3206 . . 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 215 . 2  |-  ( ph  ->  E* x  e.  A  ( C L x ) (⟂G `  G ) A )
78 reu5 2985 . 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 668 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   E!wreu 2716   E*wrmo 2717   class class class wbr 4366   ran crn 4797   ` cfv 5544  (class class class)co 6249   <"cs3 12884   Basecbs 15064   distcds 15142  TarskiGcstrkg 24420  Itvcitv 24426  LineGclng 24427  pInvGcmir 24639  ∟Gcrag 24680  ⟂Gcperpg 24682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-fzo 11867  df-hash 12466  df-word 12612  df-concat 12614  df-s1 12615  df-s2 12890  df-s3 12891  df-trkgc 24438  df-trkgb 24439  df-trkgcb 24440  df-trkg 24443  df-cgrg 24498  df-leg 24570  df-mir 24640  df-rag 24681  df-perpg 24683
This theorem is referenced by:  footeq  24708  mideulem2  24718  lmieu  24768
  Copyright terms: Public domain W3C validator