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Theorem foot 23901
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 23900 . 2  |-  ( ph  ->  E. x  e.  A  ( C L x ) (⟂G `  G ) A )
10 eqid 2467 . . . . . 6  |-  (pInvG `  G )  =  (pInvG `  G )
115ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  G  e. TarskiG )
127ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  e.  P )
135adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  G  e. TarskiG )
146adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  A  e.  ran  L )
15 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  A )
161, 4, 3, 13, 14, 15tglnpt 23761 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  P )
1716adantr 465 . . . . . 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 756 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  A )
191, 4, 3, 13, 14, 18tglnpt 23761 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  P )
2019adantr 465 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  -> 
z  e.  P )
218adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  -.  C  e.  A
)
22 nelne2 2797 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  -.  C  e.  A
)  ->  x  =/=  C )
2315, 21, 22syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  =/=  C )
2423necomd 2738 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  =/=  x )
2524adantr 465 . . . . . . . 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 23824 . . . . . . 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 465 . . . . . . 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 755 . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  e.  P )
301, 3, 4, 13, 29, 16, 24tgelrnln 23821 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( C L x )  e.  ran  L
)
311, 3, 4, 13, 29, 16, 24tglinerflx2 23825 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  ( C L x ) )
3231, 15elind 3688 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  x  e.  ( ( C L x )  i^i 
A ) )
331, 2, 3, 4, 13, 30, 14, 32isperp2 23897 . . . . . . . . 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 465 . . . . . . . 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 . . . . . . . . . . 11  |-  ( u  =  C  ->  u  =  C )
37 eqidd 2468 . . . . . . . . . . 11  |-  ( u  =  C  ->  x  =  x )
38 eqidd 2468 . . . . . . . . . . 11  |-  ( u  =  C  ->  v  =  v )
3936, 37, 38s3eqd 12794 . . . . . . . . . 10  |-  ( u  =  C  ->  <" u x v ">  =  <" C x v "> )
4039eleq1d 2536 . . . . . . . . 9  |-  ( u  =  C  ->  ( <" u x v ">  e.  (∟G `  G )  <->  <" C x v ">  e.  (∟G `  G )
) )
41 eqidd 2468 . . . . . . . . . . 11  |-  ( v  =  z  ->  C  =  C )
42 eqidd 2468 . . . . . . . . . . 11  |-  ( v  =  z  ->  x  =  x )
43 id 22 . . . . . . . . . . 11  |-  ( v  =  z  ->  v  =  z )
4441, 42, 43s3eqd 12794 . . . . . . . . . 10  |-  ( v  =  z  ->  <" C x v ">  =  <" C x z "> )
4544eleq1d 2536 . . . . . . . . 9  |-  ( v  =  z  ->  ( <" C x v ">  e.  (∟G `  G )  <->  <" C x z ">  e.  (∟G `  G )
) )
4640, 45rspc2v 3223 . . . . . . . 8  |-  ( ( 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 ) ) )
4746imp 429 . . . . . . 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 ) )
4826, 27, 35, 47syl21anc 1227 . . . . . 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 ) )
49 nelne2 2797 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  -.  C  e.  A
)  ->  z  =/=  C )
5018, 21, 49syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  =/=  C )
5150necomd 2738 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  ->  C  =/=  z )
5251adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  C  =/=  z )
531, 3, 4, 11, 12, 20, 52tglinerflx1 23824 . . . . . . 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 ) )
5415adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  x  e.  A )
55 simprr 756 . . . . . . . 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 )
561, 3, 4, 13, 29, 19, 51tgelrnln 23821 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( C L z )  e.  ran  L
)
571, 3, 4, 13, 29, 19, 51tglinerflx2 23825 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  ( C L z ) )
5857, 18elind 3688 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
z  e.  ( ( C L z )  i^i  A ) )
591, 2, 3, 4, 13, 56, 14, 58isperp2 23897 . . . . . . . . 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 ) ) )
6059adantr 465 . . . . . . . 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 ) ) )
6155, 60mpbid 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
) )
62 eqidd 2468 . . . . . . . . . . 11  |-  ( u  =  C  ->  z  =  z )
6336, 62, 38s3eqd 12794 . . . . . . . . . 10  |-  ( u  =  C  ->  <" u
z v ">  =  <" C z v "> )
6463eleq1d 2536 . . . . . . . . 9  |-  ( u  =  C  ->  ( <" u z v ">  e.  (∟G `  G )  <->  <" C
z v ">  e.  (∟G `  G )
) )
65 eqidd 2468 . . . . . . . . . . 11  |-  ( v  =  x  ->  C  =  C )
66 eqidd 2468 . . . . . . . . . . 11  |-  ( v  =  x  ->  z  =  z )
67 id 22 . . . . . . . . . . 11  |-  ( v  =  x  ->  v  =  x )
6865, 66, 67s3eqd 12794 . . . . . . . . . 10  |-  ( v  =  x  ->  <" C
z v ">  =  <" C z x "> )
6968eleq1d 2536 . . . . . . . . 9  |-  ( v  =  x  ->  ( <" C z v ">  e.  (∟G `  G )  <->  <" C
z x ">  e.  (∟G `  G )
) )
7064, 69rspc2v 3223 . . . . . . . 8  |-  ( ( 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 ) ) )
7170imp 429 . . . . . . 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 ) )
7253, 54, 61, 71syl21anc 1227 . . . . . 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 ) )
731, 2, 3, 4, 10, 11, 12, 17, 20, 48, 72ragflat 23886 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  z  e.  A )
)  /\  ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A ) )  ->  x  =  z )
7473ex 434 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  z  e.  A ) )  -> 
( ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A )  ->  x  =  z ) )
7574ralrimivva 2885 . . 3  |-  ( ph  ->  A. x  e.  A  A. z  e.  A  ( ( ( C L x ) (⟂G `  G ) A  /\  ( C L z ) (⟂G `  G ) A )  ->  x  =  z ) )
76 oveq2 6293 . . . . 5  |-  ( x  =  z  ->  ( C L x )  =  ( C L z ) )
7776breq1d 4457 . . . 4  |-  ( x  =  z  ->  (
( C L x ) (⟂G `  G
) A  <->  ( C L z ) (⟂G `  G ) A ) )
7877rmo4 3296 . . 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 ) )
7975, 78sylibr 212 . 2  |-  ( ph  ->  E* x  e.  A  ( C L x ) (⟂G `  G ) A )
80 reu5 3077 . 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 ) )
819, 79, 80sylanbrc 664 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   E*wrmo 2817   class class class wbr 4447   ran crn 5000   ` cfv 5588  (class class class)co 6285   <"cs3 12773   Basecbs 14493   distcds 14567  TarskiGcstrkg 23650  Itvcitv 23657  LineGclng 23658  pInvGcmir 23843  ∟Gcrag 23875  ⟂Gcperpg 23877
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-concat 12511  df-s1 12512  df-s2 12779  df-s3 12780  df-trkgc 23669  df-trkgb 23670  df-trkgcb 23671  df-trkg 23675  df-cgrg 23728  df-leg 23794  df-mir 23844  df-rag 23876  df-perpg 23878
This theorem is referenced by:  mideulem  23910  lmieu  23924
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