Theorem foot 25614

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.

Step	Hyp	Ref	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 )
9	1, 2, 3, 4, 5, 6, 7, 8	footex 25613	. 2 |- ( ph -> E. x e. A ( C L x ) (⟂G ` G ) A )
10		eqid 2622	. . . . . 6 |- (pInvG ` G ) = (pInvG ` G )
11	5	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ z e. A ) ) /\ ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) ) -> G e. TarskiG )
12	7	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ z e. A ) ) /\ ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) ) -> C e. P )
13	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> G e. TarskiG )
14	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> A e. ran L )
15		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> x e. A )
16	1, 4, 3, 13, 14, 15	tglnpt 25444	. . . . . . 7 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> x e. P )
17	16	adantr 481	. . . . . 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 796	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> z e. A )
19	1, 4, 3, 13, 14, 18	tglnpt 25444	. . . . . . 7 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> z e. P )
20	19	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ z e. A ) ) /\ ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) ) -> z e. P )
21	8	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> -. C e. A )
22		nelne2 2891	. . . . . . . . . . 11 |- ( ( x e. A /\ -. C e. A ) -> x =/= C )
23	15, 21, 22	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> x =/= C )
24	23	necomd 2849	. . . . . . . . 9 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> C =/= x )
25	24	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ z e. A ) ) /\ ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) ) -> C =/= x )
26	1, 3, 4, 11, 12, 17, 25	tglinerflx1 25528	. . . . . . 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 ) )
27	18	adantr 481	. . . . . . 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 794	. . . . . . . 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 )
29	7	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> C e. P )
30	1, 3, 4, 13, 29, 16, 24	tgelrnln 25525	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> ( C L x ) e. ran L )
31	1, 3, 4, 13, 29, 16, 24	tglinerflx2 25529	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> x e. ( C L x ) )
32	31, 15	elind 3798	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> x e. ( ( C L x ) i^i A ) )
33	1, 2, 3, 4, 13, 30, 14, 32	isperp2 25610	. . . . . . . . 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 ) ) )
34	33	adantr 481	. . . . . . . 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 ) ) )
35	28, 34	mpbid 222	. . . . . . 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 2623	. . . . . . . . . 10 |- ( u = C -> x = x )
38		eqidd 2623	. . . . . . . . . 10 |- ( u = C -> v = v )
39	36, 37, 38	s3eqd 13609	. . . . . . . . 9 |- ( u = C -> <" u x v "> = <" C x v "> )
40	39	eleq1d 2686	. . . . . . . 8 |- ( u = C -> ( <" u x v "> e. (∟G ` G ) <-> <" C x v "> e. (∟G ` G ) ) )
41		eqidd 2623	. . . . . . . . . 10 |- ( v = z -> C = C )
42		eqidd 2623	. . . . . . . . . 10 |- ( v = z -> x = x )
43		id 22	. . . . . . . . . 10 |- ( v = z -> v = z )
44	41, 42, 43	s3eqd 13609	. . . . . . . . 9 |- ( v = z -> <" C x v "> = <" C x z "> )
45	44	eleq1d 2686	. . . . . . . 8 |- ( v = z -> ( <" C x v "> e. (∟G ` G ) <-> <" C x z "> e. (∟G ` G ) ) )
46	40, 45	rspc2va 3323	. . . . . . 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 ) )
47	26, 27, 35, 46	syl21anc 1325	. . . . . 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 2891	. . . . . . . . . . 11 |- ( ( z e. A /\ -. C e. A ) -> z =/= C )
49	18, 21, 48	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> z =/= C )
50	49	necomd 2849	. . . . . . . . 9 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> C =/= z )
51	50	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ z e. A ) ) /\ ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) ) -> C =/= z )
52	1, 3, 4, 11, 12, 20, 51	tglinerflx1 25528	. . . . . . 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 ) )
53	15	adantr 481	. . . . . . 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 796	. . . . . . . 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 )
55	1, 3, 4, 13, 29, 19, 50	tgelrnln 25525	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> ( C L z ) e. ran L )
56	1, 3, 4, 13, 29, 19, 50	tglinerflx2 25529	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> z e. ( C L z ) )
57	56, 18	elind 3798	. . . . . . . . . 10 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> z e. ( ( C L z ) i^i A ) )
58	1, 2, 3, 4, 13, 55, 14, 57	isperp2 25610	. . . . . . . . 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 ) ) )
59	58	adantr 481	. . . . . . . 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 ) ) )
60	54, 59	mpbid 222	. . . . . . 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 2623	. . . . . . . . . 10 |- ( u = C -> z = z )
62	36, 61, 38	s3eqd 13609	. . . . . . . . 9 |- ( u = C -> <" u z v "> = <" C z v "> )
63	62	eleq1d 2686	. . . . . . . 8 |- ( u = C -> ( <" u z v "> e. (∟G ` G ) <-> <" C z v "> e. (∟G ` G ) ) )
64		eqidd 2623	. . . . . . . . . 10 |- ( v = x -> C = C )
65		eqidd 2623	. . . . . . . . . 10 |- ( v = x -> z = z )
66		id 22	. . . . . . . . . 10 |- ( v = x -> v = x )
67	64, 65, 66	s3eqd 13609	. . . . . . . . 9 |- ( v = x -> <" C z v "> = <" C z x "> )
68	67	eleq1d 2686	. . . . . . . 8 |- ( v = x -> ( <" C z v "> e. (∟G ` G ) <-> <" C z x "> e. (∟G ` G ) ) )
69	63, 68	rspc2va 3323	. . . . . . 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 ) )
70	52, 53, 60, 69	syl21anc 1325	. . . . . 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 ) )
71	1, 2, 3, 4, 10, 11, 12, 17, 20, 47, 70	ragflat 25599	. . . . 5 |- ( ( ( ph /\ ( x e. A /\ z e. A ) ) /\ ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) ) -> x = z )
72	71	ex 450	. . . 4 |- ( ( ph /\ ( x e. A /\ z e. A ) ) -> ( ( ( C L x ) (⟂G ` G ) A /\ ( C L z ) (⟂G ` G ) A ) -> x = z ) )
73	72	ralrimivva 2971	. . 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 6658	. . . . 5 |- ( x = z -> ( C L x ) = ( C L z ) )
75	74	breq1d 4663	. . . 4 |- ( x = z -> ( ( C L x ) (⟂G ` G ) A <-> ( C L z ) (⟂G ` G ) A ) )
76	75	rmo4 3399	. . 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 ) )
77	73, 76	sylibr 224	. 2 |- ( ph -> E* x e. A ( C L x ) (⟂G ` G ) A )
78		reu5 3159	. 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 ) )
79	9, 77, 78	sylanbrc 698	1 |- ( ph -> E! x e. A ( C L x ) (⟂G ` G ) A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂Gcperpg 25590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  footeq  25616  mideulem2  25626  lmieu  25676