Theorem lnperpex 25695

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.

Step	Hyp	Ref	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 )
6	5	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> G e. TarskiG )
7	6	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> G e. TarskiG )
8	7	adantr 481	. . . . . 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 794	. . . . . . 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 )
12	1, 4, 3, 5, 10, 11	tglnpt 25444	. . . . . . . . 9 |- ( ph -> A e. P )
13	12	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> A e. P )
14	13	ad3antrrr 766	. . . . . . 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 804	. . . . . . . . . 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 )
16	4, 8, 15	perpln1 25605	. . . . . . . . 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 )
17	1, 3, 4, 8, 14, 9, 16	tglnne 25523	. . . . . . . 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 )
18	17	necomd 2849	. . . . . . 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 )
19	1, 3, 4, 8, 9, 14, 18	tgelrnln 25525	. . . . . 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 )
20	10	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> D e. ran L )
21	20	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> D e. ran L )
22	21	adantr 481	. . . . . 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 )
23	1, 3, 4, 8, 9, 14, 18	tglinecom 25530	. . . . . . 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 ) )
24	23, 15	eqbrtrd 4675	. . . . . 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 )
25	1, 2, 3, 4, 8, 19, 22, 24	perpcom 25608	. . . . 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 477	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> Q O c )
27	26	adantr 481	. . . . . 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 )
30	29	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> Q e. P )
31	30	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> Q e. P )
32	31	adantr 481	. . . . . . 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 792	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> c e. P )
34	33	adantr 481	. . . . . . 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 805	. . . . . . . 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 )
36	1, 2, 3, 28, 4, 22, 8, 34, 9, 35	oppcom 25636	. . . . . . 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 )
37	1, 3, 4, 28, 8, 22, 9, 32, 34, 36	lnopp2hpgb 25655	. . . . . 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 ) )
38	27, 37	mpbid 222	. . . . 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 )
39	25, 38	jca 554	. . . 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 2622	. . . . 5 |- (hlG ` G ) = (hlG ` G )
41	11	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> A e. D )
42	41	ad2antrr 762	. . . . 5 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> A e. D )
43	1, 2, 3, 28, 4, 21, 7, 31, 33, 26	oppne2 25634	. . . . 5 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> -. c e. D )
44		lmiopp.h	. . . . . . 7 |- ( ph -> GDimTarskiG≥ 2 )
45	44	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> GDimTarskiG≥ 2 )
46	45	ad2antrr 762	. . . . 5 |- ( ( ( ( ( ph /\ d e. D ) /\ A =/= d ) /\ c e. P ) /\ Q O c ) -> GDimTarskiG≥ 2 )
47	1, 2, 3, 28, 4, 21, 7, 40, 42, 33, 43, 46	oppperpex 25645	. . . 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 ) )
48	39, 47	reximddv 3018	. . 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 )
50	1, 3, 4, 5, 10, 29, 28, 49	hpgerlem 25657	. . . 4 |- ( ph -> E. c e. P Q O c )
51	50	ad2antrr 762	. . 3 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> E. c e. P Q O c )
52	48, 51	r19.29a 3078	. 2 |- ( ( ( ph /\ d e. D ) /\ A =/= d ) -> E. p e. P ( D (⟂G ` G ) ( p L A ) /\ p ( (hpG ` G ) ` D ) Q ) )
53	1, 3, 4, 5, 10, 11	tglnpt2 25536	. 2 |- ( ph -> E. d e. D A =/= d )
54	52, 53	r19.29a 3078	1 |- ( ph -> E. p e. P ( D (⟂G ` G ) ( p L A ) /\ p ( (hpG ` G ) ` D ) Q ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   2c2 11070   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  hlGchlg 25495  ⟂Gcperpg 25590  hpGchpg 25649

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-fal 1489  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-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591  df-hpg 25650

This theorem is referenced by:  trgcopy  25696