Theorem ragperp 23830

Description: Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-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 )
ragperp.b	|- ( ph -> B e. ran L )
ragperp.x	|- ( ph -> X e. ( A i^i B ) )
ragperp.u	|- ( ph -> U e. A )
ragperp.v	|- ( ph -> V e. B )
ragperp.1	|- ( ph -> U =/= X )
ragperp.2	|- ( ph -> V =/= X )
ragperp.r	|- ( ph -> <" U X V "> e. (∟G ` G ) )

Assertion

Ref	Expression
ragperp	|- ( ph -> A (⟂G ` G ) B )

Proof of Theorem ragperp

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isperp.p	. . . 4 |- P = ( Base ` G )
2		isperp.d	. . . 4 |- .- = ( dist ` G )
3		isperp.i	. . . 4 |- I = (Itv ` G )
4		isperp.l	. . . 4 |- L = (LineG ` G )
5		eqid 2467	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
6		isperp.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	adantr 465	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> G e. TarskiG )
8		ragperp.b	. . . . . 6 |- ( ph -> B e. ran L )
9	8	adantr 465	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> B e. ran L )
10		simprr 756	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> v e. B )
11	1, 4, 3, 7, 9, 10	tglnpt 23692	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> v e. P )
12		isperp.a	. . . . . 6 |- ( ph -> A e. ran L )
13	12	adantr 465	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> A e. ran L )
14		inss1 3718	. . . . . . 7 |- ( A i^i B ) C_ A
15		ragperp.x	. . . . . . 7 |- ( ph -> X e. ( A i^i B ) )
16	14, 15	sseldi 3502	. . . . . 6 |- ( ph -> X e. A )
17	16	adantr 465	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> X e. A )
18	1, 4, 3, 7, 13, 17	tglnpt 23692	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> X e. P )
19		simprl 755	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> u e. A )
20	1, 4, 3, 7, 13, 19	tglnpt 23692	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> u e. P )
21		ragperp.v	. . . . . . 7 |- ( ph -> V e. B )
22	21	adantr 465	. . . . . 6 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> V e. B )
23	1, 4, 3, 7, 9, 22	tglnpt 23692	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> V e. P )
24		ragperp.u	. . . . . . . . 9 |- ( ph -> U e. A )
25	24	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> U e. A )
26	1, 4, 3, 7, 13, 25	tglnpt 23692	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> U e. P )
27		ragperp.r	. . . . . . . 8 |- ( ph -> <" U X V "> e. (∟G ` G ) )
28	27	adantr 465	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" U X V "> e. (∟G ` G ) )
29		ragperp.1	. . . . . . . 8 |- ( ph -> U =/= X )
30	29	adantr 465	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> U =/= X )
31	25	adantr 465	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> U e. A )
32	7	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> G e. TarskiG )
33	18	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X e. P )
34	20	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> u e. P )
35		simpr 461	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> -. X = u )
36	35	neqned 2670	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X =/= u )
37	13	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A e. ran L )
38	17	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X e. A )
39	19	adantr 465	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> u e. A )
40	1, 3, 4, 32, 33, 34, 36, 36, 37, 38, 39	tglinethru 23758	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A = ( X L u ) )
41	31, 40	eleqtrd 2557	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> U e. ( X L u ) )
42	41	ex 434	. . . . . . . . 9 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( -. X = u -> U e. ( X L u ) ) )
43	42	orrd 378	. . . . . . . 8 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( X = u \/ U e. ( X L u ) ) )
44	43	orcomd 388	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( U e. ( X L u ) \/ X = u ) )
45	1, 2, 3, 4, 5, 7, 26, 18, 23, 20, 28, 30, 44	ragcol 23812	. . . . . 6 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" u X V "> e. (∟G ` G ) )
46	1, 2, 3, 4, 5, 7, 20, 18, 23, 45	ragcom 23811	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" V X u "> e. (∟G ` G ) )
47		ragperp.2	. . . . . 6 |- ( ph -> V =/= X )
48	47	adantr 465	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> V =/= X )
49	22	adantr 465	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> V e. B )
50	7	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> G e. TarskiG )
51	18	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X e. P )
52	11	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> v e. P )
53		simpr 461	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> -. X = v )
54	53	neqned 2670	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X =/= v )
55	8	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B e. ran L )
56		inss2 3719	. . . . . . . . . . . 12 |- ( A i^i B ) C_ B
57	56, 15	sseldi 3502	. . . . . . . . . . 11 |- ( ph -> X e. B )
58	57	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X e. B )
59	10	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> v e. B )
60	1, 3, 4, 50, 51, 52, 54, 54, 55, 58, 59	tglinethru 23758	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B = ( X L v ) )
61	49, 60	eleqtrd 2557	. . . . . . . 8 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> V e. ( X L v ) )
62	61	ex 434	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( -. X = v -> V e. ( X L v ) ) )
63	62	orrd 378	. . . . . 6 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( X = v \/ V e. ( X L v ) ) )
64	63	orcomd 388	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( V e. ( X L v ) \/ X = v ) )
65	1, 2, 3, 4, 5, 7, 23, 18, 20, 11, 46, 48, 64	ragcol 23812	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" v X u "> e. (∟G ` G ) )
66	1, 2, 3, 4, 5, 7, 11, 18, 20, 65	ragcom 23811	. . 3 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" u X v "> e. (∟G ` G ) )
67	66	ralrimivva 2885	. 2 |- ( ph -> A. u e. A A. v e. B <" u X v "> e. (∟G ` G ) )
68	1, 2, 3, 4, 6, 12, 8, 15	isperp2 23828	. 2 |- ( ph -> ( A (⟂G ` G ) B <-> A. u e. A A. v e. B <" u X v "> e. (∟G ` G ) ) )
69	67, 68	mpbird 232	1 |- ( ph -> A (⟂G ` G ) B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   i^i cin 3475   class class class wbr 4447   ran crn 5000   ` cfv 5588  (class class class)co 6284   <"cs3 12770   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589  pInvGcmir 23774  ∟Gcrag 23806  ⟂Gcperpg 23808

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 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569

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 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-s2 12776  df-s3 12777  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606  df-cgrg 23659  df-mir 23775  df-rag 23807  df-perpg 23809

This theorem is referenced by:  footex  23831  colperpexlem3  23839  mideulem  23841  lmimid  23864  hypcgrlem1  23869  hypcgrlem2  23870