Theorem ragperp 24841

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 2471	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
6		isperp.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	adantr 472	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> G e. TarskiG )
8		ragperp.b	. . . . . 6 |- ( ph -> B e. ran L )
9	8	adantr 472	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> B e. ran L )
10		simprr 774	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> v e. B )
11	1, 4, 3, 7, 9, 10	tglnpt 24673	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> v e. P )
12		isperp.a	. . . . . 6 |- ( ph -> A e. ran L )
13	12	adantr 472	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> A e. ran L )
14		inss1 3643	. . . . . . 7 |- ( A i^i B ) C_ A
15		ragperp.x	. . . . . . 7 |- ( ph -> X e. ( A i^i B ) )
16	14, 15	sseldi 3416	. . . . . 6 |- ( ph -> X e. A )
17	16	adantr 472	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> X e. A )
18	1, 4, 3, 7, 13, 17	tglnpt 24673	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> X e. P )
19		simprl 772	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> u e. A )
20	1, 4, 3, 7, 13, 19	tglnpt 24673	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> u e. P )
21		ragperp.v	. . . . . . 7 |- ( ph -> V e. B )
22	21	adantr 472	. . . . . 6 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> V e. B )
23	1, 4, 3, 7, 9, 22	tglnpt 24673	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> V e. P )
24		ragperp.u	. . . . . . . . 9 |- ( ph -> U e. A )
25	24	adantr 472	. . . . . . . 8 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> U e. A )
26	1, 4, 3, 7, 13, 25	tglnpt 24673	. . . . . . 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 472	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" U X V "> e. (∟G ` G ) )
29		ragperp.1	. . . . . . . 8 |- ( ph -> U =/= X )
30	29	adantr 472	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> U =/= X )
31	24	ad2antrr 740	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> U e. A )
32	6	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> G e. TarskiG )
33	18	adantr 472	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X e. P )
34	20	adantr 472	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> u e. P )
35		simpr 468	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> -. X = u )
36	35	neqned 2650	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X =/= u )
37	12	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A e. ran L )
38	16	ad2antrr 740	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X e. A )
39	19	adantr 472	. . . . . . . . . . . 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 24760	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A = ( X L u ) )
41	31, 40	eleqtrd 2551	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> U e. ( X L u ) )
42	41	ex 441	. . . . . . . . 9 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( -. X = u -> U e. ( X L u ) ) )
43	42	orrd 385	. . . . . . . 8 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( X = u \/ U e. ( X L u ) ) )
44	43	orcomd 395	. . . . . . 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 24823	. . . . . 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 24822	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" V X u "> e. (∟G ` G ) )
47		ragperp.2	. . . . . 6 |- ( ph -> V =/= X )
48	47	adantr 472	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> V =/= X )
49	21	ad2antrr 740	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> V e. B )
50	6	ad2antrr 740	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> G e. TarskiG )
51	18	adantr 472	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X e. P )
52	11	adantr 472	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> v e. P )
53		simpr 468	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> -. X = v )
54	53	neqned 2650	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X =/= v )
55	8	ad2antrr 740	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B e. ran L )
56		inss2 3644	. . . . . . . . . . . 12 |- ( A i^i B ) C_ B
57	56, 15	sseldi 3416	. . . . . . . . . . 11 |- ( ph -> X e. B )
58	57	ad2antrr 740	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X e. B )
59	10	adantr 472	. . . . . . . . . 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 24760	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B = ( X L v ) )
61	49, 60	eleqtrd 2551	. . . . . . . 8 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> V e. ( X L v ) )
62	61	ex 441	. . . . . . 7 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( -. X = v -> V e. ( X L v ) ) )
63	62	orrd 385	. . . . . 6 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> ( X = v \/ V e. ( X L v ) ) )
64	63	orcomd 395	. . . . 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 24823	. . . 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 24822	. . 3 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" u X v "> e. (∟G ` G ) )
67	66	ralrimivva 2814	. 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 24839	. 2 |- ( ph -> ( A (⟂G ` G ) B <-> A. u e. A A. v e. B <" u X v "> e. (∟G ` G ) ) )
69	67, 68	mpbird 240	1 |- ( ph -> A (⟂G ` G ) B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   i^i cin 3389   class class class wbr 4395   ran crn 4840   ` cfv 5589  (class class class)co 6308   <"cs3 12997   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564  pInvGcmir 24776  ∟Gcrag 24817  ⟂Gcperpg 24819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580  df-cgrg 24635  df-mir 24777  df-rag 24818  df-perpg 24820

This theorem is referenced by:  footex  24842  colperpexlem3  24853  mideulem2  24855  lmimid  24915  hypcgrlem1  24920  hypcgrlem2  24921  trgcopyeulem  24926