Theorem ragperp 24484

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 2404	. . . 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 760	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> v e. B )
11	1, 4, 3, 7, 9, 10	tglnpt 24321	. . . 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 3661	. . . . . . 7 |- ( A i^i B ) C_ A
15		ragperp.x	. . . . . . 7 |- ( ph -> X e. ( A i^i B ) )
16	14, 15	sseldi 3442	. . . . . 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 24321	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> X e. P )
19		simprl 758	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> u e. A )
20	1, 4, 3, 7, 13, 19	tglnpt 24321	. . . 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 24321	. . . . 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 24321	. . . . . . 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	24	ad2antrr 726	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> U e. A )
32	6	ad2antrr 726	. . . . . . . . . . . 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 2608	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> X =/= u )
37	12	ad2antrr 726	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A e. ran L )
38	16	ad2antrr 726	. . . . . . . . . . . 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 24405	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A = ( X L u ) )
41	31, 40	eleqtrd 2494	. . . . . . . . . 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 24466	. . . . . 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 24465	. . . . 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	21	ad2antrr 726	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> V e. B )
50	6	ad2antrr 726	. . . . . . . . . 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 2608	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> X =/= v )
55	8	ad2antrr 726	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B e. ran L )
56		inss2 3662	. . . . . . . . . . . 12 |- ( A i^i B ) C_ B
57	56, 15	sseldi 3442	. . . . . . . . . . 11 |- ( ph -> X e. B )
58	57	ad2antrr 726	. . . . . . . . . 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 24405	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B = ( X L v ) )
61	49, 60	eleqtrd 2494	. . . . . . . 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 24466	. . . 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 24465	. . 3 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" u X v "> e. (∟G ` G ) )
67	66	ralrimivva 2827	. 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 24482	. 2 |- ( ph -> ( A (⟂G ` G ) B <-> A. u e. A A. v e. B <" u X v "> e. (∟G ` G ) ) )
69	67, 68	mpbird 234	1 |- ( ph -> A (⟂G ` G ) B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1407   e. wcel 1844   =/= wne 2600   A.wral 2756   i^i cin 3415   class class class wbr 4397   ran crn 4826   ` cfv 5571  (class class class)co 6280   <"cs3 12865   Basecbs 14843   distcds 14920  TarskiGcstrkg 24208  Itvcitv 24214  LineGclng 24215  pInvGcmir 24422  ∟Gcrag 24460  ⟂Gcperpg 24462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-concat 12595  df-s1 12596  df-s2 12871  df-s3 12872  df-trkgc 24226  df-trkgb 24227  df-trkgcb 24228  df-trkg 24231  df-cgrg 24286  df-mir 24423  df-rag 24461  df-perpg 24463

This theorem is referenced by:  footex  24485  colperpexlem3  24496  mideulem2  24498  lmimid  24555  hypcgrlem1  24560  hypcgrlem2  24561  trgcopyeulem  24566