Theorem ragperp 23130

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 2443	. . . 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 23005	. . . 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 3591	. . . . . . 7 |- ( A i^i B ) C_ A
15		ragperp.x	. . . . . . 7 |- ( ph -> X e. ( A i^i B ) )
16	14, 15	sseldi 3375	. . . . . 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 23005	. . . 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 23005	. . . 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 23005	. . . . 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 23005	. . . . . . 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	neneqad 2705	. . . . . . . . . . . 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 23064	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = u ) -> A = ( X L u ) )
41	31, 40	eleqtrd 2519	. . . . . . . . . 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 23115	. . . . . 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 23114	. . . . 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	neneqad 2705	. . . . . . . . . 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 3592	. . . . . . . . . . . 12 |- ( A i^i B ) C_ B
57	56, 15	sseldi 3375	. . . . . . . . . . 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 23064	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. A /\ v e. B ) ) /\ -. X = v ) -> B = ( X L v ) )
61	49, 60	eleqtrd 2519	. . . . . . . 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 23115	. . . 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 23114	. . 3 |- ( ( ph /\ ( u e. A /\ v e. B ) ) -> <" u X v "> e. (∟G ` G ) )
67	66	ralrimivva 2829	. 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 23128	. 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 1369   e. wcel 1756   =/= wne 2620   A.wral 2736   i^i cin 3348   class class class wbr 4313   ran crn 4862   ` cfv 5439  (class class class)co 6112   <"cs3 12490   Basecbs 14195   distcds 14268  TarskiGcstrkg 22911  Itvcitv 22919  LineGclng 22920  pInvGcmir 23077  ∟Gcrag 23109  ⟂Gcperpg 23111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-s2 12496  df-s3 12497  df-trkgc 22931  df-trkgb 22932  df-trkgcb 22933  df-trkg 22938  df-cgrg 22986  df-mir 23078  df-rag 23110  df-perpg 23112

This theorem is referenced by:  footex  23131