Theorem ragperp 25612

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
ragperp.b	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
ragperp.x	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
ragperp.u	⊢ (𝜑 → 𝑈 ∈ 𝐴)
ragperp.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
ragperp.1	⊢ (𝜑 → 𝑈 ≠ 𝑋)
ragperp.2	⊢ (𝜑 → 𝑉 ≠ 𝑋)
ragperp.r	⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))

Assertion

Ref	Expression
ragperp	⊢ (𝜑 → 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 2622	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		isperp.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ TarskiG)
8		ragperp.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐵 ∈ ran 𝐿)
10		simprr 796	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
11	1, 4, 3, 7, 9, 10	tglnpt 25444	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝑃)
12		isperp.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
13	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ ran 𝐿)
14		inss1 3833	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
15		ragperp.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
16	14, 15	sseldi 3601	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
17	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝐴)
18	1, 4, 3, 7, 13, 17	tglnpt 25444	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝑃)
19		simprl 794	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐴)
20	1, 4, 3, 7, 13, 19	tglnpt 25444	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝑃)
21		ragperp.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝐵)
23	1, 4, 3, 7, 9, 22	tglnpt 25444	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝑃)
24		ragperp.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
25	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝐴)
26	1, 4, 3, 7, 13, 25	tglnpt 25444	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝑃)
27		ragperp.r	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
28	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
29		ragperp.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≠ 𝑋)
30	29	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ≠ 𝑋)
31	24	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ 𝐴)
32	6	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐺 ∈ TarskiG)
33	18	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝑃)
34	20	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝑃)
35		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → ¬ 𝑋 = 𝑢)
36	35	neqned 2801	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ≠ 𝑢)
37	12	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 ∈ ran 𝐿)
38	16	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝐴)
39	19	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝐴)
40	1, 3, 4, 32, 33, 34, 36, 36, 37, 38, 39	tglinethru 25531	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 = (𝑋𝐿𝑢))
41	31, 40	eleqtrd 2703	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ (𝑋𝐿𝑢))
42	41	ex 450	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑢 → 𝑈 ∈ (𝑋𝐿𝑢)))
43	42	orrd 393	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑢 ∨ 𝑈 ∈ (𝑋𝐿𝑢)))
44	43	orcomd 403	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑈 ∈ (𝑋𝐿𝑢) ∨ 𝑋 = 𝑢))
45	1, 2, 3, 4, 5, 7, 26, 18, 23, 20, 28, 30, 44	ragcol 25594	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑉”⟩ ∈ (∟G‘𝐺))
46	1, 2, 3, 4, 5, 7, 20, 18, 23, 45	ragcom 25593	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑉𝑋𝑢”⟩ ∈ (∟G‘𝐺))
47		ragperp.2	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑋)
48	47	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ≠ 𝑋)
49	21	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ 𝐵)
50	6	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐺 ∈ TarskiG)
51	18	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝑃)
52	11	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝑃)
53		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → ¬ 𝑋 = 𝑣)
54	53	neqned 2801	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ≠ 𝑣)
55	8	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 ∈ ran 𝐿)
56		inss2 3834	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
57	56, 15	sseldi 3601	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
58	57	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝐵)
59	10	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝐵)
60	1, 3, 4, 50, 51, 52, 54, 54, 55, 58, 59	tglinethru 25531	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 = (𝑋𝐿𝑣))
61	49, 60	eleqtrd 2703	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ (𝑋𝐿𝑣))
62	61	ex 450	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑣 → 𝑉 ∈ (𝑋𝐿𝑣)))
63	62	orrd 393	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑣 ∨ 𝑉 ∈ (𝑋𝐿𝑣)))
64	63	orcomd 403	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑉 ∈ (𝑋𝐿𝑣) ∨ 𝑋 = 𝑣))
65	1, 2, 3, 4, 5, 7, 23, 18, 20, 11, 46, 48, 64	ragcol 25594	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑣𝑋𝑢”⟩ ∈ (∟G‘𝐺))
66	1, 2, 3, 4, 5, 7, 11, 18, 20, 65	ragcom 25593	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
67	66	ralrimivva 2971	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
68	1, 2, 3, 4, 6, 12, 8, 15	isperp2 25610	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))
69	67, 68	mpbird 247	1 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912   ∩ cin 3573   class class class wbr 4653  ran crn 5115  ‘cfv 5888  (class class class)co 6650  ⟨“cs3 13587  Basecbs 15857  distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂Gcperpg 25590

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-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-trkg 25352  df-cgrg 25406  df-mir 25548  df-rag 25589  df-perpg 25591

This theorem is referenced by:  footex  25613  colperpexlem3  25624  mideulem2  25626  lmimid  25686  hypcgrlem1  25691  hypcgrlem2  25692  trgcopyeulem  25697