MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foot Structured version   Visualization version   GIF version

Theorem foot 25509
Description: From a point 𝐶 outside of a line 𝐴, there exists a unique point 𝑥 on 𝐴 such that (𝐶𝐿𝑥) is perpendicular to 𝐴. That point is called the foot from 𝐶 on 𝐴. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-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 (𝜑𝐺 ∈ TarskiG)
isperp.a (𝜑𝐴 ∈ ran 𝐿)
foot.x (𝜑𝐶𝑃)
foot.y (𝜑 → ¬ 𝐶𝐴)
Assertion
Ref Expression
foot (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝜑,𝑥   𝑥,𝐶   𝑥,𝐼   𝑥,   𝑥,𝐿   𝑥,𝑃

Proof of Theorem foot
Dummy variables 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isperp.p . . 3 𝑃 = (Base‘𝐺)
2 isperp.d . . 3 = (dist‘𝐺)
3 isperp.i . . 3 𝐼 = (Itv‘𝐺)
4 isperp.l . . 3 𝐿 = (LineG‘𝐺)
5 isperp.g . . 3 (𝜑𝐺 ∈ TarskiG)
6 isperp.a . . 3 (𝜑𝐴 ∈ ran 𝐿)
7 foot.x . . 3 (𝜑𝐶𝑃)
8 foot.y . . 3 (𝜑 → ¬ 𝐶𝐴)
91, 2, 3, 4, 5, 6, 7, 8footex 25508 . 2 (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10 eqid 2626 . . . . . 6 (pInvG‘𝐺) = (pInvG‘𝐺)
115ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐺 ∈ TarskiG)
127ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑃)
135adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐺 ∈ TarskiG)
146adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐴 ∈ ran 𝐿)
15 simprl 793 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐴)
161, 4, 3, 13, 14, 15tglnpt 25339 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝑃)
1716adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝑃)
18 simprr 795 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐴)
191, 4, 3, 13, 14, 18tglnpt 25339 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝑃)
2019adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝑃)
218adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ¬ 𝐶𝐴)
22 nelne2 2893 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ¬ 𝐶𝐴) → 𝑥𝐶)
2315, 21, 22syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐶)
2423necomd 2851 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑥)
2524adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑥)
261, 3, 4, 11, 12, 17, 25tglinerflx1 25423 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
2718adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝐴)
28 simprl 793 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
297adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑃)
301, 3, 4, 13, 29, 16, 24tgelrnln 25420 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
311, 3, 4, 13, 29, 16, 24tglinerflx2 25424 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
3231, 15elind 3781 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
331, 2, 3, 4, 13, 30, 14, 32isperp2 25505 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3433adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3528, 34mpbid 222 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
36 id 22 . . . . . . . . . 10 (𝑢 = 𝐶𝑢 = 𝐶)
37 eqidd 2627 . . . . . . . . . 10 (𝑢 = 𝐶𝑥 = 𝑥)
38 eqidd 2627 . . . . . . . . . 10 (𝑢 = 𝐶𝑣 = 𝑣)
3936, 37, 38s3eqd 13541 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
4039eleq1d 2688 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41 eqidd 2627 . . . . . . . . . 10 (𝑣 = 𝑧𝐶 = 𝐶)
42 eqidd 2627 . . . . . . . . . 10 (𝑣 = 𝑧𝑥 = 𝑥)
43 id 22 . . . . . . . . . 10 (𝑣 = 𝑧𝑣 = 𝑧)
4441, 42, 43s3eqd 13541 . . . . . . . . 9 (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
4544eleq1d 2688 . . . . . . . 8 (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
4640, 45rspc2va 3312 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
4726, 27, 35, 46syl21anc 1322 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48 nelne2 2893 . . . . . . . . . . 11 ((𝑧𝐴 ∧ ¬ 𝐶𝐴) → 𝑧𝐶)
4918, 21, 48syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐶)
5049necomd 2851 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑧)
5150adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑧)
521, 3, 4, 11, 12, 20, 51tglinerflx1 25423 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
5315adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝐴)
54 simprr 795 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)
551, 3, 4, 13, 29, 19, 50tgelrnln 25420 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
561, 3, 4, 13, 29, 19, 50tglinerflx2 25424 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
5756, 18elind 3781 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
581, 2, 3, 4, 13, 55, 14, 57isperp2 25505 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
5958adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
6054, 59mpbid 222 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61 eqidd 2627 . . . . . . . . . 10 (𝑢 = 𝐶𝑧 = 𝑧)
6236, 61, 38s3eqd 13541 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
6362eleq1d 2688 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64 eqidd 2627 . . . . . . . . . 10 (𝑣 = 𝑥𝐶 = 𝐶)
65 eqidd 2627 . . . . . . . . . 10 (𝑣 = 𝑥𝑧 = 𝑧)
66 id 22 . . . . . . . . . 10 (𝑣 = 𝑥𝑣 = 𝑥)
6764, 65, 66s3eqd 13541 . . . . . . . . 9 (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
6867eleq1d 2688 . . . . . . . 8 (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
6963, 68rspc2va 3312 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
7052, 53, 60, 69syl21anc 1322 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
711, 2, 3, 4, 10, 11, 12, 17, 20, 47, 70ragflat 25494 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
7271ex 450 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7372ralrimivva 2970 . . 3 (𝜑 → ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74 oveq2 6613 . . . . 5 (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
7574breq1d 4628 . . . 4 (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
7675rmo4 3386 . . 3 (∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7773, 76sylibr 224 . 2 (𝜑 → ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
78 reu5 3153 . 2 (∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
799, 77, 78sylanbrc 697 1 (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  ∃!wreu 2914  ∃*wrmo 2915   class class class wbr 4618  ran crn 5080  cfv 5850  (class class class)co 6605  ⟨“cs3 13519  Basecbs 15776  distcds 15866  TarskiGcstrkg 25224  Itvcitv 25230  LineGclng 25231  pInvGcmir 25442  ∟Gcrag 25483  ⟂Gcperpg 25485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-concat 13235  df-s1 13236  df-s2 13525  df-s3 13526  df-trkgc 25242  df-trkgb 25243  df-trkgcb 25244  df-trkg 25247  df-cgrg 25301  df-leg 25373  df-mir 25443  df-rag 25484  df-perpg 25486
This theorem is referenced by:  footeq  25511  mideulem2  25521  lmieu  25571
  Copyright terms: Public domain W3C validator