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

Theorem foot 28745
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 28744 . 2 (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10 eqid 2735 . . . . . 6 (pInvG‘𝐺) = (pInvG‘𝐺)
115ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐺 ∈ TarskiG)
127ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑃)
135adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐺 ∈ TarskiG)
146adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐴 ∈ ran 𝐿)
15 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐴)
161, 4, 3, 13, 14, 15tglnpt 28572 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝑃)
1716adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝑃)
18 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐴)
191, 4, 3, 13, 14, 18tglnpt 28572 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝑃)
2019adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝑃)
218adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ¬ 𝐶𝐴)
22 nelne2 3038 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ¬ 𝐶𝐴) → 𝑥𝐶)
2315, 21, 22syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐶)
2423necomd 2994 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑥)
2524adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑥)
261, 3, 4, 11, 12, 17, 25tglinerflx1 28656 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
2718adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝐴)
28 simprl 771 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
297adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑃)
301, 3, 4, 13, 29, 16, 24tgelrnln 28653 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
311, 3, 4, 13, 29, 16, 24tglinerflx2 28657 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
3231, 15elind 4210 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
331, 2, 3, 4, 13, 30, 14, 32isperp2 28738 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3433adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3528, 34mpbid 232 . . . . . . 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 2736 . . . . . . . . . 10 (𝑢 = 𝐶𝑥 = 𝑥)
38 eqidd 2736 . . . . . . . . . 10 (𝑢 = 𝐶𝑣 = 𝑣)
3936, 37, 38s3eqd 14900 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
4039eleq1d 2824 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41 eqidd 2736 . . . . . . . . . 10 (𝑣 = 𝑧𝐶 = 𝐶)
42 eqidd 2736 . . . . . . . . . 10 (𝑣 = 𝑧𝑥 = 𝑥)
43 id 22 . . . . . . . . . 10 (𝑣 = 𝑧𝑣 = 𝑧)
4441, 42, 43s3eqd 14900 . . . . . . . . 9 (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
4544eleq1d 2824 . . . . . . . 8 (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
4640, 45rspc2va 3634 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
4726, 27, 35, 46syl21anc 838 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48 nelne2 3038 . . . . . . . . . . 11 ((𝑧𝐴 ∧ ¬ 𝐶𝐴) → 𝑧𝐶)
4918, 21, 48syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐶)
5049necomd 2994 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑧)
5150adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑧)
521, 3, 4, 11, 12, 20, 51tglinerflx1 28656 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
5315adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝐴)
54 simprr 773 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)
551, 3, 4, 13, 29, 19, 50tgelrnln 28653 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
561, 3, 4, 13, 29, 19, 50tglinerflx2 28657 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
5756, 18elind 4210 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
581, 2, 3, 4, 13, 55, 14, 57isperp2 28738 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
5958adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
6054, 59mpbid 232 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61 eqidd 2736 . . . . . . . . . 10 (𝑢 = 𝐶𝑧 = 𝑧)
6236, 61, 38s3eqd 14900 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
6362eleq1d 2824 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64 eqidd 2736 . . . . . . . . . 10 (𝑣 = 𝑥𝐶 = 𝐶)
65 eqidd 2736 . . . . . . . . . 10 (𝑣 = 𝑥𝑧 = 𝑧)
66 id 22 . . . . . . . . . 10 (𝑣 = 𝑥𝑣 = 𝑥)
6764, 65, 66s3eqd 14900 . . . . . . . . 9 (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
6867eleq1d 2824 . . . . . . . 8 (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
6963, 68rspc2va 3634 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
7052, 53, 60, 69syl21anc 838 . . . . . 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 28727 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
7271ex 412 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7372ralrimivva 3200 . . 3 (𝜑 → ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74 oveq2 7439 . . . . 5 (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
7574breq1d 5158 . . . 4 (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
7675rmo4 3739 . . 3 (∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7773, 76sylibr 234 . 2 (𝜑 → ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
78 reu5 3380 . 2 (∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
799, 77, 78sylanbrc 583 1 (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377   class class class wbr 5148  ran crn 5690  cfv 6563  (class class class)co 7431  ⟨“cs3 14878  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  pInvGcmir 28675  ∟Gcrag 28716  ⟂Gcperpg 28718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476  df-cgrg 28534  df-leg 28606  df-mir 28676  df-rag 28717  df-perpg 28719
This theorem is referenced by:  footeq  28747  mideulem2  28757  lmieu  28807
  Copyright terms: Public domain W3C validator