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Theorem foot 25841
 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 25840 . 2 (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10 eqid 2769 . . . . . 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 473 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐺 ∈ TarskiG)
146adantr 473 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐴 ∈ ran 𝐿)
15 simprl 808 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐴)
161, 4, 3, 13, 14, 15tglnpt 25671 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝑃)
1716adantr 473 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝑃)
18 simprr 810 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐴)
191, 4, 3, 13, 14, 18tglnpt 25671 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝑃)
2019adantr 473 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝑃)
218adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ¬ 𝐶𝐴)
22 nelne2 3038 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ¬ 𝐶𝐴) → 𝑥𝐶)
2315, 21, 22syl2anc 693 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐶)
2423necomd 2996 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑥)
2524adantr 473 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑥)
261, 3, 4, 11, 12, 17, 25tglinerflx1 25755 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
2718adantr 473 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝐴)
28 simprl 808 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
297adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑃)
301, 3, 4, 13, 29, 16, 24tgelrnln 25752 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
311, 3, 4, 13, 29, 16, 24tglinerflx2 25756 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
3231, 15elind 3946 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
331, 2, 3, 4, 13, 30, 14, 32isperp2 25837 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3433adantr 473 . . . . . . . 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 2770 . . . . . . . . . 10 (𝑢 = 𝐶𝑥 = 𝑥)
38 eqidd 2770 . . . . . . . . . 10 (𝑢 = 𝐶𝑣 = 𝑣)
3936, 37, 38s3eqd 13821 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
4039eleq1d 2833 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41 eqidd 2770 . . . . . . . . . 10 (𝑣 = 𝑧𝐶 = 𝐶)
42 eqidd 2770 . . . . . . . . . 10 (𝑣 = 𝑧𝑥 = 𝑥)
43 id 22 . . . . . . . . . 10 (𝑣 = 𝑧𝑣 = 𝑧)
4441, 42, 43s3eqd 13821 . . . . . . . . 9 (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
4544eleq1d 2833 . . . . . . . 8 (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
4640, 45rspc2va 3470 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
4726, 27, 35, 46syl21anc 1473 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48 nelne2 3038 . . . . . . . . . . 11 ((𝑧𝐴 ∧ ¬ 𝐶𝐴) → 𝑧𝐶)
4918, 21, 48syl2anc 693 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐶)
5049necomd 2996 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑧)
5150adantr 473 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑧)
521, 3, 4, 11, 12, 20, 51tglinerflx1 25755 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
5315adantr 473 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝐴)
54 simprr 810 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)
551, 3, 4, 13, 29, 19, 50tgelrnln 25752 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
561, 3, 4, 13, 29, 19, 50tglinerflx2 25756 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
5756, 18elind 3946 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
581, 2, 3, 4, 13, 55, 14, 57isperp2 25837 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
5958adantr 473 . . . . . . . 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 2770 . . . . . . . . . 10 (𝑢 = 𝐶𝑧 = 𝑧)
6236, 61, 38s3eqd 13821 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
6362eleq1d 2833 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64 eqidd 2770 . . . . . . . . . 10 (𝑣 = 𝑥𝐶 = 𝐶)
65 eqidd 2770 . . . . . . . . . 10 (𝑣 = 𝑥𝑧 = 𝑧)
66 id 22 . . . . . . . . . 10 (𝑣 = 𝑥𝑣 = 𝑥)
6764, 65, 66s3eqd 13821 . . . . . . . . 9 (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
6867eleq1d 2833 . . . . . . . 8 (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
6963, 68rspc2va 3470 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
7052, 53, 60, 69syl21anc 1473 . . . . . 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 25826 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
7271ex 448 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7372ralrimivva 3118 . . 3 (𝜑 → ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74 oveq2 6799 . . . . 5 (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
7574breq1d 4793 . . . 4 (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
7675rmo4 3548 . . 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 3306 . 2 (∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
799, 77, 78sylanbrc 698 1 (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1629   ∈ wcel 2143   ≠ wne 2941  ∀wral 3059  ∃wrex 3060  ∃!wreu 3061  ∃*wrmo 3062   class class class wbr 4783  ran crn 5249  ‘cfv 6030  (class class class)co 6791  ⟨“cs3 13799  Basecbs 16070  distcds 16164  TarskiGcstrkg 25556  Itvcitv 25562  LineGclng 25563  pInvGcmir 25774  ∟Gcrag 25815  ⟂Gcperpg 25817 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-cda 9190  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-3 11280  df-n0 11493  df-xnn0 11564  df-z 11578  df-uz 11888  df-fz 12533  df-fzo 12673  df-hash 13325  df-word 13498  df-concat 13500  df-s1 13501  df-s2 13805  df-s3 13806  df-trkgc 25574  df-trkgb 25575  df-trkgcb 25576  df-trkg 25579  df-cgrg 25633  df-leg 25705  df-mir 25775  df-rag 25816  df-perpg 25818 This theorem is referenced by:  footeq  25843  mideulem2  25853  lmieu  25903
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