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Theorem foot 25659
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 25658 . 2 (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10 eqid 2651 . . . . . 6 (pInvG‘𝐺) = (pInvG‘𝐺)
115ad2antrr 762 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐺 ∈ TarskiG)
127ad2antrr 762 . . . . . 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 809 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐴)
161, 4, 3, 13, 14, 15tglnpt 25489 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝑃)
1716adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝑃)
18 simprr 811 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐴)
191, 4, 3, 13, 14, 18tglnpt 25489 . . . . . . 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 2920 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ¬ 𝐶𝐴) → 𝑥𝐶)
2315, 21, 22syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐶)
2423necomd 2878 . . . . . . . . 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 25573 . . . . . . 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 809 . . . . . . . 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 25570 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
311, 3, 4, 13, 29, 16, 24tglinerflx2 25574 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
3231, 15elind 3831 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
331, 2, 3, 4, 13, 30, 14, 32isperp2 25655 . . . . . . . . 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 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 2652 . . . . . . . . . 10 (𝑢 = 𝐶𝑥 = 𝑥)
38 eqidd 2652 . . . . . . . . . 10 (𝑢 = 𝐶𝑣 = 𝑣)
3936, 37, 38s3eqd 13655 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
4039eleq1d 2715 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41 eqidd 2652 . . . . . . . . . 10 (𝑣 = 𝑧𝐶 = 𝐶)
42 eqidd 2652 . . . . . . . . . 10 (𝑣 = 𝑧𝑥 = 𝑥)
43 id 22 . . . . . . . . . 10 (𝑣 = 𝑧𝑣 = 𝑧)
4441, 42, 43s3eqd 13655 . . . . . . . . 9 (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
4544eleq1d 2715 . . . . . . . 8 (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
4640, 45rspc2va 3354 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
4726, 27, 35, 46syl21anc 1365 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48 nelne2 2920 . . . . . . . . . . 11 ((𝑧𝐴 ∧ ¬ 𝐶𝐴) → 𝑧𝐶)
4918, 21, 48syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐶)
5049necomd 2878 . . . . . . . . 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 25573 . . . . . . 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 811 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)
551, 3, 4, 13, 29, 19, 50tgelrnln 25570 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
561, 3, 4, 13, 29, 19, 50tglinerflx2 25574 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
5756, 18elind 3831 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
581, 2, 3, 4, 13, 55, 14, 57isperp2 25655 . . . . . . . . 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 222 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61 eqidd 2652 . . . . . . . . . 10 (𝑢 = 𝐶𝑧 = 𝑧)
6236, 61, 38s3eqd 13655 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
6362eleq1d 2715 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64 eqidd 2652 . . . . . . . . . 10 (𝑣 = 𝑥𝐶 = 𝐶)
65 eqidd 2652 . . . . . . . . . 10 (𝑣 = 𝑥𝑧 = 𝑧)
66 id 22 . . . . . . . . . 10 (𝑣 = 𝑥𝑣 = 𝑥)
6764, 65, 66s3eqd 13655 . . . . . . . . 9 (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
6867eleq1d 2715 . . . . . . . 8 (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
6963, 68rspc2va 3354 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
7052, 53, 60, 69syl21anc 1365 . . . . . 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 25644 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
7271ex 449 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7372ralrimivva 3000 . . 3 (𝜑 → ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74 oveq2 6698 . . . . 5 (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
7574breq1d 4695 . . . 4 (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
7675rmo4 3432 . . 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 3189 . 2 (∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
799, 77, 78sylanbrc 699 1 (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  ∃*wrmo 2944   class class class wbr 4685  ran crn 5144  cfv 5926  (class class class)co 6690  ⟨“cs3 13633  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  pInvGcmir 25592  ∟Gcrag 25633  ⟂Gcperpg 25635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkg 25397  df-cgrg 25451  df-leg 25523  df-mir 25593  df-rag 25634  df-perpg 25636
This theorem is referenced by:  footeq  25661  mideulem2  25671  lmieu  25721
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