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Theorem foot 26505
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 26504 . 2 (𝜑 → ∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10 eqid 2824 . . . . . 6 (pInvG‘𝐺) = (pInvG‘𝐺)
115ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐺 ∈ TarskiG)
127ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑃)
135adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐺 ∈ TarskiG)
146adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐴 ∈ ran 𝐿)
15 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐴)
161, 4, 3, 13, 14, 15tglnpt 26332 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝑃)
1716adantr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝑃)
18 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐴)
191, 4, 3, 13, 14, 18tglnpt 26332 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝑃)
2019adantr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝑃)
218adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ¬ 𝐶𝐴)
22 nelne2 3110 . . . . . . . . . . 11 ((𝑥𝐴 ∧ ¬ 𝐶𝐴) → 𝑥𝐶)
2315, 21, 22syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥𝐶)
2423necomd 3068 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑥)
2524adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑥)
261, 3, 4, 11, 12, 17, 25tglinerflx1 26416 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
2718adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧𝐴)
28 simprl 770 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
297adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑃)
301, 3, 4, 13, 29, 16, 24tgelrnln 26413 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
311, 3, 4, 13, 29, 16, 24tglinerflx2 26417 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
3231, 15elind 4154 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
331, 2, 3, 4, 13, 30, 14, 32isperp2 26498 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3433adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
3528, 34mpbid 235 . . . . . . 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 2825 . . . . . . . . . 10 (𝑢 = 𝐶𝑥 = 𝑥)
38 eqidd 2825 . . . . . . . . . 10 (𝑢 = 𝐶𝑣 = 𝑣)
3936, 37, 38s3eqd 14215 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
4039eleq1d 2900 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41 eqidd 2825 . . . . . . . . . 10 (𝑣 = 𝑧𝐶 = 𝐶)
42 eqidd 2825 . . . . . . . . . 10 (𝑣 = 𝑧𝑥 = 𝑥)
43 id 22 . . . . . . . . . 10 (𝑣 = 𝑧𝑣 = 𝑧)
4441, 42, 43s3eqd 14215 . . . . . . . . 9 (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
4544eleq1d 2900 . . . . . . . 8 (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
4640, 45rspc2va 3619 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
4726, 27, 35, 46syl21anc 836 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48 nelne2 3110 . . . . . . . . . . 11 ((𝑧𝐴 ∧ ¬ 𝐶𝐴) → 𝑧𝐶)
4918, 21, 48syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧𝐶)
5049necomd 3068 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝐶𝑧)
5150adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶𝑧)
521, 3, 4, 11, 12, 20, 51tglinerflx1 26416 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
5315adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥𝐴)
54 simprr 772 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)
551, 3, 4, 13, 29, 19, 50tgelrnln 26413 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
561, 3, 4, 13, 29, 19, 50tglinerflx2 26417 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
5756, 18elind 4154 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
581, 2, 3, 4, 13, 55, 14, 57isperp2 26498 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
5958adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
6054, 59mpbid 235 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61 eqidd 2825 . . . . . . . . . 10 (𝑢 = 𝐶𝑧 = 𝑧)
6236, 61, 38s3eqd 14215 . . . . . . . . 9 (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
6362eleq1d 2900 . . . . . . . 8 (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64 eqidd 2825 . . . . . . . . . 10 (𝑣 = 𝑥𝐶 = 𝐶)
65 eqidd 2825 . . . . . . . . . 10 (𝑣 = 𝑥𝑧 = 𝑧)
66 id 22 . . . . . . . . . 10 (𝑣 = 𝑥𝑣 = 𝑥)
6764, 65, 66s3eqd 14215 . . . . . . . . 9 (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
6867eleq1d 2900 . . . . . . . 8 (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
6963, 68rspc2va 3619 . . . . . . 7 (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
7052, 53, 60, 69syl21anc 836 . . . . . 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 26487 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑧𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
7271ex 416 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑧𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7372ralrimivva 3185 . . 3 (𝜑 → ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74 oveq2 7146 . . . . 5 (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
7574breq1d 5057 . . . 4 (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
7675rmo4 3706 . . 3 (∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑥𝐴𝑧𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
7773, 76sylibr 237 . 2 (𝜑 → ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
78 reu5 3413 . 2 (∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
799, 77, 78sylanbrc 586 1 (𝜑 → ∃!𝑥𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3013  wral 3132  wrex 3133  ∃!wreu 3134  ∃*wrmo 3135   class class class wbr 5047  ran crn 5537  cfv 6336  (class class class)co 7138  ⟨“cs3 14193  Basecbs 16472  distcds 16563  TarskiGcstrkg 26213  Itvcitv 26219  LineGclng 26220  pInvGcmir 26435  ∟Gcrag 26476  ⟂Gcperpg 26478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-3 11687  df-n0 11884  df-xnn0 11954  df-z 11968  df-uz 12230  df-fz 12884  df-fzo 13027  df-hash 13685  df-word 13856  df-concat 13912  df-s1 13939  df-s2 14199  df-s3 14200  df-trkgc 26231  df-trkgb 26232  df-trkgcb 26233  df-trkg 26236  df-cgrg 26294  df-leg 26366  df-mir 26436  df-rag 26477  df-perpg 26479
This theorem is referenced by:  footeq  26507  mideulem2  26517  lmieu  26567
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