Step
*
2
1
2
2
1
1
of Lemma
ip-line-circle-1
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. (s ≤ r0) ∧ ((||a - p|| < ||a - b||) ⇒ (s < r0))
27. qq - pp ≡ q - p
⊢ ∃u:{u:Point(rv)| ab=au ∧ q_u_p}
∃v:{v:Point(rv)| ab=av ∧ q_p_v} . ((||a - p|| < ||a - b||) ⇒ (p # v ∧ ((||a - b|| < ||a - q||) ⇒ q # u)))
BY
{ (InstConcl [⌜p + t*q - p⌝;⌜p + s*q - p⌝]⋅ THEN Auto THEN Try ((MemTypeCD THEN Auto))) }
1
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. s ≤ r0
27. (||a - p|| < ||a - b||) ⇒ (s < r0)
28. qq - pp ≡ q - p
⊢ ab=ap + t*q - p
2
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. s ≤ r0
27. (||a - p|| < ||a - b||) ⇒ (s < r0)
28. qq - pp ≡ q - p
29. ab=ap + t*q - p
⊢ q_p + t*q - p_p
3
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. s ≤ r0
27. (||a - p|| < ||a - b||) ⇒ (s < r0)
28. qq - pp ≡ q - p
⊢ ab=ap + s*q - p
4
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. s ≤ r0
27. (||a - p|| < ||a - b||) ⇒ (s < r0)
28. qq - pp ≡ q - p
29. ab=ap + s*q - p
⊢ q_p_p + s*q - p
5
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. s ≤ r0
27. qq - pp ≡ q - p
28. ||a - p|| < ||a - b||
29. s < r0
⊢ p # p + s*q - p
6
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a # b
7. p # q
8. pp : Point(rv)
9. p - a = pp ∈ Point(rv)
10. qq : Point(rv)
11. q - a = qq ∈ Point(rv)
12. ||pp|| ≤ ||a - b||
13. ||a - b|| ≤ ||qq||
14. pp # qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||^2
17. r0 ≤ (((r(2) * pp ⋅ qq - pp) * r(2) * pp ⋅ qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - ||a - b||^2))
18. t : ℝ
19. quadratic1(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = t ∈ ℝ
20. s : ℝ
21. quadratic2(||qq - pp||^2;r(2) * pp ⋅ qq - pp;||pp||^2 - ||a - b||^2) = s ∈ ℝ
22. ∀X:Point(rv). pp + X ≡ p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t ∈ [r0, r1]
26. s ≤ r0
27. qq - pp ≡ q - p
28. ||a - p|| < ||a - b||
29. p # p + s*q - p
30. ||a - b|| < ||a - q||
31. s < r0
⊢ q # p + t*q - p
Latex:
Latex:
1. rv : InnerProductSpace
2. a : Point(rv)
3. b : Point(rv)
4. p : Point(rv)
5. q : Point(rv)
6. a \# b
7. p \# q
8. pp : Point(rv)
9. p - a = pp
10. qq : Point(rv)
11. q - a = qq
12. ||pp|| \mleq{} ||a - b||
13. ||a - b|| \mleq{} ||qq||
14. pp \# qq
15. r0 < ||qq - pp||
16. r0 < ||qq - pp||\^{}2
17. r0 \mleq{} (((r(2) * pp \mcdot{} qq - pp) * r(2) * pp \mcdot{} qq - pp) - r(4)
* ||qq - pp||\^{}2
* (||pp||\^{}2 - ||a - b||\^{}2))
18. t : \mBbbR{}
19. quadratic1(||qq - pp||\^{}2;r(2) * pp \mcdot{} qq - pp;||pp||\^{}2 - ||a - b||\^{}2) = t
20. s : \mBbbR{}
21. quadratic2(||qq - pp||\^{}2;r(2) * pp \mcdot{} qq - pp;||pp||\^{}2 - ||a - b||\^{}2) = s
22. \mforall{}X:Point(rv). pp + X \mequiv{} p + X - a
23. ||p + t*q - p - a|| = ||a - b||
24. ||p + s*q - p - a|| = ||a - b||
25. t \mmember{} [r0, r1]
26. (s \mleq{} r0) \mwedge{} ((||a - p|| < ||a - b||) {}\mRightarrow{} (s < r0))
27. qq - pp \mequiv{} q - p
\mvdash{} \mexists{}u:\{u:Point(rv)| ab=au \mwedge{} q\_u\_p\}
\mexists{}v:\{v:Point(rv)| ab=av \mwedge{} q\_p\_v\}
((||a - p|| < ||a - b||) {}\mRightarrow{} (p \# v \mwedge{} ((||a - b|| < ||a - q||) {}\mRightarrow{} q \# u)))
By
Latex:
(InstConcl [\mkleeneopen{}p + t*q - p\mkleeneclose{};\mkleeneopen{}p + s*q - p\mkleeneclose{}]\mcdot{} THEN Auto THEN Try ((MemTypeCD THEN Auto)))
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