Step
*
1
1
1
2
1
1
2
1
3
of Lemma
rv-line-circle-0
1. n : ℕ
2. a : ℝ^n
3. b : ℝ^n
4. p : ℝ^n
5. q : ℝ^n
6. p ≠ q
7. d(a;p) ≤ d(a;b)
8. d(a;b) ≤ d(a;q)
9. pp : ℝ^n
10. p - a = pp ∈ ℝ^n
11. qq : ℝ^n
12. q - a = qq ∈ ℝ^n
13. ||pp|| ≤ d(a;b)
14. d(a;b) ≤ ||qq||
15. pp ≠ qq
16. r0 < ||qq - pp||
17. r0 < ||qq - pp||^2
18. r0 ≤ (((r(2) * pp⋅qq - pp) * r(2) * pp⋅qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - d(a;b)^2))
19. d(a;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
20. d(a;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
21. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ [r0, r1]
22. ∀X:ℝ^n. req-vec(n;pp + X;p + X - a)
23. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp)
24. d(a;p) < d(a;b)
25. q-p-p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
26. d(a;b) < d(a;q)
27. d(a;b) < ||qq||
28. ||pp|| < d(a;b)
⊢ q-p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp-p
BY
{ Assert ⌜quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ (r0, r1)⌝⋅ }
1
.....assertion.....
1. n : ℕ
2. a : ℝ^n
3. b : ℝ^n
4. p : ℝ^n
5. q : ℝ^n
6. p ≠ q
7. d(a;p) ≤ d(a;b)
8. d(a;b) ≤ d(a;q)
9. pp : ℝ^n
10. p - a = pp ∈ ℝ^n
11. qq : ℝ^n
12. q - a = qq ∈ ℝ^n
13. ||pp|| ≤ d(a;b)
14. d(a;b) ≤ ||qq||
15. pp ≠ qq
16. r0 < ||qq - pp||
17. r0 < ||qq - pp||^2
18. r0 ≤ (((r(2) * pp⋅qq - pp) * r(2) * pp⋅qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - d(a;b)^2))
19. d(a;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
20. d(a;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
21. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ [r0, r1]
22. ∀X:ℝ^n. req-vec(n;pp + X;p + X - a)
23. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp)
24. d(a;p) < d(a;b)
25. q-p-p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
26. d(a;b) < d(a;q)
27. d(a;b) < ||qq||
28. ||pp|| < d(a;b)
⊢ quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ (r0, r1)
2
1. n : ℕ
2. a : ℝ^n
3. b : ℝ^n
4. p : ℝ^n
5. q : ℝ^n
6. p ≠ q
7. d(a;p) ≤ d(a;b)
8. d(a;b) ≤ d(a;q)
9. pp : ℝ^n
10. p - a = pp ∈ ℝ^n
11. qq : ℝ^n
12. q - a = qq ∈ ℝ^n
13. ||pp|| ≤ d(a;b)
14. d(a;b) ≤ ||qq||
15. pp ≠ qq
16. r0 < ||qq - pp||
17. r0 < ||qq - pp||^2
18. r0 ≤ (((r(2) * pp⋅qq - pp) * r(2) * pp⋅qq - pp) - r(4) * ||qq - pp||^2 * (||pp||^2 - d(a;b)^2))
19. d(a;p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
20. d(a;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp) = d(a;b)
21. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ [r0, r1]
22. ∀X:ℝ^n. req-vec(n;pp + X;p + X - a)
23. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp)
24. d(a;p) < d(a;b)
25. q-p-p + quadratic2(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp
26. d(a;b) < d(a;q)
27. d(a;b) < ||qq||
28. ||pp|| < d(a;b)
29. quadratic1(||qq - pp||^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2) ∈ (r0, r1)
⊢ q-p + quadratic1(d(qq;pp)^2;r(2) * pp⋅qq - pp;||pp||^2 - d(a;b)^2)*qq - pp-p
Latex:
Latex:
1. n : \mBbbN{}
2. a : \mBbbR{}\^{}n
3. b : \mBbbR{}\^{}n
4. p : \mBbbR{}\^{}n
5. q : \mBbbR{}\^{}n
6. p \mneq{} q
7. d(a;p) \mleq{} d(a;b)
8. d(a;b) \mleq{} d(a;q)
9. pp : \mBbbR{}\^{}n
10. p - a = pp
11. qq : \mBbbR{}\^{}n
12. q - a = qq
13. ||pp|| \mleq{} d(a;b)
14. d(a;b) \mleq{} ||qq||
15. pp \mneq{} qq
16. r0 < ||qq - pp||
17. r0 < ||qq - pp||\^{}2
18. r0 \mleq{} (((r(2) * pp\mcdot{}qq - pp) * r(2) * pp\mcdot{}qq - pp) - r(4) * ||qq - pp||\^{}2 * (||pp||\^{}2 - d(a;b)\^{}2))
19. d(a;p + quadratic1(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp) = d(a;b)
20. d(a;p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp) = d(a;b)
21. quadratic1(||qq - pp||\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2) \mmember{} [r0, r1]
22. \mforall{}X:\mBbbR{}\^{}n. req-vec(n;pp + X;p + X - a)
23. real-vec-be(n;q;p;p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp)
24. d(a;p) < d(a;b)
25. q-p-p + quadratic2(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp
26. d(a;b) < d(a;q)
27. d(a;b) < ||qq||
28. ||pp|| < d(a;b)
\mvdash{} q-p + quadratic1(d(qq;pp)\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2)*qq - pp-p
By
Latex:
Assert \mkleeneopen{}quadratic1(||qq - pp||\^{}2;r(2) * pp\mcdot{}qq - pp;||pp||\^{}2 - d(a;b)\^{}2) \mmember{} (r0, r1)\mkleeneclose{}\mcdot{}
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