Step
*
1
2
2
2
5
of Lemma
rv-circle-circle-lemma3'
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. p : {p:ℝ^2| d(a;b) = d(a;p)}
6. q : {q:ℝ^2| d(c;d) = d(c;q)}
7. i : {x:ℝ^2| (d(c;p) = d(c;x)) ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))}
8. o : {y:ℝ^2| (d(a;q) = d(a;y)) ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))}
9. a ≠ c
10. d(c;p) = d(c;i)
11. ¬(c ≠ i ∧ i ≠ d ∧ (¬c-i-d))
12. d(a;q) = d(a;o)
13. ¬(a ≠ o ∧ o ≠ b ∧ (¬a-o-b))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
15. d(c;p) ≤ d(c;d)
16. d(a;q) ≤ d(a;b)
17. u : ℝ^2
18. v : ℝ^2
19. ||u|| = d(a;b)
20. ||u - c - a|| = d(c;d)
21. ||v|| = d(a;b)
22. ||v - c - a|| = d(c;d)
23. ((d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2))
⇒ (r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a))
⊢ ((d(a;o) < d(a;b)) ∧ (d(c;i) < d(c;d))) ⇒ (r2-left(u + a;c;a) ∧ r2-left(v + a;a;c))
BY
{ ParallelLast }
1
.....antecedent.....
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. p : {p:ℝ^2| d(a;b) = d(a;p)}
6. q : {q:ℝ^2| d(c;d) = d(c;q)}
7. i : {x:ℝ^2| (d(c;p) = d(c;x)) ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))}
8. o : {y:ℝ^2| (d(a;q) = d(a;y)) ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))}
9. a ≠ c
10. d(c;p) = d(c;i)
11. ¬(c ≠ i ∧ i ≠ d ∧ (¬c-i-d))
12. d(a;q) = d(a;o)
13. ¬(a ≠ o ∧ o ≠ b ∧ (¬a-o-b))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
15. d(c;p) ≤ d(c;d)
16. d(a;q) ≤ d(a;b)
17. u : ℝ^2
18. v : ℝ^2
19. ||u|| = d(a;b)
20. ||u - c - a|| = d(c;d)
21. ||v|| = d(a;b)
22. ||v - c - a|| = d(c;d)
23. (d(a;o) < d(a;b)) ∧ (d(c;i) < d(c;d))
⊢ (d(a;b)^2 - d(c;d)^2) + ||c - a||^2^2 < (r(4) * ||c - a||^2 * d(a;b)^2)
2
1. a : ℝ^2
2. b : ℝ^2
3. c : ℝ^2
4. d : ℝ^2
5. p : {p:ℝ^2| d(a;b) = d(a;p)}
6. q : {q:ℝ^2| d(c;d) = d(c;q)}
7. i : {x:ℝ^2| (d(c;p) = d(c;x)) ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))}
8. o : {y:ℝ^2| (d(a;q) = d(a;y)) ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))}
9. a ≠ c
10. d(c;p) = d(c;i)
11. ¬(c ≠ i ∧ i ≠ d ∧ (¬c-i-d))
12. d(a;q) = d(a;o)
13. ¬(a ≠ o ∧ o ≠ b ∧ (¬a-o-b))
14. ∀x,y,z:ℝ^2. (d(x;y) = d(x;z) ⇐⇒ ||z - x|| = d(x;y))
15. d(c;p) ≤ d(c;d)
16. d(a;q) ≤ d(a;b)
17. u : ℝ^2
18. v : ℝ^2
19. ||u|| = d(a;b)
20. ||u - c - a|| = d(c;d)
21. ||v|| = d(a;b)
22. ||v - c - a|| = d(c;d)
23. (d(a;o) < d(a;b)) ∧ (d(c;i) < d(c;d))
24. r2-left(u;c - a;λi.r0) ∧ r2-left(v;λi.r0;c - a)
⊢ r2-left(u + a;c;a) ∧ r2-left(v + a;a;c)
Latex:
Latex:
1. a : \mBbbR{}\^{}2
2. b : \mBbbR{}\^{}2
3. c : \mBbbR{}\^{}2
4. d : \mBbbR{}\^{}2
5. p : \{p:\mBbbR{}\^{}2| d(a;b) = d(a;p)\}
6. q : \{q:\mBbbR{}\^{}2| d(c;d) = d(c;q)\}
7. i : \{x:\mBbbR{}\^{}2| (d(c;p) = d(c;x)) \mwedge{} (\mneg{}(c \mneq{} x \mwedge{} x \mneq{} d \mwedge{} (\mneg{}c-x-d)))\}
8. o : \{y:\mBbbR{}\^{}2| (d(a;q) = d(a;y)) \mwedge{} (\mneg{}(a \mneq{} y \mwedge{} y \mneq{} b \mwedge{} (\mneg{}a-y-b)))\}
9. a \mneq{} c
10. d(c;p) = d(c;i)
11. \mneg{}(c \mneq{} i \mwedge{} i \mneq{} d \mwedge{} (\mneg{}c-i-d))
12. d(a;q) = d(a;o)
13. \mneg{}(a \mneq{} o \mwedge{} o \mneq{} b \mwedge{} (\mneg{}a-o-b))
14. \mforall{}x,y,z:\mBbbR{}\^{}2. (d(x;y) = d(x;z) \mLeftarrow{}{}\mRightarrow{} ||z - x|| = d(x;y))
15. d(c;p) \mleq{} d(c;d)
16. d(a;q) \mleq{} d(a;b)
17. u : \mBbbR{}\^{}2
18. v : \mBbbR{}\^{}2
19. ||u|| = d(a;b)
20. ||u - c - a|| = d(c;d)
21. ||v|| = d(a;b)
22. ||v - c - a|| = d(c;d)
23. ((d(a;b)\^{}2 - d(c;d)\^{}2) + ||c - a||\^{}2\^{}2 < (r(4) * ||c - a||\^{}2 * d(a;b)\^{}2))
{}\mRightarrow{} (r2-left(u;c - a;\mlambda{}i.r0) \mwedge{} r2-left(v;\mlambda{}i.r0;c - a))
\mvdash{} ((d(a;o) < d(a;b)) \mwedge{} (d(c;i) < d(c;d))) {}\mRightarrow{} (r2-left(u + a;c;a) \mwedge{} r2-left(v + a;a;c))
By
Latex:
ParallelLast
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