:: POLYEQ_4 semantic presentation
theorem Th1: :: POLYEQ_4:1
theorem Th2: :: POLYEQ_4:2
theorem Th3: :: POLYEQ_4:3
for b
1, b
2, b
3 being
Real holds
not ( b
1 <> 0 & b
2 / b
1 < 0 & not (
((- b3) + (sqrt (delta b1,b3,b2))) / (2 * b1) > 0 &
((- b3) - (sqrt (delta b1,b3,b2))) / (2 * b1) < 0 ) & not (
((- b3) + (sqrt (delta b1,b3,b2))) / (2 * b1) < 0 &
((- b3) - (sqrt (delta b1,b3,b2))) / (2 * b1) > 0 ) )
theorem Th4: :: POLYEQ_4:4
for b
1, b
2 being
Realfor b
3 being
Nat holds
not ( b
1 > 0 & ex b
4 being
Nat st
( b
3 = 2
* b
4 & b
4 >= 1 ) & b
2 |^ b
3 = b
1 & not b
2 = b
3 -root b
1 & not b
2 = - (b3 -root b1) )
theorem Th5: :: POLYEQ_4:5
for b
1, b
2, b
3 being
Real holds
not ( b
1 <> 0 &
Polynom b
1,b
2,0,b
3 = 0 & not b
3 = 0 & not b
3 = - (b2 / b1) )
theorem Th6: :: POLYEQ_4:6
for b
1, b
2 being
Real holds
( b
1 <> 0 &
Polynom b
1,0,0,b
2 = 0 implies b
2 = 0 )
theorem Th7: :: POLYEQ_4:7
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Nat holds
not ( b
1 <> 0 & ex b
6 being
Nat st b
5 = (2 * b6) + 1 &
delta b
1,b
2,b
3 >= 0 &
Polynom b
1,b
2,b
3,
(b4 |^ b5) = 0 & not b
4 = b
5 -root (((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1)) & not b
4 = b
5 -root (((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1)) )
theorem Th8: :: POLYEQ_4:8
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Nat holds
not ( b
1 <> 0 & b
2 / b
1 < 0 & b
3 / b
1 > 0 & ex b
6 being
Nat st
( b
5 = 2
* b
6 & b
6 >= 1 ) &
delta b
1,b
2,b
3 >= 0 &
Polynom b
1,b
2,b
3,
(b4 |^ b5) = 0 & not b
4 = b
5 -root (((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1)) & not b
4 = - (b5 -root (((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1))) & not b
4 = b
5 -root (((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1)) & not b
4 = - (b5 -root (((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1))) )
theorem Th9: :: POLYEQ_4:9
for b
1, b
2, b
3 being
Realfor b
4 being
Nat holds
not ( b
1 <> 0 & ex b
5 being
Nat st b
4 = (2 * b5) + 1 &
Polynom b
1,b
2,0,
(b3 |^ b4) = 0 & not b
3 = 0 & not b
3 = b
4 -root (- (b2 / b1)) )
theorem Th10: :: POLYEQ_4:10
for b
1, b
2, b
3 being
Realfor b
4 being
Nat holds
not ( b
1 <> 0 & b
2 / b
1 < 0 & ex b
5 being
Nat st
( b
4 = 2
* b
5 & b
5 >= 1 ) &
Polynom b
1,b
2,0,
(b3 |^ b4) = 0 & not b
3 = 0 & not b
3 = b
4 -root (- (b2 / b1)) & not b
3 = - (b4 -root (- (b2 / b1))) )
theorem Th11: :: POLYEQ_4:11
for b
1, b
2 being
Real holds
(
(b1 |^ 3) + (b2 |^ 3) = (b1 + b2) * (((b1 ^2 ) - (b1 * b2)) + (b2 ^2 )) &
(b1 |^ 5) + (b2 |^ 5) = (b1 + b2) * (((((b1 |^ 4) - ((b1 |^ 3) * b2)) + ((b1 |^ 2) * (b2 |^ 2))) - (b1 * (b2 |^ 3))) + (b2 |^ 4)) )
theorem Th12: :: POLYEQ_4:12
for b
1, b
2, b
3 being
Real holds
not ( b
1 <> 0 &
((b2 ^2 ) - ((2 * b1) * b2)) - (3 * (b1 ^2 )) >= 0 &
Polynom b
1,b
2,b
2,b
1,b
3 = 0 & not b
3 = - 1 & not b
3 = ((b1 - b2) + (sqrt (((b2 ^2 ) - ((2 * b1) * b2)) - (3 * (b1 ^2 ))))) / (2 * b1) & not b
3 = ((b1 - b2) - (sqrt (((b2 ^2 ) - ((2 * b1) * b2)) - (3 * (b1 ^2 ))))) / (2 * b1) )
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
complex number ;
func Polynom c
1,c
2,c
3,c
4,c
5,c
6,c
7 -> set equals :: POLYEQ_4:def 1
(((((a1 * (a7 |^ 5)) + (a2 * (a7 |^ 4))) + (a3 * (a7 |^ 3))) + (a4 * (a7 ^2 ))) + (a5 * a7)) + a
6;
coherence
(((((c1 * (c7 |^ 5)) + (c2 * (c7 |^ 4))) + (c3 * (c7 |^ 3))) + (c4 * (c7 ^2 ))) + (c5 * c7)) + c6 is set
;
end;
:: deftheorem Def1 defines Polynom POLYEQ_4:def 1 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
complex number holds
Polynom b
1,b
2,b
3,b
4,b
5,b
6,b
7 = (((((b1 * (b7 |^ 5)) + (b2 * (b7 |^ 4))) + (b3 * (b7 |^ 3))) + (b4 * (b7 ^2 ))) + (b5 * b7)) + b
6;
registration
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
complex number ;
cluster Polynom a
1,a
2,a
3,a
4,a
5,a
6,a
7 -> complex ;
coherence
Polynom c1,c2,c3,c4,c5,c6,c7 is complex
;
end;
registration
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
real number ;
cluster Polynom a
1,a
2,a
3,a
4,a
5,a
6,a
7 -> complex real ;
coherence
Polynom c1,c2,c3,c4,c5,c6,c7 is real
;
end;
theorem Th13: :: POLYEQ_4:13
for b
1, b
2, b
3, b
4 being
Real holds
( b
1 <> 0 &
(((b2 ^2 ) + ((2 * b1) * b2)) + (5 * (b1 ^2 ))) - ((4 * b1) * b3) > 0 &
Polynom b
1,b
2,b
3,b
3,b
2,b
1,b
4 = 0 implies for b
5, b
6 being
Real holds
not ( b
5 = ((b1 - b2) + (sqrt ((((b2 ^2 ) + ((2 * b1) * b2)) + (5 * (b1 ^2 ))) - ((4 * b1) * b3)))) / (2 * b1) & b
6 = ((b1 - b2) - (sqrt ((((b2 ^2 ) + ((2 * b1) * b2)) + (5 * (b1 ^2 ))) - ((4 * b1) * b3)))) / (2 * b1) & not b
4 = - 1 & not b
4 = (b5 + (sqrt (delta 1,(- b5),1))) / 2 & not b
4 = (b6 + (sqrt (delta 1,(- b6),1))) / 2 & not b
4 = (b5 - (sqrt (delta 1,(- b5),1))) / 2 & not b
4 = (b6 - (sqrt (delta 1,(- b6),1))) / 2 ) )
theorem Th14: :: POLYEQ_4:14
for b
1, b
2, b
3, b
4 being
Real holds
not ( b
1 + b
2 = b
3 & b
1 * b
2 = b
4 &
(b3 ^2 ) - (4 * b4) >= 0 & not ( b
1 = (b3 + (sqrt ((b3 ^2 ) - (4 * b4)))) / 2 & b
2 = (b3 - (sqrt ((b3 ^2 ) - (4 * b4)))) / 2 ) & not ( b
1 = (b3 - (sqrt ((b3 ^2 ) - (4 * b4)))) / 2 & b
2 = (b3 + (sqrt ((b3 ^2 ) - (4 * b4)))) / 2 ) )
theorem Th15: :: POLYEQ_4:15
theorem Th16: :: POLYEQ_4:16
theorem Th17: :: POLYEQ_4:17
canceled;
theorem Th18: :: POLYEQ_4:18
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Nat holds
not (
(b1 |^ b5) + (b2 |^ b5) = b
3 &
(b1 |^ b5) - (b2 |^ b5) = b
4 & ex b
6 being
Nat st
( b
5 = 2
* b
6 & b
6 >= 1 ) & b
3 > 0 & b
3 + b
4 > 0 & b
3 - b
4 > 0 & not ( b
1 = b
5 -root ((b3 + b4) / 2) & b
2 = b
5 -root ((b3 - b4) / 2) ) & not ( b
1 = b
5 -root ((b3 + b4) / 2) & b
2 = - (b5 -root ((b3 - b4) / 2)) ) & not ( b
1 = - (b5 -root ((b3 + b4) / 2)) & b
2 = b
5 -root ((b3 - b4) / 2) ) & not ( b
1 = - (b5 -root ((b3 + b4) / 2)) & b
2 = - (b5 -root ((b3 - b4) / 2)) ) )
theorem Th19: :: POLYEQ_4:19
for b
1, b
2, b
3, b
4, b
5 being
Realfor b
6 being
Nat holds
not (
(b1 * (b2 |^ b6)) + (b3 * (b4 |^ b6)) = b
5 & b
2 * b
4 = 0 & ex b
7 being
Nat st b
6 = (2 * b7) + 1 & b
1 * b
3 <> 0 & not ( b
2 = 0 & b
4 = b
6 -root (b5 / b3) ) & not ( b
2 = b
6 -root (b5 / b1) & b
4 = 0 ) )
theorem Th20: :: POLYEQ_4:20
for b
1, b
2, b
3, b
4, b
5 being
Realfor b
6 being
Nat holds
not (
(b1 * (b2 |^ b6)) + (b3 * (b4 |^ b6)) = b
5 & b
2 * b
4 = 0 & ex b
7 being
Nat st
( b
6 = 2
* b
7 & b
7 >= 1 ) & b
5 / b
3 > 0 & b
5 / b
1 > 0 & b
1 * b
3 <> 0 & not ( b
2 = 0 & b
4 = b
6 -root (b5 / b3) ) & not ( b
2 = 0 & b
4 = - (b6 -root (b5 / b3)) ) & not ( b
2 = b
6 -root (b5 / b1) & b
4 = 0 ) & not ( b
2 = - (b6 -root (b5 / b1)) & b
4 = 0 ) )
theorem Th21: :: POLYEQ_4:21
for b
1, b
2, b
3, b
4, b
5 being
Realfor b
6 being
Nat holds
( b
1 * (b2 |^ b6) = b
3 & b
2 * b
4 = b
5 & ex b
7 being
Nat st b
6 = (2 * b7) + 1 & b
3 * b
1 <> 0 implies ( b
2 = b
6 -root (b3 / b1) & b
4 = b
5 * (b6 -root (b1 / b3)) ) )
theorem Th22: :: POLYEQ_4:22
for b
1, b
2, b
3, b
4, b
5 being
Realfor b
6 being
Nat holds
not ( b
1 * (b2 |^ b6) = b
3 & b
2 * b
4 = b
5 & ex b
7 being
Nat st
( b
6 = 2
* b
7 & b
7 >= 1 ) & b
3 / b
1 > 0 & b
1 <> 0 & not ( b
2 = b
6 -root (b3 / b1) & b
4 = b
5 * (b6 -root (b1 / b3)) ) & not ( b
2 = - (b6 -root (b3 / b1)) & b
4 = - (b5 * (b6 -root (b1 / b3))) ) )
theorem Th23: :: POLYEQ_4:23
canceled;
theorem Th24: :: POLYEQ_4:24
theorem Th25: :: POLYEQ_4:25
for b
1, b
2 being
Real holds
( b
1 > 0 & b
1 <> 1 & b
1 to_power b
2 = b
1 implies b
2 = 1 )
theorem Th26: :: POLYEQ_4:26
canceled;
theorem Th27: :: POLYEQ_4:27
for b
1, b
2, b
3 being
Real holds
( b
1 > 0 & b
1 <> 1 & b
3 > 0 &
log b
1,b
3 = 0 implies b
3 = 1 )
theorem Th28: :: POLYEQ_4:28
for b
1, b
2, b
3 being
Real holds
( b
1 > 0 & b
1 <> 1 & b
3 > 0 &
log b
1,b
3 = 1 implies b
3 = b
1 )