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Theorem quart 20159
Description: The quartic equation, writing out all roots using square and cube root functions so that only direct substitutions remain, and we can actually claim to have a "quartic equation". Naturally, this theorem is ridiculously long (see quartfull 23688) if all the substitutions are performed. (Contributed by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
quart.a  |-  ( ph  ->  A  e.  CC )
quart.b  |-  ( ph  ->  B  e.  CC )
quart.c  |-  ( ph  ->  C  e.  CC )
quart.d  |-  ( ph  ->  D  e.  CC )
quart.x  |-  ( ph  ->  X  e.  CC )
quart.e  |-  ( ph  ->  E  =  -u ( A  /  4 ) )
quart.p  |-  ( ph  ->  P  =  ( B  -  ( ( 3  /  8 )  x.  ( A ^ 2 ) ) ) )
quart.q  |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  /  2 ) )  +  ( ( A ^ 3 )  / 
8 ) ) )
quart.r  |-  ( ph  ->  R  =  ( ( D  -  ( ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2 )  x.  B )  / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
4 ) ) ) ) )
quart.u  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
quart.v  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
quart.w  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
quart.s  |-  ( ph  ->  S  =  ( ( sqr `  M )  /  2 ) )
quart.m  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  T )  +  ( U  /  T ) )  /  3 ) )
quart.t  |-  ( ph  ->  T  =  ( ( ( V  +  W
)  /  2 )  ^ c  ( 1  /  3 ) ) )
quart.t0  |-  ( ph  ->  T  =/=  0 )
quart.m0  |-  ( ph  ->  M  =/=  0 )
quart.i  |-  ( ph  ->  I  =  ( sqr `  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) ) )
quart.j  |-  ( ph  ->  J  =  ( sqr `  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) ) )
Assertion
Ref Expression
quart  |-  ( ph  ->  ( ( ( ( X ^ 4 )  +  ( A  x.  ( X ^ 3 ) ) )  +  ( ( B  x.  ( X ^ 2 ) )  +  ( ( C  x.  X )  +  D ) ) )  =  0  <->  ( ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) )  \/  ( X  =  ( ( E  +  S
)  +  J )  \/  X  =  ( ( E  +  S
)  -  J ) ) ) ) )

Proof of Theorem quart
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 quart.a . . . 4  |-  ( ph  ->  A  e.  CC )
2 quart.b . . . 4  |-  ( ph  ->  B  e.  CC )
3 quart.c . . . 4  |-  ( ph  ->  C  e.  CC )
4 quart.d . . . 4  |-  ( ph  ->  D  e.  CC )
5 quart.p . . . 4  |-  ( ph  ->  P  =  ( B  -  ( ( 3  /  8 )  x.  ( A ^ 2 ) ) ) )
6 quart.q . . . 4  |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  /  2 ) )  +  ( ( A ^ 3 )  / 
8 ) ) )
7 quart.r . . . 4  |-  ( ph  ->  R  =  ( ( D  -  ( ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2 )  x.  B )  / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
4 ) ) ) ) )
8 quart.x . . . 4  |-  ( ph  ->  X  e.  CC )
9 quart.e . . . . . 6  |-  ( ph  ->  E  =  -u ( A  /  4 ) )
109oveq2d 5876 . . . . 5  |-  ( ph  ->  ( X  -  E
)  =  ( X  -  -u ( A  / 
4 ) ) )
11 4cn 9822 . . . . . . . 8  |-  4  e.  CC
1211a1i 10 . . . . . . 7  |-  ( ph  ->  4  e.  CC )
13 4nn 9881 . . . . . . . . 9  |-  4  e.  NN
1413nnne0i 9782 . . . . . . . 8  |-  4  =/=  0
1514a1i 10 . . . . . . 7  |-  ( ph  ->  4  =/=  0 )
161, 12, 15divcld 9538 . . . . . 6  |-  ( ph  ->  ( A  /  4
)  e.  CC )
178, 16subnegd 9166 . . . . 5  |-  ( ph  ->  ( X  -  -u ( A  /  4 ) )  =  ( X  +  ( A  /  4
) ) )
1810, 17eqtrd 2317 . . . 4  |-  ( ph  ->  ( X  -  E
)  =  ( X  +  ( A  / 
4 ) ) )
191, 2, 3, 4, 5, 6, 7, 8, 18quart1 20154 . . 3  |-  ( ph  ->  ( ( ( X ^ 4 )  +  ( A  x.  ( X ^ 3 ) ) )  +  ( ( B  x.  ( X ^ 2 ) )  +  ( ( C  x.  X )  +  D ) ) )  =  ( ( ( ( X  -  E
) ^ 4 )  +  ( P  x.  ( ( X  -  E ) ^ 2 ) ) )  +  ( ( Q  x.  ( X  -  E
) )  +  R
) ) )
2019eqeq1d 2293 . 2  |-  ( ph  ->  ( ( ( ( X ^ 4 )  +  ( A  x.  ( X ^ 3 ) ) )  +  ( ( B  x.  ( X ^ 2 ) )  +  ( ( C  x.  X )  +  D ) ) )  =  0  <->  ( (
( ( X  -  E ) ^ 4 )  +  ( P  x.  ( ( X  -  E ) ^
2 ) ) )  +  ( ( Q  x.  ( X  -  E ) )  +  R ) )  =  0 ) )
211, 2, 3, 4, 5, 6, 7quart1cl 20152 . . . 4  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
2221simp1d 967 . . 3  |-  ( ph  ->  P  e.  CC )
2321simp2d 968 . . 3  |-  ( ph  ->  Q  e.  CC )
2416negcld 9146 . . . . 5  |-  ( ph  -> 
-u ( A  / 
4 )  e.  CC )
259, 24eqeltrd 2359 . . . 4  |-  ( ph  ->  E  e.  CC )
268, 25subcld 9159 . . 3  |-  ( ph  ->  ( X  -  E
)  e.  CC )
27 quart.u . . . . 5  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
28 quart.v . . . . 5  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
29 quart.w . . . . 5  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
30 quart.s . . . . 5  |-  ( ph  ->  S  =  ( ( sqr `  M )  /  2 ) )
31 quart.m . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  T )  +  ( U  /  T ) )  /  3 ) )
32 quart.t . . . . 5  |-  ( ph  ->  T  =  ( ( ( V  +  W
)  /  2 )  ^ c  ( 1  /  3 ) ) )
33 quart.t0 . . . . 5  |-  ( ph  ->  T  =/=  0 )
341, 2, 3, 4, 1, 9, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33quartlem3 20157 . . . 4  |-  ( ph  ->  ( S  e.  CC  /\  M  e.  CC  /\  T  e.  CC )
)
3534simp1d 967 . . 3  |-  ( ph  ->  S  e.  CC )
3630oveq2d 5876 . . . . . 6  |-  ( ph  ->  ( 2  x.  S
)  =  ( 2  x.  ( ( sqr `  M )  /  2
) ) )
3734simp2d 968 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
3837sqrcld 11921 . . . . . . 7  |-  ( ph  ->  ( sqr `  M
)  e.  CC )
39 2cn 9818 . . . . . . . 8  |-  2  e.  CC
4039a1i 10 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
41 2ne0 9831 . . . . . . . 8  |-  2  =/=  0
4241a1i 10 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
4338, 40, 42divcan2d 9540 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( sqr `  M
)  /  2 ) )  =  ( sqr `  M ) )
4436, 43eqtrd 2317 . . . . 5  |-  ( ph  ->  ( 2  x.  S
)  =  ( sqr `  M ) )
4544oveq1d 5875 . . . 4  |-  ( ph  ->  ( ( 2  x.  S ) ^ 2 )  =  ( ( sqr `  M ) ^ 2 ) )
4637sqsqrd 11923 . . . 4  |-  ( ph  ->  ( ( sqr `  M
) ^ 2 )  =  M )
4745, 46eqtr2d 2318 . . 3  |-  ( ph  ->  M  =  ( ( 2  x.  S ) ^ 2 ) )
48 quart.m0 . . 3  |-  ( ph  ->  M  =/=  0 )
49 quart.i . . . . 5  |-  ( ph  ->  I  =  ( sqr `  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) ) )
50 quart.j . . . . 5  |-  ( ph  ->  J  =  ( sqr `  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) ) )
511, 2, 3, 4, 1, 9, 5, 6, 7, 27, 28, 29, 30, 31, 32, 33, 48, 49, 50quartlem4 20158 . . . 4  |-  ( ph  ->  ( S  =/=  0  /\  I  e.  CC  /\  J  e.  CC ) )
5251simp2d 968 . . 3  |-  ( ph  ->  I  e.  CC )
5349oveq1d 5875 . . . 4  |-  ( ph  ->  ( I ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
5435sqcld 11245 . . . . . . . 8  |-  ( ph  ->  ( S ^ 2 )  e.  CC )
5554negcld 9146 . . . . . . 7  |-  ( ph  -> 
-u ( S ^
2 )  e.  CC )
5622halfcld 9958 . . . . . . 7  |-  ( ph  ->  ( P  /  2
)  e.  CC )
5755, 56subcld 9159 . . . . . 6  |-  ( ph  ->  ( -u ( S ^ 2 )  -  ( P  /  2
) )  e.  CC )
5823, 12, 15divcld 9538 . . . . . . 7  |-  ( ph  ->  ( Q  /  4
)  e.  CC )
5951simp1d 967 . . . . . . 7  |-  ( ph  ->  S  =/=  0 )
6058, 35, 59divcld 9538 . . . . . 6  |-  ( ph  ->  ( ( Q  / 
4 )  /  S
)  e.  CC )
6157, 60addcld 8856 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) )  e.  CC )
6261sqsqrd 11923 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) )
6353, 62eqtrd 2317 . . 3  |-  ( ph  ->  ( I ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) )
6421simp3d 969 . . 3  |-  ( ph  ->  R  e.  CC )
65 ax-1cn 8797 . . . . . 6  |-  1  e.  CC
6665a1i 10 . . . . 5  |-  ( ph  ->  1  e.  CC )
67 3nn 9880 . . . . . . 7  |-  3  e.  NN
6867nnzi 10049 . . . . . 6  |-  3  e.  ZZ
69 1exp 11133 . . . . . 6  |-  ( 3  e.  ZZ  ->  (
1 ^ 3 )  =  1 )
7068, 69mp1i 11 . . . . 5  |-  ( ph  ->  ( 1 ^ 3 )  =  1 )
7134simp3d 969 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  CC )
7271mulid2d 8855 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  T
)  =  T )
7372oveq2d 5876 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  +  ( 1  x.  T ) )  =  ( ( 2  x.  P )  +  T ) )
7472oveq2d 5876 . . . . . . . . 9  |-  ( ph  ->  ( U  /  (
1  x.  T ) )  =  ( U  /  T ) )
7573, 74oveq12d 5878 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  =  ( ( ( 2  x.  P
)  +  T )  +  ( U  /  T ) ) )
7675oveq1d 5875 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
)  =  ( ( ( ( 2  x.  P )  +  T
)  +  ( U  /  T ) )  /  3 ) )
7776negeqd 9048 . . . . . 6  |-  ( ph  -> 
-u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 )  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T
) )  /  3
) )
7831, 77eqtr4d 2320 . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  /  3 ) )
79 oveq1 5867 . . . . . . . 8  |-  ( x  =  1  ->  (
x ^ 3 )  =  ( 1 ^ 3 ) )
8079eqeq1d 2293 . . . . . . 7  |-  ( x  =  1  ->  (
( x ^ 3 )  =  1  <->  (
1 ^ 3 )  =  1 ) )
81 oveq1 5867 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
x  x.  T )  =  ( 1  x.  T ) )
8281oveq2d 5876 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
( 2  x.  P
)  +  ( x  x.  T ) )  =  ( ( 2  x.  P )  +  ( 1  x.  T
) ) )
8381oveq2d 5876 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( U  /  ( x  x.  T ) )  =  ( U  /  (
1  x.  T ) ) )
8482, 83oveq12d 5878 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  ( x  x.  T ) ) )  =  ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) ) )
8584oveq1d 5875 . . . . . . . . 9  |-  ( x  =  1  ->  (
( ( ( 2  x.  P )  +  ( x  x.  T
) )  +  ( U  /  ( x  x.  T ) ) )  /  3 )  =  ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 ) )
8685negeqd 9048 . . . . . . . 8  |-  ( x  =  1  ->  -u (
( ( ( 2  x.  P )  +  ( x  x.  T
) )  +  ( U  /  ( x  x.  T ) ) )  /  3 )  =  -u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 ) )
8786eqeq2d 2296 . . . . . . 7  |-  ( x  =  1  ->  ( M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  ( x  x.  T ) ) )  /  3 )  <->  M  =  -u ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
) ) )
8880, 87anbi12d 691 . . . . . 6  |-  ( x  =  1  ->  (
( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) )  <->  ( (
1 ^ 3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 ) ) ) )
8988rspcev 2886 . . . . 5  |-  ( ( 1  e.  CC  /\  ( ( 1 ^ 3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
) ) )  ->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) )
9066, 70, 78, 89syl12anc 1180 . . . 4  |-  ( ph  ->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) )
91 mulcl 8823 . . . . . 6  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
9239, 22, 91sylancr 644 . . . . 5  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
9322sqcld 11245 . . . . . 6  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
94 mulcl 8823 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
9511, 64, 94sylancr 644 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
9693, 95subcld 9159 . . . . 5  |-  ( ph  ->  ( ( P ^
2 )  -  (
4  x.  R ) )  e.  CC )
9723sqcld 11245 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
9897negcld 9146 . . . . 5  |-  ( ph  -> 
-u ( Q ^
2 )  e.  CC )
9932oveq1d 5875 . . . . . 6  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( V  +  W )  /  2
)  ^ c  ( 1  /  3 ) ) ^ 3 ) )
1001, 2, 3, 4, 1, 9, 5, 6, 7, 27, 28, 29quartlem2 20156 . . . . . . . . . 10  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
101100simp2d 968 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
102100simp3d 969 . . . . . . . . 9  |-  ( ph  ->  W  e.  CC )
103101, 102addcld 8856 . . . . . . . 8  |-  ( ph  ->  ( V  +  W
)  e.  CC )
104103halfcld 9958 . . . . . . 7  |-  ( ph  ->  ( ( V  +  W )  /  2
)  e.  CC )
105 cxproot 20039 . . . . . . 7  |-  ( ( ( ( V  +  W )  /  2
)  e.  CC  /\  3  e.  NN )  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
106104, 67, 105sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
10799, 106eqtrd 2317 . . . . 5  |-  ( ph  ->  ( T ^ 3 )  =  ( ( V  +  W )  /  2 ) )
10829oveq1d 5875 . . . . . 6  |-  ( ph  ->  ( W ^ 2 )  =  ( ( sqr `  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) ) ^ 2 ) )
109101sqcld 11245 . . . . . . . 8  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
110100simp1d 967 . . . . . . . . . 10  |-  ( ph  ->  U  e.  CC )
111 3nn0 9985 . . . . . . . . . 10  |-  3  e.  NN0
112 expcl 11123 . . . . . . . . . 10  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
113110, 111, 112sylancl 643 . . . . . . . . 9  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
114 mulcl 8823 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
11511, 113, 114sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
116109, 115subcld 9159 . . . . . . 7  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
117116sqsqrd 11923 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) ) ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
118108, 117eqtrd 2317 . . . . 5  |-  ( ph  ->  ( W ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
11922, 23, 64, 27, 28quartlem1 20155 . . . . . 6  |-  ( ph  ->  ( U  =  ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) )  /\  V  =  ( ( ( 2  x.  ( ( 2  x.  P ) ^ 3 ) )  -  (
9  x.  ( ( 2  x.  P )  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) ) )  +  (; 2 7  x.  -u ( Q ^ 2 ) ) ) ) )
120119simpld 445 . . . . 5  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
121119simprd 449 . . . . 5  |-  ( ph  ->  V  =  ( ( ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  -  ( 9  x.  ( ( 2  x.  P )  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )  +  (; 2 7  x.  -u ( Q ^ 2 ) ) ) )
12292, 96, 98, 37, 71, 107, 102, 118, 120, 121, 33mcubic 20145 . . . 4  |-  ( ph  ->  ( ( ( ( M ^ 3 )  +  ( ( 2  x.  P )  x.  ( M ^ 2 ) ) )  +  ( ( ( ( P ^ 2 )  -  ( 4  x.  R ) )  x.  M )  +  -u ( Q ^ 2 ) ) )  =  0  <->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) ) )
12390, 122mpbird 223 . . 3  |-  ( ph  ->  ( ( ( M ^ 3 )  +  ( ( 2  x.  P )  x.  ( M ^ 2 ) ) )  +  ( ( ( ( P ^
2 )  -  (
4  x.  R ) )  x.  M )  +  -u ( Q ^
2 ) ) )  =  0 )
12451simp3d 969 . . 3  |-  ( ph  ->  J  e.  CC )
12550oveq1d 5875 . . . 4  |-  ( ph  ->  ( J ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
12657, 60subcld 9159 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) )  e.  CC )
127126sqsqrd 11923 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) )
128125, 127eqtrd 2317 . . 3  |-  ( ph  ->  ( J ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) )
12922, 23, 26, 35, 47, 48, 52, 63, 64, 123, 124, 128dquart 20151 . 2  |-  ( ph  ->  ( ( ( ( ( X  -  E
) ^ 4 )  +  ( P  x.  ( ( X  -  E ) ^ 2 ) ) )  +  ( ( Q  x.  ( X  -  E
) )  +  R
) )  =  0  <-> 
( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  \/  ( ( X  -  E )  =  ( S  +  J )  \/  ( X  -  E )  =  ( S  -  J ) ) ) ) )
13035negcld 9146 . . . . . . . 8  |-  ( ph  -> 
-u S  e.  CC )
131130, 52addcld 8856 . . . . . . 7  |-  ( ph  ->  ( -u S  +  I )  e.  CC )
1328, 25, 131subaddd 9177 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( E  +  ( -u S  +  I
) )  =  X ) )
13325, 35negsubd 9165 . . . . . . . . 9  |-  ( ph  ->  ( E  +  -u S )  =  ( E  -  S ) )
134133oveq1d 5875 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( ( E  -  S )  +  I ) )
13525, 130, 52addassd 8859 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
136134, 135eqtr3d 2319 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
137136eqeq1d 2293 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  +  I )  =  X  <-> 
( E  +  (
-u S  +  I
) )  =  X ) )
138132, 137bitr4d 247 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( ( E  -  S )  +  I )  =  X ) )
139 eqcom 2287 . . . . 5  |-  ( ( ( E  -  S
)  +  I )  =  X  <->  X  =  ( ( E  -  S )  +  I
) )
140138, 139syl6bb 252 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  X  =  (
( E  -  S
)  +  I ) ) )
141130, 52subcld 9159 . . . . . . 7  |-  ( ph  ->  ( -u S  -  I )  e.  CC )
1428, 25, 141subaddd 9177 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( E  +  ( -u S  -  I
) )  =  X ) )
143133oveq1d 5875 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( ( E  -  S )  -  I ) )
14425, 130, 52addsubassd 9179 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
145143, 144eqtr3d 2319 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
146145eqeq1d 2293 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  -  I )  =  X  <-> 
( E  +  (
-u S  -  I
) )  =  X ) )
147142, 146bitr4d 247 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( ( E  -  S )  -  I )  =  X ) )
148 eqcom 2287 . . . . 5  |-  ( ( ( E  -  S
)  -  I )  =  X  <->  X  =  ( ( E  -  S )  -  I
) )
149147, 148syl6bb 252 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  X  =  (
( E  -  S
)  -  I ) ) )
150140, 149orbi12d 690 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  <->  ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) ) ) )
15135, 124addcld 8856 . . . . . . 7  |-  ( ph  ->  ( S  +  J
)  e.  CC )
1528, 25, 151subaddd 9177 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( E  +  ( S  +  J ) )  =  X ) )
15325, 35, 124addassd 8859 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  +  J
)  =  ( E  +  ( S  +  J ) ) )
154153eqeq1d 2293 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  +  J )  =  X  <-> 
( E  +  ( S  +  J ) )  =  X ) )
155152, 154bitr4d 247 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( ( E  +  S )  +  J
)  =  X ) )
156 eqcom 2287 . . . . 5  |-  ( ( ( E  +  S
)  +  J )  =  X  <->  X  =  ( ( E  +  S )  +  J
) )
157155, 156syl6bb 252 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
X  =  ( ( E  +  S )  +  J ) ) )
15835, 124subcld 9159 . . . . . . 7  |-  ( ph  ->  ( S  -  J
)  e.  CC )
1598, 25, 158subaddd 9177 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( E  +  ( S  -  J ) )  =  X ) )
16025, 35, 124addsubassd 9179 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  -  J
)  =  ( E  +  ( S  -  J ) ) )
161160eqeq1d 2293 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  -  J )  =  X  <-> 
( E  +  ( S  -  J ) )  =  X ) )
162159, 161bitr4d 247 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( ( E  +  S )  -  J
)  =  X ) )
163 eqcom 2287 . . . . 5  |-  ( ( ( E  +  S
)  -  J )  =  X  <->  X  =  ( ( E  +  S )  -  J
) )
164162, 163syl6bb 252 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
X  =  ( ( E  +  S )  -  J ) ) )
165157, 164orbi12d 690 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( S  +  J
)  \/  ( X  -  E )  =  ( S  -  J
) )  <->  ( X  =  ( ( E  +  S )  +  J )  \/  X  =  ( ( E  +  S )  -  J ) ) ) )
166150, 165orbi12d 690 . 2  |-  ( ph  ->  ( ( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  \/  ( ( X  -  E )  =  ( S  +  J )  \/  ( X  -  E )  =  ( S  -  J ) ) )  <->  ( ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) )  \/  ( X  =  ( ( E  +  S
)  +  J )  \/  X  =  ( ( E  +  S
)  -  J ) ) ) ) )
16720, 129, 1663bitrd 270 1  |-  ( ph  ->  ( ( ( ( X ^ 4 )  +  ( A  x.  ( X ^ 3 ) ) )  +  ( ( B  x.  ( X ^ 2 ) )  +  ( ( C  x.  X )  +  D ) ) )  =  0  <->  ( ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) )  \/  ( X  =  ( ( E  +  S
)  +  J )  \/  X  =  ( ( E  +  S
)  -  J ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   E.wrex 2546   ` cfv 5257  (class class class)co 5860   CCcc 8737   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    - cmin 9039   -ucneg 9040    / cdiv 9425   NNcn 9748   2c2 9797   3c3 9798   4c4 9799   5c5 9800   6c6 9801   7c7 9802   8c8 9803   9c9 9804   NN0cn0 9967   ZZcz 10026  ;cdc 10126   ^cexp 11106   sqrcsqr 11720    ^ c ccxp 19915
This theorem is referenced by:  quartfull  23688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-dvds 12534  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917
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