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Theorem quart 20654
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 24749) 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 6056 . . . . 5  |-  ( ph  ->  ( X  -  E
)  =  ( X  -  -u ( A  / 
4 ) ) )
11 4cn 10030 . . . . . . . 8  |-  4  e.  CC
1211a1i 11 . . . . . . 7  |-  ( ph  ->  4  e.  CC )
13 4nn 10091 . . . . . . . . 9  |-  4  e.  NN
1413nnne0i 9990 . . . . . . . 8  |-  4  =/=  0
1514a1i 11 . . . . . . 7  |-  ( ph  ->  4  =/=  0 )
161, 12, 15divcld 9746 . . . . . 6  |-  ( ph  ->  ( A  /  4
)  e.  CC )
178, 16subnegd 9374 . . . . 5  |-  ( ph  ->  ( X  -  -u ( A  /  4 ) )  =  ( X  +  ( A  /  4
) ) )
1810, 17eqtrd 2436 . . . 4  |-  ( ph  ->  ( X  -  E
)  =  ( X  +  ( A  / 
4 ) ) )
191, 2, 3, 4, 5, 6, 7, 8, 18quart1 20649 . . 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 2412 . 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 20647 . . . 4  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
2221simp1d 969 . . 3  |-  ( ph  ->  P  e.  CC )
2321simp2d 970 . . 3  |-  ( ph  ->  Q  e.  CC )
2416negcld 9354 . . . . 5  |-  ( ph  -> 
-u ( A  / 
4 )  e.  CC )
259, 24eqeltrd 2478 . . . 4  |-  ( ph  ->  E  e.  CC )
268, 25subcld 9367 . . 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 20652 . . . 4  |-  ( ph  ->  ( S  e.  CC  /\  M  e.  CC  /\  T  e.  CC )
)
3534simp1d 969 . . 3  |-  ( ph  ->  S  e.  CC )
3630oveq2d 6056 . . . . . 6  |-  ( ph  ->  ( 2  x.  S
)  =  ( 2  x.  ( ( sqr `  M )  /  2
) ) )
3734simp2d 970 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
3837sqrcld 12194 . . . . . . 7  |-  ( ph  ->  ( sqr `  M
)  e.  CC )
39 2cn 10026 . . . . . . . 8  |-  2  e.  CC
4039a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
41 2ne0 10039 . . . . . . . 8  |-  2  =/=  0
4241a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
4338, 40, 42divcan2d 9748 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( sqr `  M
)  /  2 ) )  =  ( sqr `  M ) )
4436, 43eqtrd 2436 . . . . 5  |-  ( ph  ->  ( 2  x.  S
)  =  ( sqr `  M ) )
4544oveq1d 6055 . . . 4  |-  ( ph  ->  ( ( 2  x.  S ) ^ 2 )  =  ( ( sqr `  M ) ^ 2 ) )
4637sqsqrd 12196 . . . 4  |-  ( ph  ->  ( ( sqr `  M
) ^ 2 )  =  M )
4745, 46eqtr2d 2437 . . 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 20653 . . . 4  |-  ( ph  ->  ( S  =/=  0  /\  I  e.  CC  /\  J  e.  CC ) )
5251simp2d 970 . . 3  |-  ( ph  ->  I  e.  CC )
5349oveq1d 6055 . . . 4  |-  ( ph  ->  ( I ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
5435sqcld 11476 . . . . . . . 8  |-  ( ph  ->  ( S ^ 2 )  e.  CC )
5554negcld 9354 . . . . . . 7  |-  ( ph  -> 
-u ( S ^
2 )  e.  CC )
5622halfcld 10168 . . . . . . 7  |-  ( ph  ->  ( P  /  2
)  e.  CC )
5755, 56subcld 9367 . . . . . 6  |-  ( ph  ->  ( -u ( S ^ 2 )  -  ( P  /  2
) )  e.  CC )
5823, 12, 15divcld 9746 . . . . . . 7  |-  ( ph  ->  ( Q  /  4
)  e.  CC )
5951simp1d 969 . . . . . . 7  |-  ( ph  ->  S  =/=  0 )
6058, 35, 59divcld 9746 . . . . . 6  |-  ( ph  ->  ( ( Q  / 
4 )  /  S
)  e.  CC )
6157, 60addcld 9063 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) )  e.  CC )
6261sqsqrd 12196 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) )
6353, 62eqtrd 2436 . . 3  |-  ( ph  ->  ( I ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) )
6421simp3d 971 . . 3  |-  ( ph  ->  R  e.  CC )
65 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
6665a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
67 3nn 10090 . . . . . . 7  |-  3  e.  NN
6867nnzi 10261 . . . . . 6  |-  3  e.  ZZ
69 1exp 11364 . . . . . 6  |-  ( 3  e.  ZZ  ->  (
1 ^ 3 )  =  1 )
7068, 69mp1i 12 . . . . 5  |-  ( ph  ->  ( 1 ^ 3 )  =  1 )
7134simp3d 971 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  CC )
7271mulid2d 9062 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  T
)  =  T )
7372oveq2d 6056 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  +  ( 1  x.  T ) )  =  ( ( 2  x.  P )  +  T ) )
7472oveq2d 6056 . . . . . . . . 9  |-  ( ph  ->  ( U  /  (
1  x.  T ) )  =  ( U  /  T ) )
7573, 74oveq12d 6058 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  =  ( ( ( 2  x.  P
)  +  T )  +  ( U  /  T ) ) )
7675oveq1d 6055 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
)  =  ( ( ( ( 2  x.  P )  +  T
)  +  ( U  /  T ) )  /  3 ) )
7776negeqd 9256 . . . . . 6  |-  ( ph  -> 
-u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 )  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T
) )  /  3
) )
7831, 77eqtr4d 2439 . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  /  3 ) )
79 oveq1 6047 . . . . . . . 8  |-  ( x  =  1  ->  (
x ^ 3 )  =  ( 1 ^ 3 ) )
8079eqeq1d 2412 . . . . . . 7  |-  ( x  =  1  ->  (
( x ^ 3 )  =  1  <->  (
1 ^ 3 )  =  1 ) )
81 oveq1 6047 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
x  x.  T )  =  ( 1  x.  T ) )
8281oveq2d 6056 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
( 2  x.  P
)  +  ( x  x.  T ) )  =  ( ( 2  x.  P )  +  ( 1  x.  T
) ) )
8381oveq2d 6056 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( U  /  ( x  x.  T ) )  =  ( U  /  (
1  x.  T ) ) )
8482, 83oveq12d 6058 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  ( x  x.  T ) ) )  =  ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) ) )
8584oveq1d 6055 . . . . . . . . 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 9256 . . . . . . . 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 2415 . . . . . . 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 692 . . . . . 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 3012 . . . . 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 1182 . . . 4  |-  ( ph  ->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) )
91 mulcl 9030 . . . . . 6  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
9239, 22, 91sylancr 645 . . . . 5  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
9322sqcld 11476 . . . . . 6  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
94 mulcl 9030 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
9511, 64, 94sylancr 645 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
9693, 95subcld 9367 . . . . 5  |-  ( ph  ->  ( ( P ^
2 )  -  (
4  x.  R ) )  e.  CC )
9723sqcld 11476 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
9897negcld 9354 . . . . 5  |-  ( ph  -> 
-u ( Q ^
2 )  e.  CC )
9932oveq1d 6055 . . . . . 6  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( V  +  W )  /  2
)  ^ c  ( 1  /  3 ) ) ^ 3 ) )
1001, 2, 3, 4, 1, 9, 5, 6, 7, 27, 28, 29quartlem2 20651 . . . . . . . . . 10  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
101100simp2d 970 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
102100simp3d 971 . . . . . . . . 9  |-  ( ph  ->  W  e.  CC )
103101, 102addcld 9063 . . . . . . . 8  |-  ( ph  ->  ( V  +  W
)  e.  CC )
104103halfcld 10168 . . . . . . 7  |-  ( ph  ->  ( ( V  +  W )  /  2
)  e.  CC )
105 cxproot 20534 . . . . . . 7  |-  ( ( ( ( V  +  W )  /  2
)  e.  CC  /\  3  e.  NN )  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
106104, 67, 105sylancl 644 . . . . . 6  |-  ( ph  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
10799, 106eqtrd 2436 . . . . 5  |-  ( ph  ->  ( T ^ 3 )  =  ( ( V  +  W )  /  2 ) )
10829oveq1d 6055 . . . . . 6  |-  ( ph  ->  ( W ^ 2 )  =  ( ( sqr `  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) ) ^ 2 ) )
109101sqcld 11476 . . . . . . . 8  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
110100simp1d 969 . . . . . . . . . 10  |-  ( ph  ->  U  e.  CC )
111 3nn0 10195 . . . . . . . . . 10  |-  3  e.  NN0
112 expcl 11354 . . . . . . . . . 10  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
113110, 111, 112sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
114 mulcl 9030 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
11511, 113, 114sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
116109, 115subcld 9367 . . . . . . 7  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
117116sqsqrd 12196 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) ) ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
118108, 117eqtrd 2436 . . . . 5  |-  ( ph  ->  ( W ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
11922, 23, 64, 27, 28quartlem1 20650 . . . . . 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 446 . . . . 5  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
121119simprd 450 . . . . 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 20640 . . . 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 224 . . 3  |-  ( ph  ->  ( ( ( M ^ 3 )  +  ( ( 2  x.  P )  x.  ( M ^ 2 ) ) )  +  ( ( ( ( P ^
2 )  -  (
4  x.  R ) )  x.  M )  +  -u ( Q ^
2 ) ) )  =  0 )
12451simp3d 971 . . 3  |-  ( ph  ->  J  e.  CC )
12550oveq1d 6055 . . . 4  |-  ( ph  ->  ( J ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
12657, 60subcld 9367 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) )  e.  CC )
127126sqsqrd 12196 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) )
128125, 127eqtrd 2436 . . 3  |-  ( ph  ->  ( J ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) )
12922, 23, 26, 35, 47, 48, 52, 63, 64, 123, 124, 128dquart 20646 . 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 9354 . . . . . . . 8  |-  ( ph  -> 
-u S  e.  CC )
131130, 52addcld 9063 . . . . . . 7  |-  ( ph  ->  ( -u S  +  I )  e.  CC )
1328, 25, 131subaddd 9385 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( E  +  ( -u S  +  I
) )  =  X ) )
13325, 35negsubd 9373 . . . . . . . . 9  |-  ( ph  ->  ( E  +  -u S )  =  ( E  -  S ) )
134133oveq1d 6055 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( ( E  -  S )  +  I ) )
13525, 130, 52addassd 9066 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
136134, 135eqtr3d 2438 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
137136eqeq1d 2412 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  +  I )  =  X  <-> 
( E  +  (
-u S  +  I
) )  =  X ) )
138132, 137bitr4d 248 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( ( E  -  S )  +  I )  =  X ) )
139 eqcom 2406 . . . . 5  |-  ( ( ( E  -  S
)  +  I )  =  X  <->  X  =  ( ( E  -  S )  +  I
) )
140138, 139syl6bb 253 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  X  =  (
( E  -  S
)  +  I ) ) )
141130, 52subcld 9367 . . . . . . 7  |-  ( ph  ->  ( -u S  -  I )  e.  CC )
1428, 25, 141subaddd 9385 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( E  +  ( -u S  -  I
) )  =  X ) )
143133oveq1d 6055 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( ( E  -  S )  -  I ) )
14425, 130, 52addsubassd 9387 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
145143, 144eqtr3d 2438 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
146145eqeq1d 2412 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  -  I )  =  X  <-> 
( E  +  (
-u S  -  I
) )  =  X ) )
147142, 146bitr4d 248 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( ( E  -  S )  -  I )  =  X ) )
148 eqcom 2406 . . . . 5  |-  ( ( ( E  -  S
)  -  I )  =  X  <->  X  =  ( ( E  -  S )  -  I
) )
149147, 148syl6bb 253 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  X  =  (
( E  -  S
)  -  I ) ) )
150140, 149orbi12d 691 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  <->  ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) ) ) )
15135, 124addcld 9063 . . . . . . 7  |-  ( ph  ->  ( S  +  J
)  e.  CC )
1528, 25, 151subaddd 9385 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( E  +  ( S  +  J ) )  =  X ) )
15325, 35, 124addassd 9066 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  +  J
)  =  ( E  +  ( S  +  J ) ) )
154153eqeq1d 2412 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  +  J )  =  X  <-> 
( E  +  ( S  +  J ) )  =  X ) )
155152, 154bitr4d 248 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( ( E  +  S )  +  J
)  =  X ) )
156 eqcom 2406 . . . . 5  |-  ( ( ( E  +  S
)  +  J )  =  X  <->  X  =  ( ( E  +  S )  +  J
) )
157155, 156syl6bb 253 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
X  =  ( ( E  +  S )  +  J ) ) )
15835, 124subcld 9367 . . . . . . 7  |-  ( ph  ->  ( S  -  J
)  e.  CC )
1598, 25, 158subaddd 9385 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( E  +  ( S  -  J ) )  =  X ) )
16025, 35, 124addsubassd 9387 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  -  J
)  =  ( E  +  ( S  -  J ) ) )
161160eqeq1d 2412 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  -  J )  =  X  <-> 
( E  +  ( S  -  J ) )  =  X ) )
162159, 161bitr4d 248 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( ( E  +  S )  -  J
)  =  X ) )
163 eqcom 2406 . . . . 5  |-  ( ( ( E  +  S
)  -  J )  =  X  <->  X  =  ( ( E  +  S )  -  J
) )
164162, 163syl6bb 253 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
X  =  ( ( E  +  S )  -  J ) ) )
165157, 164orbi12d 691 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( S  +  J
)  \/  ( X  -  E )  =  ( S  -  J
) )  <->  ( X  =  ( ( E  +  S )  +  J )  \/  X  =  ( ( E  +  S )  -  J ) ) ) )
166150, 165orbi12d 691 . 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 271 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   3c3 10006   4c4 10007   5c5 10008   6c6 10009   7c7 10010   8c8 10011   9c9 10012   NN0cn0 10177   ZZcz 10238  ;cdc 10338   ^cexp 11337   sqrcsqr 11993    ^ c ccxp 20406
This theorem is referenced by:  quartfull  24749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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