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Theorem quart 20703
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 24798) 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 6099 . . . . 5  |-  ( ph  ->  ( X  -  E
)  =  ( X  -  -u ( A  / 
4 ) ) )
11 4cn 10076 . . . . . . . 8  |-  4  e.  CC
1211a1i 11 . . . . . . 7  |-  ( ph  ->  4  e.  CC )
13 4nn 10137 . . . . . . . . 9  |-  4  e.  NN
1413nnne0i 10036 . . . . . . . 8  |-  4  =/=  0
1514a1i 11 . . . . . . 7  |-  ( ph  ->  4  =/=  0 )
161, 12, 15divcld 9792 . . . . . 6  |-  ( ph  ->  ( A  /  4
)  e.  CC )
178, 16subnegd 9420 . . . . 5  |-  ( ph  ->  ( X  -  -u ( A  /  4 ) )  =  ( X  +  ( A  /  4
) ) )
1810, 17eqtrd 2470 . . . 4  |-  ( ph  ->  ( X  -  E
)  =  ( X  +  ( A  / 
4 ) ) )
191, 2, 3, 4, 5, 6, 7, 8, 18quart1 20698 . . 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 2446 . 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 20696 . . . 4  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
2221simp1d 970 . . 3  |-  ( ph  ->  P  e.  CC )
2321simp2d 971 . . 3  |-  ( ph  ->  Q  e.  CC )
2416negcld 9400 . . . . 5  |-  ( ph  -> 
-u ( A  / 
4 )  e.  CC )
259, 24eqeltrd 2512 . . . 4  |-  ( ph  ->  E  e.  CC )
268, 25subcld 9413 . . 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 20701 . . . 4  |-  ( ph  ->  ( S  e.  CC  /\  M  e.  CC  /\  T  e.  CC )
)
3534simp1d 970 . . 3  |-  ( ph  ->  S  e.  CC )
3630oveq2d 6099 . . . . . 6  |-  ( ph  ->  ( 2  x.  S
)  =  ( 2  x.  ( ( sqr `  M )  /  2
) ) )
3734simp2d 971 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
3837sqrcld 12241 . . . . . . 7  |-  ( ph  ->  ( sqr `  M
)  e.  CC )
39 2cn 10072 . . . . . . . 8  |-  2  e.  CC
4039a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
41 2ne0 10085 . . . . . . . 8  |-  2  =/=  0
4241a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
4338, 40, 42divcan2d 9794 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( sqr `  M
)  /  2 ) )  =  ( sqr `  M ) )
4436, 43eqtrd 2470 . . . . 5  |-  ( ph  ->  ( 2  x.  S
)  =  ( sqr `  M ) )
4544oveq1d 6098 . . . 4  |-  ( ph  ->  ( ( 2  x.  S ) ^ 2 )  =  ( ( sqr `  M ) ^ 2 ) )
4637sqsqrd 12243 . . . 4  |-  ( ph  ->  ( ( sqr `  M
) ^ 2 )  =  M )
4745, 46eqtr2d 2471 . . 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 20702 . . . 4  |-  ( ph  ->  ( S  =/=  0  /\  I  e.  CC  /\  J  e.  CC ) )
5251simp2d 971 . . 3  |-  ( ph  ->  I  e.  CC )
5349oveq1d 6098 . . . 4  |-  ( ph  ->  ( I ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
5435sqcld 11523 . . . . . . . 8  |-  ( ph  ->  ( S ^ 2 )  e.  CC )
5554negcld 9400 . . . . . . 7  |-  ( ph  -> 
-u ( S ^
2 )  e.  CC )
5622halfcld 10214 . . . . . . 7  |-  ( ph  ->  ( P  /  2
)  e.  CC )
5755, 56subcld 9413 . . . . . 6  |-  ( ph  ->  ( -u ( S ^ 2 )  -  ( P  /  2
) )  e.  CC )
5823, 12, 15divcld 9792 . . . . . . 7  |-  ( ph  ->  ( Q  /  4
)  e.  CC )
5951simp1d 970 . . . . . . 7  |-  ( ph  ->  S  =/=  0 )
6058, 35, 59divcld 9792 . . . . . 6  |-  ( ph  ->  ( ( Q  / 
4 )  /  S
)  e.  CC )
6157, 60addcld 9109 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) )  e.  CC )
6261sqsqrd 12243 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) )
6353, 62eqtrd 2470 . . 3  |-  ( ph  ->  ( I ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) )
6421simp3d 972 . . 3  |-  ( ph  ->  R  e.  CC )
65 ax-1cn 9050 . . . . . 6  |-  1  e.  CC
6665a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
67 3nn 10136 . . . . . . 7  |-  3  e.  NN
6867nnzi 10307 . . . . . 6  |-  3  e.  ZZ
69 1exp 11411 . . . . . 6  |-  ( 3  e.  ZZ  ->  (
1 ^ 3 )  =  1 )
7068, 69mp1i 12 . . . . 5  |-  ( ph  ->  ( 1 ^ 3 )  =  1 )
7134simp3d 972 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  CC )
7271mulid2d 9108 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  T
)  =  T )
7372oveq2d 6099 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  +  ( 1  x.  T ) )  =  ( ( 2  x.  P )  +  T ) )
7472oveq2d 6099 . . . . . . . . 9  |-  ( ph  ->  ( U  /  (
1  x.  T ) )  =  ( U  /  T ) )
7573, 74oveq12d 6101 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  =  ( ( ( 2  x.  P
)  +  T )  +  ( U  /  T ) ) )
7675oveq1d 6098 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
)  =  ( ( ( ( 2  x.  P )  +  T
)  +  ( U  /  T ) )  /  3 ) )
7776negeqd 9302 . . . . . 6  |-  ( ph  -> 
-u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 )  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T
) )  /  3
) )
7831, 77eqtr4d 2473 . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  /  3 ) )
79 oveq1 6090 . . . . . . . 8  |-  ( x  =  1  ->  (
x ^ 3 )  =  ( 1 ^ 3 ) )
8079eqeq1d 2446 . . . . . . 7  |-  ( x  =  1  ->  (
( x ^ 3 )  =  1  <->  (
1 ^ 3 )  =  1 ) )
81 oveq1 6090 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
x  x.  T )  =  ( 1  x.  T ) )
8281oveq2d 6099 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
( 2  x.  P
)  +  ( x  x.  T ) )  =  ( ( 2  x.  P )  +  ( 1  x.  T
) ) )
8381oveq2d 6099 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( U  /  ( x  x.  T ) )  =  ( U  /  (
1  x.  T ) ) )
8482, 83oveq12d 6101 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  ( x  x.  T ) ) )  =  ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) ) )
8584oveq1d 6098 . . . . . . . . 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 9302 . . . . . . . 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 2449 . . . . . . 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 693 . . . . . 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 3054 . . . . 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 1183 . . . 4  |-  ( ph  ->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) )
91 mulcl 9076 . . . . . 6  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
9239, 22, 91sylancr 646 . . . . 5  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
9322sqcld 11523 . . . . . 6  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
94 mulcl 9076 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
9511, 64, 94sylancr 646 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
9693, 95subcld 9413 . . . . 5  |-  ( ph  ->  ( ( P ^
2 )  -  (
4  x.  R ) )  e.  CC )
9723sqcld 11523 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
9897negcld 9400 . . . . 5  |-  ( ph  -> 
-u ( Q ^
2 )  e.  CC )
9932oveq1d 6098 . . . . . 6  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( V  +  W )  /  2
)  ^ c  ( 1  /  3 ) ) ^ 3 ) )
1001, 2, 3, 4, 1, 9, 5, 6, 7, 27, 28, 29quartlem2 20700 . . . . . . . . . 10  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
101100simp2d 971 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
102100simp3d 972 . . . . . . . . 9  |-  ( ph  ->  W  e.  CC )
103101, 102addcld 9109 . . . . . . . 8  |-  ( ph  ->  ( V  +  W
)  e.  CC )
104103halfcld 10214 . . . . . . 7  |-  ( ph  ->  ( ( V  +  W )  /  2
)  e.  CC )
105 cxproot 20583 . . . . . . 7  |-  ( ( ( ( V  +  W )  /  2
)  e.  CC  /\  3  e.  NN )  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
106104, 67, 105sylancl 645 . . . . . 6  |-  ( ph  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
10799, 106eqtrd 2470 . . . . 5  |-  ( ph  ->  ( T ^ 3 )  =  ( ( V  +  W )  /  2 ) )
10829oveq1d 6098 . . . . . 6  |-  ( ph  ->  ( W ^ 2 )  =  ( ( sqr `  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) ) ^ 2 ) )
109101sqcld 11523 . . . . . . . 8  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
110100simp1d 970 . . . . . . . . . 10  |-  ( ph  ->  U  e.  CC )
111 3nn0 10241 . . . . . . . . . 10  |-  3  e.  NN0
112 expcl 11401 . . . . . . . . . 10  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
113110, 111, 112sylancl 645 . . . . . . . . 9  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
114 mulcl 9076 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
11511, 113, 114sylancr 646 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
116109, 115subcld 9413 . . . . . . 7  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
117116sqsqrd 12243 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) ) ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
118108, 117eqtrd 2470 . . . . 5  |-  ( ph  ->  ( W ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
11922, 23, 64, 27, 28quartlem1 20699 . . . . . 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 447 . . . . 5  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
121119simprd 451 . . . . 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 20689 . . . 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 225 . . 3  |-  ( ph  ->  ( ( ( M ^ 3 )  +  ( ( 2  x.  P )  x.  ( M ^ 2 ) ) )  +  ( ( ( ( P ^
2 )  -  (
4  x.  R ) )  x.  M )  +  -u ( Q ^
2 ) ) )  =  0 )
12451simp3d 972 . . 3  |-  ( ph  ->  J  e.  CC )
12550oveq1d 6098 . . . 4  |-  ( ph  ->  ( J ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
12657, 60subcld 9413 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) )  e.  CC )
127126sqsqrd 12243 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) )
128125, 127eqtrd 2470 . . 3  |-  ( ph  ->  ( J ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) )
12922, 23, 26, 35, 47, 48, 52, 63, 64, 123, 124, 128dquart 20695 . 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 9400 . . . . . . . 8  |-  ( ph  -> 
-u S  e.  CC )
131130, 52addcld 9109 . . . . . . 7  |-  ( ph  ->  ( -u S  +  I )  e.  CC )
1328, 25, 131subaddd 9431 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( E  +  ( -u S  +  I
) )  =  X ) )
13325, 35negsubd 9419 . . . . . . . . 9  |-  ( ph  ->  ( E  +  -u S )  =  ( E  -  S ) )
134133oveq1d 6098 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( ( E  -  S )  +  I ) )
13525, 130, 52addassd 9112 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
136134, 135eqtr3d 2472 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
137136eqeq1d 2446 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  +  I )  =  X  <-> 
( E  +  (
-u S  +  I
) )  =  X ) )
138132, 137bitr4d 249 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( ( E  -  S )  +  I )  =  X ) )
139 eqcom 2440 . . . . 5  |-  ( ( ( E  -  S
)  +  I )  =  X  <->  X  =  ( ( E  -  S )  +  I
) )
140138, 139syl6bb 254 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  X  =  (
( E  -  S
)  +  I ) ) )
141130, 52subcld 9413 . . . . . . 7  |-  ( ph  ->  ( -u S  -  I )  e.  CC )
1428, 25, 141subaddd 9431 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( E  +  ( -u S  -  I
) )  =  X ) )
143133oveq1d 6098 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( ( E  -  S )  -  I ) )
14425, 130, 52addsubassd 9433 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
145143, 144eqtr3d 2472 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
146145eqeq1d 2446 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  -  I )  =  X  <-> 
( E  +  (
-u S  -  I
) )  =  X ) )
147142, 146bitr4d 249 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( ( E  -  S )  -  I )  =  X ) )
148 eqcom 2440 . . . . 5  |-  ( ( ( E  -  S
)  -  I )  =  X  <->  X  =  ( ( E  -  S )  -  I
) )
149147, 148syl6bb 254 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  X  =  (
( E  -  S
)  -  I ) ) )
150140, 149orbi12d 692 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  <->  ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) ) ) )
15135, 124addcld 9109 . . . . . . 7  |-  ( ph  ->  ( S  +  J
)  e.  CC )
1528, 25, 151subaddd 9431 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( E  +  ( S  +  J ) )  =  X ) )
15325, 35, 124addassd 9112 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  +  J
)  =  ( E  +  ( S  +  J ) ) )
154153eqeq1d 2446 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  +  J )  =  X  <-> 
( E  +  ( S  +  J ) )  =  X ) )
155152, 154bitr4d 249 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( ( E  +  S )  +  J
)  =  X ) )
156 eqcom 2440 . . . . 5  |-  ( ( ( E  +  S
)  +  J )  =  X  <->  X  =  ( ( E  +  S )  +  J
) )
157155, 156syl6bb 254 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
X  =  ( ( E  +  S )  +  J ) ) )
15835, 124subcld 9413 . . . . . . 7  |-  ( ph  ->  ( S  -  J
)  e.  CC )
1598, 25, 158subaddd 9431 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( E  +  ( S  -  J ) )  =  X ) )
16025, 35, 124addsubassd 9433 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  -  J
)  =  ( E  +  ( S  -  J ) ) )
161160eqeq1d 2446 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  -  J )  =  X  <-> 
( E  +  ( S  -  J ) )  =  X ) )
162159, 161bitr4d 249 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( ( E  +  S )  -  J
)  =  X ) )
163 eqcom 2440 . . . . 5  |-  ( ( ( E  +  S
)  -  J )  =  X  <->  X  =  ( ( E  +  S )  -  J
) )
164162, 163syl6bb 254 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
X  =  ( ( E  +  S )  -  J ) ) )
165157, 164orbi12d 692 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( S  +  J
)  \/  ( X  -  E )  =  ( S  -  J
) )  <->  ( X  =  ( ( E  +  S )  +  J )  \/  X  =  ( ( E  +  S )  -  J ) ) ) )
166150, 165orbi12d 692 . 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 272 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   NNcn 10002   2c2 10051   3c3 10052   4c4 10053   5c5 10054   6c6 10055   7c7 10056   8c8 10057   9c9 10058   NN0cn0 10223   ZZcz 10284  ;cdc 10384   ^cexp 11384   sqrcsqr 12040    ^ c ccxp 20455
This theorem is referenced by:  quartfull  24798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-dvds 12855  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456  df-cxp 20457
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