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Theorem quart 20120
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 23059) 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
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 5808 . . . . 5  |-  ( ph  ->  ( X  -  E
)  =  ( X  -  -u ( A  / 
4 ) ) )
11 4cn 9788 . . . . . . . 8  |-  4  e.  CC
1211a1i 12 . . . . . . 7  |-  ( ph  ->  4  e.  CC )
13 4nn 9847 . . . . . . . . 9  |-  4  e.  NN
1413nnne0i 9748 . . . . . . . 8  |-  4  =/=  0
1514a1i 12 . . . . . . 7  |-  ( ph  ->  4  =/=  0 )
161, 12, 15divcld 9504 . . . . . 6  |-  ( ph  ->  ( A  /  4
)  e.  CC )
178, 16subnegd 9132 . . . . 5  |-  ( ph  ->  ( X  -  -u ( A  /  4 ) )  =  ( X  +  ( A  /  4
) ) )
1810, 17eqtrd 2290 . . . 4  |-  ( ph  ->  ( X  -  E
)  =  ( X  +  ( A  / 
4 ) ) )
191, 2, 3, 4, 5, 6, 7, 8, 18quart1 20115 . . 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 2266 . 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 20113 . . . 4  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
2221simp1d 972 . . 3  |-  ( ph  ->  P  e.  CC )
2321simp2d 973 . . 3  |-  ( ph  ->  Q  e.  CC )
2416negcld 9112 . . . . 5  |-  ( ph  -> 
-u ( A  / 
4 )  e.  CC )
259, 24eqeltrd 2332 . . . 4  |-  ( ph  ->  E  e.  CC )
268, 25subcld 9125 . . 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 20118 . . . 4  |-  ( ph  ->  ( S  e.  CC  /\  M  e.  CC  /\  T  e.  CC )
)
3534simp1d 972 . . 3  |-  ( ph  ->  S  e.  CC )
3630oveq2d 5808 . . . . . 6  |-  ( ph  ->  ( 2  x.  S
)  =  ( 2  x.  ( ( sqr `  M )  /  2
) ) )
3734simp2d 973 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
3837sqrcld 11885 . . . . . . 7  |-  ( ph  ->  ( sqr `  M
)  e.  CC )
39 2cn 9784 . . . . . . . 8  |-  2  e.  CC
4039a1i 12 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
41 2ne0 9797 . . . . . . . 8  |-  2  =/=  0
4241a1i 12 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
4338, 40, 42divcan2d 9506 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( sqr `  M
)  /  2 ) )  =  ( sqr `  M ) )
4436, 43eqtrd 2290 . . . . 5  |-  ( ph  ->  ( 2  x.  S
)  =  ( sqr `  M ) )
4544oveq1d 5807 . . . 4  |-  ( ph  ->  ( ( 2  x.  S ) ^ 2 )  =  ( ( sqr `  M ) ^ 2 ) )
4637sqsqrd 11887 . . . 4  |-  ( ph  ->  ( ( sqr `  M
) ^ 2 )  =  M )
4745, 46eqtr2d 2291 . . 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 20119 . . . 4  |-  ( ph  ->  ( S  =/=  0  /\  I  e.  CC  /\  J  e.  CC ) )
5251simp2d 973 . . 3  |-  ( ph  ->  I  e.  CC )
5349oveq1d 5807 . . . 4  |-  ( ph  ->  ( I ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
5435sqcld 11210 . . . . . . . 8  |-  ( ph  ->  ( S ^ 2 )  e.  CC )
5554negcld 9112 . . . . . . 7  |-  ( ph  -> 
-u ( S ^
2 )  e.  CC )
5622halfcld 9924 . . . . . . 7  |-  ( ph  ->  ( P  /  2
)  e.  CC )
5755, 56subcld 9125 . . . . . 6  |-  ( ph  ->  ( -u ( S ^ 2 )  -  ( P  /  2
) )  e.  CC )
5823, 12, 15divcld 9504 . . . . . . 7  |-  ( ph  ->  ( Q  /  4
)  e.  CC )
5951simp1d 972 . . . . . . 7  |-  ( ph  ->  S  =/=  0 )
6058, 35, 59divcld 9504 . . . . . 6  |-  ( ph  ->  ( ( Q  / 
4 )  /  S
)  e.  CC )
6157, 60addcld 8822 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) )  e.  CC )
6261sqsqrd 11887 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) )
6353, 62eqtrd 2290 . . 3  |-  ( ph  ->  ( I ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) )
6421simp3d 974 . . 3  |-  ( ph  ->  R  e.  CC )
65 ax-1cn 8763 . . . . . 6  |-  1  e.  CC
6665a1i 12 . . . . 5  |-  ( ph  ->  1  e.  CC )
67 3nn 9846 . . . . . . 7  |-  3  e.  NN
6867nnzi 10015 . . . . . 6  |-  3  e.  ZZ
69 1exp 11098 . . . . . 6  |-  ( 3  e.  ZZ  ->  (
1 ^ 3 )  =  1 )
7068, 69mp1i 13 . . . . 5  |-  ( ph  ->  ( 1 ^ 3 )  =  1 )
7134simp3d 974 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  CC )
7271mulid2d 8821 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  T
)  =  T )
7372oveq2d 5808 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  +  ( 1  x.  T ) )  =  ( ( 2  x.  P )  +  T ) )
7472oveq2d 5808 . . . . . . . . 9  |-  ( ph  ->  ( U  /  (
1  x.  T ) )  =  ( U  /  T ) )
7573, 74oveq12d 5810 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  =  ( ( ( 2  x.  P
)  +  T )  +  ( U  /  T ) ) )
7675oveq1d 5807 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
)  =  ( ( ( ( 2  x.  P )  +  T
)  +  ( U  /  T ) )  /  3 ) )
7776negeqd 9014 . . . . . 6  |-  ( ph  -> 
-u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 )  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T
) )  /  3
) )
7831, 77eqtr4d 2293 . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  /  3 ) )
79 oveq1 5799 . . . . . . . 8  |-  ( x  =  1  ->  (
x ^ 3 )  =  ( 1 ^ 3 ) )
8079eqeq1d 2266 . . . . . . 7  |-  ( x  =  1  ->  (
( x ^ 3 )  =  1  <->  (
1 ^ 3 )  =  1 ) )
81 oveq1 5799 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
x  x.  T )  =  ( 1  x.  T ) )
8281oveq2d 5808 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
( 2  x.  P
)  +  ( x  x.  T ) )  =  ( ( 2  x.  P )  +  ( 1  x.  T
) ) )
8381oveq2d 5808 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( U  /  ( x  x.  T ) )  =  ( U  /  (
1  x.  T ) ) )
8482, 83oveq12d 5810 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  ( x  x.  T ) ) )  =  ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) ) )
8584oveq1d 5807 . . . . . . . . 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 9014 . . . . . . . 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 2269 . . . . . . 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 694 . . . . . 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 ) ) ) )
8988rcla4ev 2859 . . . . 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 1185 . . . 4  |-  ( ph  ->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) )
91 mulcl 8789 . . . . . 6  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
9239, 22, 91sylancr 647 . . . . 5  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
9322sqcld 11210 . . . . . 6  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
94 mulcl 8789 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
9511, 64, 94sylancr 647 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
9693, 95subcld 9125 . . . . 5  |-  ( ph  ->  ( ( P ^
2 )  -  (
4  x.  R ) )  e.  CC )
9723sqcld 11210 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
9897negcld 9112 . . . . 5  |-  ( ph  -> 
-u ( Q ^
2 )  e.  CC )
9932oveq1d 5807 . . . . . 6  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( V  +  W )  /  2
)  ^ c  ( 1  /  3 ) ) ^ 3 ) )
1001, 2, 3, 4, 1, 9, 5, 6, 7, 27, 28, 29quartlem2 20117 . . . . . . . . . 10  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
101100simp2d 973 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
102100simp3d 974 . . . . . . . . 9  |-  ( ph  ->  W  e.  CC )
103101, 102addcld 8822 . . . . . . . 8  |-  ( ph  ->  ( V  +  W
)  e.  CC )
104103halfcld 9924 . . . . . . 7  |-  ( ph  ->  ( ( V  +  W )  /  2
)  e.  CC )
105 cxproot 20000 . . . . . . 7  |-  ( ( ( ( V  +  W )  /  2
)  e.  CC  /\  3  e.  NN )  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
106104, 67, 105sylancl 646 . . . . . 6  |-  ( ph  ->  ( ( ( ( V  +  W )  /  2 )  ^ c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
10799, 106eqtrd 2290 . . . . 5  |-  ( ph  ->  ( T ^ 3 )  =  ( ( V  +  W )  /  2 ) )
10829oveq1d 5807 . . . . . 6  |-  ( ph  ->  ( W ^ 2 )  =  ( ( sqr `  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) ) ^ 2 ) )
109101sqcld 11210 . . . . . . . 8  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
110100simp1d 972 . . . . . . . . . 10  |-  ( ph  ->  U  e.  CC )
111 3nn0 9951 . . . . . . . . . 10  |-  3  e.  NN0
112 expcl 11088 . . . . . . . . . 10  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
113110, 111, 112sylancl 646 . . . . . . . . 9  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
114 mulcl 8789 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
11511, 113, 114sylancr 647 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
116109, 115subcld 9125 . . . . . . 7  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
117116sqsqrd 11887 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) ) ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
118108, 117eqtrd 2290 . . . . 5  |-  ( ph  ->  ( W ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
11922, 23, 64, 27, 28quartlem1 20116 . . . . . 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 20106 . . . 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 974 . . 3  |-  ( ph  ->  J  e.  CC )
12550oveq1d 5807 . . . 4  |-  ( ph  ->  ( J ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
12657, 60subcld 9125 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) )  e.  CC )
127126sqsqrd 11887 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) )
128125, 127eqtrd 2290 . . 3  |-  ( ph  ->  ( J ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) )
12922, 23, 26, 35, 47, 48, 52, 63, 64, 123, 124, 128dquart 20112 . 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 9112 . . . . . . . 8  |-  ( ph  -> 
-u S  e.  CC )
131130, 52addcld 8822 . . . . . . 7  |-  ( ph  ->  ( -u S  +  I )  e.  CC )
1328, 25, 131subaddd 9143 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( E  +  ( -u S  +  I
) )  =  X ) )
13325, 35negsubd 9131 . . . . . . . . 9  |-  ( ph  ->  ( E  +  -u S )  =  ( E  -  S ) )
134133oveq1d 5807 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( ( E  -  S )  +  I ) )
13525, 130, 52addassd 8825 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
136134, 135eqtr3d 2292 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
137136eqeq1d 2266 . . . . . 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 2260 . . . . 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 9125 . . . . . . 7  |-  ( ph  ->  ( -u S  -  I )  e.  CC )
1428, 25, 141subaddd 9143 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( E  +  ( -u S  -  I
) )  =  X ) )
143133oveq1d 5807 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( ( E  -  S )  -  I ) )
14425, 130, 52addsubassd 9145 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
145143, 144eqtr3d 2292 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
146145eqeq1d 2266 . . . . . 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 2260 . . . . 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 693 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  <->  ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) ) ) )
15135, 124addcld 8822 . . . . . . 7  |-  ( ph  ->  ( S  +  J
)  e.  CC )
1528, 25, 151subaddd 9143 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( E  +  ( S  +  J ) )  =  X ) )
15325, 35, 124addassd 8825 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  +  J
)  =  ( E  +  ( S  +  J ) ) )
154153eqeq1d 2266 . . . . . 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 2260 . . . . 5  |-  ( ( ( E  +  S
)  +  J )  =  X  <->  X  =  ( ( E  +  S )  +  J
) )
157155, 156syl6bb 254 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
X  =  ( ( E  +  S )  +  J ) ) )
15835, 124subcld 9125 . . . . . . 7  |-  ( ph  ->  ( S  -  J
)  e.  CC )
1598, 25, 158subaddd 9143 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( E  +  ( S  -  J ) )  =  X ) )
16025, 35, 124addsubassd 9145 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  -  J
)  =  ( E  +  ( S  -  J ) ) )
161160eqeq1d 2266 . . . . . 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 2260 . . . . 5  |-  ( ( ( E  +  S
)  -  J )  =  X  <->  X  =  ( ( E  +  S )  -  J
) )
164162, 163syl6bb 254 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
X  =  ( ( E  +  S )  -  J ) ) )
165157, 164orbi12d 693 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( S  +  J
)  \/  ( X  -  E )  =  ( S  -  J
) )  <->  ( X  =  ( ( E  +  S )  +  J )  \/  X  =  ( ( E  +  S )  -  J ) ) ) )
166150, 165orbi12d 693 . 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 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   ` cfv 4673  (class class class)co 5792   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    - cmin 9005   -ucneg 9006    / cdiv 9391   NNcn 9714   2c2 9763   3c3 9764   4c4 9765   5c5 9766   6c6 9767   7c7 9768   8c8 9769   9c9 9770   NN0cn0 9933   ZZcz 9992  ;cdc 10092   ^cexp 11071   sqrcsqr 11684    ^ c ccxp 19876
This theorem is referenced by:  quartfull  23059
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ioc 10628  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-shft 11528  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-limsup 11911  df-clim 11928  df-rlim 11929  df-sum 12125  df-ef 12312  df-sin 12314  df-cos 12315  df-pi 12317  df-divides 12495  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-lp 16831  df-perf 16832  df-cn 16920  df-cnp 16921  df-haus 17006  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cncf 18345  df-limc 19179  df-dv 19180  df-log 19877  df-cxp 19878
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