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Theorem quart 22920
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 28194) if all the substitutions are performed. This is Metamath 100 proof #46. (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 6298 . . . . 5  |-  ( ph  ->  ( X  -  E
)  =  ( X  -  -u ( A  / 
4 ) ) )
11 4cn 10609 . . . . . . . 8  |-  4  e.  CC
1211a1i 11 . . . . . . 7  |-  ( ph  ->  4  e.  CC )
13 4ne0 10628 . . . . . . . 8  |-  4  =/=  0
1413a1i 11 . . . . . . 7  |-  ( ph  ->  4  =/=  0 )
151, 12, 14divcld 10316 . . . . . 6  |-  ( ph  ->  ( A  /  4
)  e.  CC )
168, 15subnegd 9933 . . . . 5  |-  ( ph  ->  ( X  -  -u ( A  /  4 ) )  =  ( X  +  ( A  /  4
) ) )
1710, 16eqtrd 2508 . . . 4  |-  ( ph  ->  ( X  -  E
)  =  ( X  +  ( A  / 
4 ) ) )
181, 2, 3, 4, 5, 6, 7, 8, 17quart1 22915 . . 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
) ) )
1918eqeq1d 2469 . 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 ) )
201, 2, 3, 4, 5, 6, 7quart1cl 22913 . . . 4  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
2120simp1d 1008 . . 3  |-  ( ph  ->  P  e.  CC )
2220simp2d 1009 . . 3  |-  ( ph  ->  Q  e.  CC )
2315negcld 9913 . . . . 5  |-  ( ph  -> 
-u ( A  / 
4 )  e.  CC )
249, 23eqeltrd 2555 . . . 4  |-  ( ph  ->  E  e.  CC )
258, 24subcld 9926 . . 3  |-  ( ph  ->  ( X  -  E
)  e.  CC )
26 quart.u . . . . 5  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
27 quart.v . . . . 5  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
28 quart.w . . . . 5  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
29 quart.s . . . . 5  |-  ( ph  ->  S  =  ( ( sqr `  M )  /  2 ) )
30 quart.m . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  T )  +  ( U  /  T ) )  /  3 ) )
31 quart.t . . . . 5  |-  ( ph  ->  T  =  ( ( ( V  +  W
)  /  2 )  ^c  ( 1  /  3 ) ) )
32 quart.t0 . . . . 5  |-  ( ph  ->  T  =/=  0 )
331, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32quartlem3 22918 . . . 4  |-  ( ph  ->  ( S  e.  CC  /\  M  e.  CC  /\  T  e.  CC )
)
3433simp1d 1008 . . 3  |-  ( ph  ->  S  e.  CC )
3529oveq2d 6298 . . . . . 6  |-  ( ph  ->  ( 2  x.  S
)  =  ( 2  x.  ( ( sqr `  M )  /  2
) ) )
3633simp2d 1009 . . . . . . . 8  |-  ( ph  ->  M  e.  CC )
3736sqrtcld 13227 . . . . . . 7  |-  ( ph  ->  ( sqr `  M
)  e.  CC )
38 2cnd 10604 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
39 2ne0 10624 . . . . . . . 8  |-  2  =/=  0
4039a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
4137, 38, 40divcan2d 10318 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( sqr `  M
)  /  2 ) )  =  ( sqr `  M ) )
4235, 41eqtrd 2508 . . . . 5  |-  ( ph  ->  ( 2  x.  S
)  =  ( sqr `  M ) )
4342oveq1d 6297 . . . 4  |-  ( ph  ->  ( ( 2  x.  S ) ^ 2 )  =  ( ( sqr `  M ) ^ 2 ) )
4436sqsqrtd 13229 . . . 4  |-  ( ph  ->  ( ( sqr `  M
) ^ 2 )  =  M )
4543, 44eqtr2d 2509 . . 3  |-  ( ph  ->  M  =  ( ( 2  x.  S ) ^ 2 ) )
46 quart.m0 . . 3  |-  ( ph  ->  M  =/=  0 )
47 quart.i . . . . 5  |-  ( ph  ->  I  =  ( sqr `  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) ) )
48 quart.j . . . . 5  |-  ( ph  ->  J  =  ( sqr `  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) ) )
491, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28, 29, 30, 31, 32, 46, 47, 48quartlem4 22919 . . . 4  |-  ( ph  ->  ( S  =/=  0  /\  I  e.  CC  /\  J  e.  CC ) )
5049simp2d 1009 . . 3  |-  ( ph  ->  I  e.  CC )
5147oveq1d 6297 . . . 4  |-  ( ph  ->  ( I ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
5234sqcld 12272 . . . . . . . 8  |-  ( ph  ->  ( S ^ 2 )  e.  CC )
5352negcld 9913 . . . . . . 7  |-  ( ph  -> 
-u ( S ^
2 )  e.  CC )
5421halfcld 10779 . . . . . . 7  |-  ( ph  ->  ( P  /  2
)  e.  CC )
5553, 54subcld 9926 . . . . . 6  |-  ( ph  ->  ( -u ( S ^ 2 )  -  ( P  /  2
) )  e.  CC )
5622, 12, 14divcld 10316 . . . . . . 7  |-  ( ph  ->  ( Q  /  4
)  e.  CC )
5749simp1d 1008 . . . . . . 7  |-  ( ph  ->  S  =/=  0 )
5856, 34, 57divcld 10316 . . . . . 6  |-  ( ph  ->  ( ( Q  / 
4 )  /  S
)  e.  CC )
5955, 58addcld 9611 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) )  e.  CC )
6059sqsqrtd 13229 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  +  ( ( Q  /  4
)  /  S ) ) )
6151, 60eqtrd 2508 . . 3  |-  ( ph  ->  ( I ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  +  ( ( Q  /  4 )  /  S ) ) )
6220simp3d 1010 . . 3  |-  ( ph  ->  R  e.  CC )
63 1cnd 9608 . . . . 5  |-  ( ph  ->  1  e.  CC )
64 3z 10893 . . . . . 6  |-  3  e.  ZZ
65 1exp 12159 . . . . . 6  |-  ( 3  e.  ZZ  ->  (
1 ^ 3 )  =  1 )
6664, 65mp1i 12 . . . . 5  |-  ( ph  ->  ( 1 ^ 3 )  =  1 )
6733simp3d 1010 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  CC )
6867mulid2d 9610 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  T
)  =  T )
6968oveq2d 6298 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  +  ( 1  x.  T ) )  =  ( ( 2  x.  P )  +  T ) )
7068oveq2d 6298 . . . . . . . . 9  |-  ( ph  ->  ( U  /  (
1  x.  T ) )  =  ( U  /  T ) )
7169, 70oveq12d 6300 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  =  ( ( ( 2  x.  P
)  +  T )  +  ( U  /  T ) ) )
7271oveq1d 6297 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) )  /  3
)  =  ( ( ( ( 2  x.  P )  +  T
)  +  ( U  /  T ) )  /  3 ) )
7372negeqd 9810 . . . . . 6  |-  ( ph  -> 
-u ( ( ( ( 2  x.  P
)  +  ( 1  x.  T ) )  +  ( U  / 
( 1  x.  T
) ) )  / 
3 )  =  -u ( ( ( ( 2  x.  P )  +  T )  +  ( U  /  T
) )  /  3
) )
7430, 73eqtr4d 2511 . . . . 5  |-  ( ph  ->  M  =  -u (
( ( ( 2  x.  P )  +  ( 1  x.  T
) )  +  ( U  /  ( 1  x.  T ) ) )  /  3 ) )
75 oveq1 6289 . . . . . . . 8  |-  ( x  =  1  ->  (
x ^ 3 )  =  ( 1 ^ 3 ) )
7675eqeq1d 2469 . . . . . . 7  |-  ( x  =  1  ->  (
( x ^ 3 )  =  1  <->  (
1 ^ 3 )  =  1 ) )
77 oveq1 6289 . . . . . . . . . . . 12  |-  ( x  =  1  ->  (
x  x.  T )  =  ( 1  x.  T ) )
7877oveq2d 6298 . . . . . . . . . . 11  |-  ( x  =  1  ->  (
( 2  x.  P
)  +  ( x  x.  T ) )  =  ( ( 2  x.  P )  +  ( 1  x.  T
) ) )
7977oveq2d 6298 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( U  /  ( x  x.  T ) )  =  ( U  /  (
1  x.  T ) ) )
8078, 79oveq12d 6300 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  ( x  x.  T ) ) )  =  ( ( ( 2  x.  P )  +  ( 1  x.  T ) )  +  ( U  /  (
1  x.  T ) ) ) )
8180oveq1d 6297 . . . . . . . . 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 ) )
8281negeqd 9810 . . . . . . . 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 ) )
8382eqeq2d 2481 . . . . . . 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
) ) )
8476, 83anbi12d 710 . . . . . 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 ) ) ) )
8584rspcev 3214 . . . . 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
) ) )
8663, 66, 74, 85syl12anc 1226 . . . 4  |-  ( ph  ->  E. x  e.  CC  ( ( x ^
3 )  =  1  /\  M  =  -u ( ( ( ( 2  x.  P )  +  ( x  x.  T ) )  +  ( U  /  (
x  x.  T ) ) )  /  3
) ) )
87 2cn 10602 . . . . . 6  |-  2  e.  CC
88 mulcl 9572 . . . . . 6  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
8987, 21, 88sylancr 663 . . . . 5  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
9021sqcld 12272 . . . . . 6  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
91 mulcl 9572 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
9211, 62, 91sylancr 663 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
9390, 92subcld 9926 . . . . 5  |-  ( ph  ->  ( ( P ^
2 )  -  (
4  x.  R ) )  e.  CC )
9422sqcld 12272 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
9594negcld 9913 . . . . 5  |-  ( ph  -> 
-u ( Q ^
2 )  e.  CC )
9631oveq1d 6297 . . . . . 6  |-  ( ph  ->  ( T ^ 3 )  =  ( ( ( ( V  +  W )  /  2
)  ^c  ( 1  /  3 ) ) ^ 3 ) )
971, 2, 3, 4, 1, 9, 5, 6, 7, 26, 27, 28quartlem2 22917 . . . . . . . . . 10  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
9897simp2d 1009 . . . . . . . . 9  |-  ( ph  ->  V  e.  CC )
9997simp3d 1010 . . . . . . . . 9  |-  ( ph  ->  W  e.  CC )
10098, 99addcld 9611 . . . . . . . 8  |-  ( ph  ->  ( V  +  W
)  e.  CC )
101100halfcld 10779 . . . . . . 7  |-  ( ph  ->  ( ( V  +  W )  /  2
)  e.  CC )
102 3nn 10690 . . . . . . 7  |-  3  e.  NN
103 cxproot 22799 . . . . . . 7  |-  ( ( ( ( V  +  W )  /  2
)  e.  CC  /\  3  e.  NN )  ->  ( ( ( ( V  +  W )  /  2 )  ^c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
104101, 102, 103sylancl 662 . . . . . 6  |-  ( ph  ->  ( ( ( ( V  +  W )  /  2 )  ^c  ( 1  / 
3 ) ) ^
3 )  =  ( ( V  +  W
)  /  2 ) )
10596, 104eqtrd 2508 . . . . 5  |-  ( ph  ->  ( T ^ 3 )  =  ( ( V  +  W )  /  2 ) )
10628oveq1d 6297 . . . . . 6  |-  ( ph  ->  ( W ^ 2 )  =  ( ( sqr `  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) ) ^ 2 ) )
10798sqcld 12272 . . . . . . . 8  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
10897simp1d 1008 . . . . . . . . . 10  |-  ( ph  ->  U  e.  CC )
109 3nn0 10809 . . . . . . . . . 10  |-  3  e.  NN0
110 expcl 12148 . . . . . . . . . 10  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
111108, 109, 110sylancl 662 . . . . . . . . 9  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
112 mulcl 9572 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
11311, 111, 112sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
114107, 113subcld 9926 . . . . . . 7  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
115114sqsqrtd 13229 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) ) ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
116106, 115eqtrd 2508 . . . . 5  |-  ( ph  ->  ( W ^ 2 )  =  ( ( V ^ 2 )  -  ( 4  x.  ( U ^ 3 ) ) ) )
11721, 22, 62, 26, 27quartlem1 22916 . . . . . 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 ) ) ) ) )
118117simpld 459 . . . . 5  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
119117simprd 463 . . . . 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 ) ) ) )
12089, 93, 95, 36, 67, 105, 99, 116, 118, 119, 32mcubic 22906 . . . 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
) ) ) )
12186, 120mpbird 232 . . 3  |-  ( ph  ->  ( ( ( M ^ 3 )  +  ( ( 2  x.  P )  x.  ( M ^ 2 ) ) )  +  ( ( ( ( P ^
2 )  -  (
4  x.  R ) )  x.  M )  +  -u ( Q ^
2 ) ) )  =  0 )
12249simp3d 1010 . . 3  |-  ( ph  ->  J  e.  CC )
12348oveq1d 6297 . . . 4  |-  ( ph  ->  ( J ^ 2 )  =  ( ( sqr `  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 ) )
12455, 58subcld 9926 . . . . 5  |-  ( ph  ->  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) )  e.  CC )
125124sqsqrtd 13229 . . . 4  |-  ( ph  ->  ( ( sqr `  (
( -u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) ) ^ 2 )  =  ( ( -u ( S ^ 2 )  -  ( P  /  2
) )  -  (
( Q  /  4
)  /  S ) ) )
126123, 125eqtrd 2508 . . 3  |-  ( ph  ->  ( J ^ 2 )  =  ( (
-u ( S ^
2 )  -  ( P  /  2 ) )  -  ( ( Q  /  4 )  /  S ) ) )
12721, 22, 25, 34, 45, 46, 50, 61, 62, 121, 122, 126dquart 22912 . 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 ) ) ) ) )
12834negcld 9913 . . . . . . . 8  |-  ( ph  -> 
-u S  e.  CC )
129128, 50addcld 9611 . . . . . . 7  |-  ( ph  ->  ( -u S  +  I )  e.  CC )
1308, 24, 129subaddd 9944 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( E  +  ( -u S  +  I
) )  =  X ) )
13124, 34negsubd 9932 . . . . . . . . 9  |-  ( ph  ->  ( E  +  -u S )  =  ( E  -  S ) )
132131oveq1d 6297 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( ( E  -  S )  +  I ) )
13324, 128, 50addassd 9614 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
134132, 133eqtr3d 2510 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  +  I
)  =  ( E  +  ( -u S  +  I ) ) )
135134eqeq1d 2469 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  +  I )  =  X  <-> 
( E  +  (
-u S  +  I
) )  =  X ) )
136130, 135bitr4d 256 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  ( ( E  -  S )  +  I )  =  X ) )
137 eqcom 2476 . . . . 5  |-  ( ( ( E  -  S
)  +  I )  =  X  <->  X  =  ( ( E  -  S )  +  I
) )
138136, 137syl6bb 261 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  +  I
)  <->  X  =  (
( E  -  S
)  +  I ) ) )
139128, 50subcld 9926 . . . . . . 7  |-  ( ph  ->  ( -u S  -  I )  e.  CC )
1408, 24, 139subaddd 9944 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( E  +  ( -u S  -  I
) )  =  X ) )
141131oveq1d 6297 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( ( E  -  S )  -  I ) )
14224, 128, 50addsubassd 9946 . . . . . . . 8  |-  ( ph  ->  ( ( E  +  -u S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
143141, 142eqtr3d 2510 . . . . . . 7  |-  ( ph  ->  ( ( E  -  S )  -  I
)  =  ( E  +  ( -u S  -  I ) ) )
144143eqeq1d 2469 . . . . . 6  |-  ( ph  ->  ( ( ( E  -  S )  -  I )  =  X  <-> 
( E  +  (
-u S  -  I
) )  =  X ) )
145140, 144bitr4d 256 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  ( ( E  -  S )  -  I )  =  X ) )
146 eqcom 2476 . . . . 5  |-  ( ( ( E  -  S
)  -  I )  =  X  <->  X  =  ( ( E  -  S )  -  I
) )
147145, 146syl6bb 261 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  (
-u S  -  I
)  <->  X  =  (
( E  -  S
)  -  I ) ) )
148138, 147orbi12d 709 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( -u S  +  I )  \/  ( X  -  E )  =  ( -u S  -  I ) )  <->  ( X  =  ( ( E  -  S )  +  I )  \/  X  =  ( ( E  -  S )  -  I ) ) ) )
14934, 122addcld 9611 . . . . . . 7  |-  ( ph  ->  ( S  +  J
)  e.  CC )
1508, 24, 149subaddd 9944 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( E  +  ( S  +  J ) )  =  X ) )
15124, 34, 122addassd 9614 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  +  J
)  =  ( E  +  ( S  +  J ) ) )
152151eqeq1d 2469 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  +  J )  =  X  <-> 
( E  +  ( S  +  J ) )  =  X ) )
153150, 152bitr4d 256 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
( ( E  +  S )  +  J
)  =  X ) )
154 eqcom 2476 . . . . 5  |-  ( ( ( E  +  S
)  +  J )  =  X  <->  X  =  ( ( E  +  S )  +  J
) )
155153, 154syl6bb 261 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  +  J )  <-> 
X  =  ( ( E  +  S )  +  J ) ) )
15634, 122subcld 9926 . . . . . . 7  |-  ( ph  ->  ( S  -  J
)  e.  CC )
1578, 24, 156subaddd 9944 . . . . . 6  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( E  +  ( S  -  J ) )  =  X ) )
15824, 34, 122addsubassd 9946 . . . . . . 7  |-  ( ph  ->  ( ( E  +  S )  -  J
)  =  ( E  +  ( S  -  J ) ) )
159158eqeq1d 2469 . . . . . 6  |-  ( ph  ->  ( ( ( E  +  S )  -  J )  =  X  <-> 
( E  +  ( S  -  J ) )  =  X ) )
160157, 159bitr4d 256 . . . . 5  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
( ( E  +  S )  -  J
)  =  X ) )
161 eqcom 2476 . . . . 5  |-  ( ( ( E  +  S
)  -  J )  =  X  <->  X  =  ( ( E  +  S )  -  J
) )
162160, 161syl6bb 261 . . . 4  |-  ( ph  ->  ( ( X  -  E )  =  ( S  -  J )  <-> 
X  =  ( ( E  +  S )  -  J ) ) )
163155, 162orbi12d 709 . . 3  |-  ( ph  ->  ( ( ( X  -  E )  =  ( S  +  J
)  \/  ( X  -  E )  =  ( S  -  J
) )  <->  ( X  =  ( ( E  +  S )  +  J )  \/  X  =  ( ( E  +  S )  -  J ) ) ) )
164148, 163orbi12d 709 . 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 ) ) ) ) )
16519, 127, 1643bitrd 279 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802    / cdiv 10202   NNcn 10532   2c2 10581   3c3 10582   4c4 10583   5c5 10584   6c6 10585   7c7 10586   8c8 10587   9c9 10588   NN0cn0 10791   ZZcz 10860  ;cdc 10972   ^cexp 12130   sqrcsqrt 13025    ^c ccxp 22671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-dvds 13844  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672  df-cxp 22673
This theorem is referenced by:  quartfull  28194
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