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Theorem dcubic2 22198
Description: Reverse direction of dcubic 22200. Given a solution  U to the "substitution" quadratic equation  X  =  U  -  M  /  U, show that  X is in the desired form. (Contributed by Mario Carneiro, 25-Apr-2015.)
Hypotheses
Ref Expression
dcubic.c  |-  ( ph  ->  P  e.  CC )
dcubic.d  |-  ( ph  ->  Q  e.  CC )
dcubic.x  |-  ( ph  ->  X  e.  CC )
dcubic.t  |-  ( ph  ->  T  e.  CC )
dcubic.3  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
dcubic.g  |-  ( ph  ->  G  e.  CC )
dcubic.2  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
dcubic.m  |-  ( ph  ->  M  =  ( P  /  3 ) )
dcubic.n  |-  ( ph  ->  N  =  ( Q  /  2 ) )
dcubic.0  |-  ( ph  ->  T  =/=  0 )
dcubic2.u  |-  ( ph  ->  U  e.  CC )
dcubic2.z  |-  ( ph  ->  U  =/=  0 )
dcubic2.2  |-  ( ph  ->  X  =  ( U  -  ( M  /  U ) ) )
dcubic2.x  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )
Assertion
Ref Expression
dcubic2  |-  ( ph  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
Distinct variable groups:    M, r    P, r    ph, r    Q, r    T, r    U, r    X, r
Allowed substitution hints:    G( r)    N( r)

Proof of Theorem dcubic2
StepHypRef Expression
1 dcubic2.u . . . . 5  |-  ( ph  ->  U  e.  CC )
2 dcubic.t . . . . 5  |-  ( ph  ->  T  e.  CC )
3 dcubic.0 . . . . 5  |-  ( ph  ->  T  =/=  0 )
41, 2, 3divcld 10103 . . . 4  |-  ( ph  ->  ( U  /  T
)  e.  CC )
54adantr 462 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  ( U  /  T )  e.  CC )
6 3nn0 10593 . . . . . . 7  |-  3  e.  NN0
76a1i 11 . . . . . 6  |-  ( ph  ->  3  e.  NN0 )
81, 2, 3, 7expdivd 12018 . . . . 5  |-  ( ph  ->  ( ( U  /  T ) ^ 3 )  =  ( ( U ^ 3 )  /  ( T ^
3 ) ) )
98adantr 462 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  ( ( U ^ 3 )  / 
( T ^ 3 ) ) )
10 oveq1 6097 . . . . 5  |-  ( ( U ^ 3 )  =  ( G  -  N )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
11 dcubic.3 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
1211oveq1d 6105 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
13 expcl 11879 . . . . . . . 8  |-  ( ( T  e.  CC  /\  3  e.  NN0 )  -> 
( T ^ 3 )  e.  CC )
142, 6, 13sylancl 657 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  e.  CC )
15 3z 10675 . . . . . . . . 9  |-  3  e.  ZZ
1615a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  ZZ )
172, 3, 16expne0d 12010 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =/=  0 )
1814, 17dividd 10101 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  1 )
1912, 18eqtr3d 2475 . . . . 5  |-  ( ph  ->  ( ( G  -  N )  /  ( T ^ 3 ) )  =  1 )
2010, 19sylan9eqr 2495 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  1 )
219, 20eqtrd 2473 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  1 )
22 dcubic2.2 . . . . 5  |-  ( ph  ->  X  =  ( U  -  ( M  /  U ) ) )
231, 2, 3divcan1d 10104 . . . . . 6  |-  ( ph  ->  ( ( U  /  T )  x.  T
)  =  U )
2423oveq2d 6106 . . . . . 6  |-  ( ph  ->  ( M  /  (
( U  /  T
)  x.  T ) )  =  ( M  /  U ) )
2523, 24oveq12d 6108 . . . . 5  |-  ( ph  ->  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) )  =  ( U  -  ( M  /  U
) ) )
2622, 25eqtr4d 2476 . . . 4  |-  ( ph  ->  X  =  ( ( ( U  /  T
)  x.  T )  -  ( M  / 
( ( U  /  T )  x.  T
) ) ) )
2726adantr 462 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
28 oveq1 6097 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
r ^ 3 )  =  ( ( U  /  T ) ^
3 ) )
2928eqeq1d 2449 . . . . 5  |-  ( r  =  ( U  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( U  /  T
) ^ 3 )  =  1 ) )
30 oveq1 6097 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  (
r  x.  T )  =  ( ( U  /  T )  x.  T ) )
3130oveq2d 6106 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( U  /  T
)  x.  T ) ) )
3230, 31oveq12d 6108 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) )  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
3332eqeq2d 2452 . . . . 5  |-  ( r  =  ( U  /  T )  ->  ( X  =  ( (
r  x.  T )  -  ( M  / 
( r  x.  T
) ) )  <->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) ) ) )
3429, 33anbi12d 705 . . . 4  |-  ( r  =  ( U  /  T )  ->  (
( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) )  <->  ( ( ( U  /  T ) ^ 3 )  =  1  /\  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) ) ) ) )
3534rspcev 3070 . . 3  |-  ( ( ( U  /  T
)  e.  CC  /\  ( ( ( U  /  T ) ^
3 )  =  1  /\  X  =  ( ( ( U  /  T )  x.  T
)  -  ( M  /  ( ( U  /  T )  x.  T ) ) ) ) )  ->  E. r  e.  CC  ( ( r ^ 3 )  =  1  /\  X  =  ( ( r  x.  T )  -  ( M  /  ( r  x.  T ) ) ) ) )
365, 21, 27, 35syl12anc 1211 . 2  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  E. r  e.  CC  ( ( r ^ 3 )  =  1  /\  X  =  ( ( r  x.  T )  -  ( M  /  ( r  x.  T ) ) ) ) )
37 dcubic.m . . . . . . . 8  |-  ( ph  ->  M  =  ( P  /  3 ) )
38 dcubic.c . . . . . . . . 9  |-  ( ph  ->  P  e.  CC )
39 3cn 10392 . . . . . . . . . 10  |-  3  e.  CC
4039a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  e.  CC )
41 3ne0 10412 . . . . . . . . . 10  |-  3  =/=  0
4241a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  =/=  0 )
4338, 40, 42divcld 10103 . . . . . . . 8  |-  ( ph  ->  ( P  /  3
)  e.  CC )
4437, 43eqeltrd 2515 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
45 dcubic2.z . . . . . . 7  |-  ( ph  ->  U  =/=  0 )
4644, 1, 45divcld 10103 . . . . . 6  |-  ( ph  ->  ( M  /  U
)  e.  CC )
4746negcld 9702 . . . . 5  |-  ( ph  -> 
-u ( M  /  U )  e.  CC )
4847, 2, 3divcld 10103 . . . 4  |-  ( ph  ->  ( -u ( M  /  U )  /  T )  e.  CC )
4948adantr 462 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( M  /  U )  /  T
)  e.  CC )
5047, 2, 3, 7expdivd 12018 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( (
-u ( M  /  U ) ^ 3 )  /  ( T ^ 3 ) ) )
5144, 1, 45divnegd 10116 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  (
-u M  /  U
) )
5251oveq1d 6105 . . . . . . . 8  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u M  /  U ) ^ 3 ) )
5344negcld 9702 . . . . . . . . 9  |-  ( ph  -> 
-u M  e.  CC )
5453, 1, 45, 7expdivd 12018 . . . . . . . 8  |-  ( ph  ->  ( ( -u M  /  U ) ^ 3 )  =  ( (
-u M ^ 3 )  /  ( U ^ 3 ) ) )
5511oveq2d 6106 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
56 dcubic.g . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  e.  CC )
57 dcubic.n . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  =  ( Q  /  2 ) )
58 dcubic.d . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  Q  e.  CC )
5958halfcld 10565 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
6057, 59eqeltrd 2515 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  CC )
61 subsq 11969 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6256, 60, 61syl2anc 656 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6355, 62eqtr4d 2476 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( ( G ^ 2 )  -  ( N ^ 2 ) ) )
64 dcubic.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
6564oveq1d 6105 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( M ^
3 ) )  -  ( N ^ 2 ) ) )
6660sqcld 12002 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
67 expcl 11879 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
6844, 6, 67sylancl 657 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
6966, 68pncan2d 9717 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( N ^ 2 ) )  =  ( M ^
3 ) )
7063, 65, 693eqtrd 2477 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( M ^
3 ) )
7170negeqd 9600 . . . . . . . . . . 11  |-  ( ph  -> 
-u ( ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
7256, 60addcld 9401 . . . . . . . . . . . 12  |-  ( ph  ->  ( G  +  N
)  e.  CC )
7372, 14mulneg1d 9793 . . . . . . . . . . 11  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( ( G  +  N )  x.  ( T ^ 3 ) ) )
74 3nn 10476 . . . . . . . . . . . . 13  |-  3  e.  NN
7574a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  3  e.  NN )
76 2nn 10475 . . . . . . . . . . . . . 14  |-  2  e.  NN
77 1nn0 10591 . . . . . . . . . . . . . 14  |-  1  e.  NN0
78 1nn 10329 . . . . . . . . . . . . . 14  |-  1  e.  NN
79 2t1e2 10466 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  1 )  =  2
8079oveq1i 6100 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
81 2p1e3 10441 . . . . . . . . . . . . . . 15  |-  ( 2  +  1 )  =  3
8280, 81eqtri 2461 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  1 )  +  1 )  =  3
83 1lt2 10484 . . . . . . . . . . . . . 14  |-  1  <  2
8476, 77, 78, 82, 83ndvdsi 13610 . . . . . . . . . . . . 13  |-  -.  2  ||  3
8584a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  -.  2  ||  3
)
86 oexpneg 13591 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN  /\  -.  2  ||  3 )  -> 
( -u M ^ 3 )  =  -u ( M ^ 3 ) )
8744, 75, 85, 86syl3anc 1213 . . . . . . . . . . 11  |-  ( ph  ->  ( -u M ^
3 )  =  -u ( M ^ 3 ) )
8871, 73, 873eqtr4d 2483 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  (
-u M ^ 3 ) )
8988oveq1d 6105 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u M ^ 3 )  / 
( U ^ 3 ) ) )
9072negcld 9702 . . . . . . . . . 10  |-  ( ph  -> 
-u ( G  +  N )  e.  CC )
91 expcl 11879 . . . . . . . . . . 11  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
921, 6, 91sylancl 657 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
931, 45, 16expne0d 12010 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  =/=  0 )
9490, 14, 92, 93div23d 10140 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) ) )
9589, 94eqtr3d 2475 . . . . . . . 8  |-  ( ph  ->  ( ( -u M ^ 3 )  / 
( U ^ 3 ) )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9652, 54, 953eqtrd 2477 . . . . . . 7  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9796oveq1d 6105 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U ) ^
3 )  /  ( T ^ 3 ) )  =  ( ( (
-u ( G  +  N )  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) ) )
9890, 92, 93divcld 10103 . . . . . . 7  |-  ( ph  ->  ( -u ( G  +  N )  / 
( U ^ 3 ) )  e.  CC )
9998, 14, 17divcan4d 10109 . . . . . 6  |-  ( ph  ->  ( ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) )  =  (
-u ( G  +  N )  /  ( U ^ 3 ) ) )
10050, 97, 993eqtrd 2477 . . . . 5  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
101100adantr 462 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
102 oveq1 6097 . . . . . 6  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  (
( U ^ 3 )  /  ( U ^ 3 ) )  =  ( -u ( G  +  N )  /  ( U ^
3 ) ) )
103102eqcomd 2446 . . . . 5  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  ( -u ( G  +  N
)  /  ( U ^ 3 ) )  =  ( ( U ^ 3 )  / 
( U ^ 3 ) ) )
10492, 93dividd 10101 . . . . 5  |-  ( ph  ->  ( ( U ^
3 )  /  ( U ^ 3 ) )  =  1 )
105103, 104sylan9eqr 2495 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  /  ( U ^ 3 ) )  =  1 )
106101, 105eqtrd 2473 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  1 )
10746, 1neg2subd 9732 . . . . . 6  |-  ( ph  ->  ( -u ( M  /  U )  -  -u U )  =  ( U  -  ( M  /  U ) ) )
10822, 107eqtr4d 2476 . . . . 5  |-  ( ph  ->  X  =  ( -u ( M  /  U
)  -  -u U
) )
109108adantr 462 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( -u ( M  /  U )  -  -u U ) )
11047, 2, 3divcan1d 10104 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
111110adantr 462 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
11244, 1, 45divneg2d 10117 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  ( M  /  -u U
) )
113110, 112eqtrd 2473 . . . . . . . 8  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
114113adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
115114oveq2d 6106 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  ( M  /  ( M  /  -u U ) ) )
11644adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  e.  CC )
1171negcld 9702 . . . . . . . 8  |-  ( ph  -> 
-u U  e.  CC )
118117adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  e.  CC )
11973, 71eqtrd 2473 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
120119adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^
3 ) )
12190adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  e.  CC )
12214adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  e.  CC )
123 simpr 458 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =  -u ( G  +  N )
)
12493adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =/=  0 )
125123, 124eqnetrrd 2626 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  =/=  0 )
12617adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  =/=  0 )
127121, 122, 125, 126mulne0d 9984 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =/=  0 )
128120, 127eqnetrrd 2626 . . . . . . . 8  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( M ^ 3 )  =/=  0 )
129 oveq1 6097 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ 3 )  =  ( 0 ^ 3 ) )
130 0exp 11895 . . . . . . . . . . . . 13  |-  ( 3  e.  NN  ->  (
0 ^ 3 )  =  0 )
13174, 130ax-mp 5 . . . . . . . . . . . 12  |-  ( 0 ^ 3 )  =  0
132129, 131syl6eq 2489 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( M ^ 3 )  =  0 )
133132negeqd 9600 . . . . . . . . . 10  |-  ( M  =  0  ->  -u ( M ^ 3 )  = 
-u 0 )
134 neg0 9651 . . . . . . . . . 10  |-  -u 0  =  0
135133, 134syl6eq 2489 . . . . . . . . 9  |-  ( M  =  0  ->  -u ( M ^ 3 )  =  0 )
136135necon3i 2648 . . . . . . . 8  |-  ( -u ( M ^ 3 )  =/=  0  ->  M  =/=  0 )
137128, 136syl 16 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  =/=  0 )
1381, 45negne0d 9713 . . . . . . . 8  |-  ( ph  -> 
-u U  =/=  0
)
139138adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  =/=  0 )
140116, 118, 137, 139ddcand 10123 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  ( M  /  -u U ) )  =  -u U )
141115, 140eqtrd 2473 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  -u U
)
142111, 141oveq12d 6108 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( ( -u ( M  /  U
)  /  T )  x.  T )  -  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) )  =  (
-u ( M  /  U )  -  -u U
) )
143109, 142eqtr4d 2476 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( (
( -u ( M  /  U )  /  T
)  x.  T )  -  ( M  / 
( ( -u ( M  /  U )  /  T )  x.  T
) ) ) )
144 oveq1 6097 . . . . . 6  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r ^ 3 )  =  ( ( -u ( M  /  U
)  /  T ) ^ 3 ) )
145144eqeq1d 2449 . . . . 5  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( -u ( M  /  U )  /  T
) ^ 3 )  =  1 ) )
146 oveq1 6097 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r  x.  T )  =  ( ( -u ( M  /  U
)  /  T )  x.  T ) )
147146oveq2d 6106 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) )
148146, 147oveq12d 6108 . . . . . 6  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) )  =  ( ( (
-u ( M  /  U )  /  T
)  x.  T )  -  ( M  / 
( ( -u ( M  /  U )  /  T )  x.  T
) ) ) )
149148eqeq2d 2452 . . . . 5  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  ( X  =  ( (
r  x.  T )  -  ( M  / 
( r  x.  T
) ) )  <->  X  =  ( ( ( -u ( M  /  U
)  /  T )  x.  T )  -  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) ) ) )
150145, 149anbi12d 705 . . . 4  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) )  <->  ( ( (
-u ( M  /  U )  /  T
) ^ 3 )  =  1  /\  X  =  ( ( (
-u ( M  /  U )  /  T
)  x.  T )  -  ( M  / 
( ( -u ( M  /  U )  /  T )  x.  T
) ) ) ) ) )
151150rspcev 3070 . . 3  |-  ( ( ( -u ( M  /  U )  /  T )  e.  CC  /\  ( ( ( -u ( M  /  U
)  /  T ) ^ 3 )  =  1  /\  X  =  ( ( ( -u ( M  /  U
)  /  T )  x.  T )  -  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) ) ) )  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
15249, 106, 143, 151syl12anc 1211 . 2  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
15392sqcld 12002 . . . . . . 7  |-  ( ph  ->  ( ( U ^
3 ) ^ 2 )  e.  CC )
154153mulid2d 9400 . . . . . 6  |-  ( ph  ->  ( 1  x.  (
( U ^ 3 ) ^ 2 ) )  =  ( ( U ^ 3 ) ^ 2 ) )
15558, 92mulcld 9402 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( U ^ 3 ) )  e.  CC )
156155, 68negsubd 9721 . . . . . 6  |-  ( ph  ->  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^ 3 ) )  =  ( ( Q  x.  ( U ^
3 ) )  -  ( M ^ 3 ) ) )
157154, 156oveq12d 6108 . . . . 5  |-  ( ph  ->  ( ( 1  x.  ( ( U ^
3 ) ^ 2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^
3 ) ) )  =  ( ( ( U ^ 3 ) ^ 2 )  +  ( ( Q  x.  ( U ^ 3 ) )  -  ( M ^ 3 ) ) ) )
158 dcubic2.x . . . . . 6  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )
159 dcubic.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
16038, 58, 159, 2, 11, 56, 64, 37, 57, 3, 1, 45, 22dcubic1lem 22197 . . . . . 6  |-  ( ph  ->  ( ( ( X ^ 3 )  +  ( ( P  x.  X )  +  Q
) )  =  0  <-> 
( ( ( U ^ 3 ) ^
2 )  +  ( ( Q  x.  ( U ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 ) )
161158, 160mpbid 210 . . . . 5  |-  ( ph  ->  ( ( ( U ^ 3 ) ^
2 )  +  ( ( Q  x.  ( U ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 )
162157, 161eqtrd 2473 . . . 4  |-  ( ph  ->  ( ( 1  x.  ( ( U ^
3 ) ^ 2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^
3 ) ) )  =  0 )
163 ax-1cn 9336 . . . . . 6  |-  1  e.  CC
164163a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
165 ax-1ne0 9347 . . . . . 6  |-  1  =/=  0
166165a1i 11 . . . . 5  |-  ( ph  ->  1  =/=  0 )
16768negcld 9702 . . . . 5  |-  ( ph  -> 
-u ( M ^
3 )  e.  CC )
168 2cn 10388 . . . . . 6  |-  2  e.  CC
169 mulcl 9362 . . . . . 6  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( 2  x.  G
)  e.  CC )
170168, 56, 169sylancr 658 . . . . 5  |-  ( ph  ->  ( 2  x.  G
)  e.  CC )
171 sqmul 11925 . . . . . . 7  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
172168, 56, 171sylancr 658 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
17364oveq2d 6106 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( G ^ 2 ) )  =  ( ( 2 ^ 2 )  x.  ( ( N ^
2 )  +  ( M ^ 3 ) ) ) )
174168sqcli 11942 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
175 mulcl 9362 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
176174, 66, 175sylancr 658 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
177 mulcl 9362 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
178174, 68, 177sylancr 658 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
179176, 178subnegd 9722 . . . . . . 7  |-  ( ph  ->  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  -  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
18057oveq2d 6106 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
181168a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  CC )
182 2ne0 10410 . . . . . . . . . . . . 13  |-  2  =/=  0
183182a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
18458, 181, 183divcan2d 10105 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
185180, 184eqtrd 2473 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
186185oveq1d 6105 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( Q ^ 2 ) )
187 sqmul 11925 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  N  e.  CC )  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
188168, 60, 187sylancr 658 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
189186, 188eqtr3d 2475 . . . . . . . 8  |-  ( ph  ->  ( Q ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
190167mulid2d 9400 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  x.  -u ( M ^ 3 ) )  =  -u ( M ^
3 ) )
191190oveq2d 6106 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  ( 4  x.  -u ( M ^
3 ) ) )
192 4cn 10395 . . . . . . . . . . 11  |-  4  e.  CC
193 mulneg2 9778 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
194192, 68, 193sylancr 658 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
195191, 194eqtrd 2473 . . . . . . . . 9  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
4  x.  ( M ^ 3 ) ) )
196 sq2 11958 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
197196oveq1i 6100 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( M ^
3 ) )  =  ( 4  x.  ( M ^ 3 ) )
198197negeqi 9599 . . . . . . . . 9  |-  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) )
199195, 198syl6eqr 2491 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )
200189, 199oveq12d 6108 . . . . . . 7  |-  ( ph  ->  ( ( Q ^
2 )  -  (
4  x.  ( 1  x.  -u ( M ^
3 ) ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  -  -u ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
201174a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ 2 )  e.  CC )
202201, 66, 68adddid 9406 . . . . . . 7  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
203179, 200, 2023eqtr4rd 2484 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
204172, 173, 2033eqtrd 2477 . . . . 5  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
205164, 166, 58, 167, 92, 170, 204quad2 22193 . . . 4  |-  ( ph  ->  ( ( ( 1  x.  ( ( U ^ 3 ) ^
2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^ 3 ) ) )  =  0  <->  (
( U ^ 3 )  =  ( (
-u Q  +  ( 2  x.  G ) )  /  ( 2  x.  1 ) )  \/  ( U ^
3 )  =  ( ( -u Q  -  ( 2  x.  G
) )  /  (
2  x.  1 ) ) ) ) )
206162, 205mpbid 210 . . 3  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  \/  ( U ^ 3 )  =  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) ) ) )
20779oveq2i 6101 . . . . . 6  |-  ( (
-u Q  +  ( 2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  +  ( 2  x.  G ) )  /  2 )
20858negcld 9702 . . . . . . . 8  |-  ( ph  -> 
-u Q  e.  CC )
209208, 170, 181, 183divdird 10141 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  +  ( ( 2  x.  G
)  /  2 ) ) )
21057negeqd 9600 . . . . . . . . 9  |-  ( ph  -> 
-u N  =  -u ( Q  /  2
) )
21158, 181, 183divnegd 10116 . . . . . . . . 9  |-  ( ph  -> 
-u ( Q  / 
2 )  =  (
-u Q  /  2
) )
212210, 211eqtr2d 2474 . . . . . . . 8  |-  ( ph  ->  ( -u Q  / 
2 )  =  -u N )
21356, 181, 183divcan3d 10108 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  G )  /  2
)  =  G )
214212, 213oveq12d 6108 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  +  ( ( 2  x.  G )  /  2
) )  =  (
-u N  +  G
) )
21560negcld 9702 . . . . . . . . 9  |-  ( ph  -> 
-u N  e.  CC )
216215, 56addcomd 9567 . . . . . . . 8  |-  ( ph  ->  ( -u N  +  G )  =  ( G  +  -u N
) )
21756, 60negsubd 9721 . . . . . . . 8  |-  ( ph  ->  ( G  +  -u N )  =  ( G  -  N ) )
218216, 217eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( -u N  +  G )  =  ( G  -  N ) )
219209, 214, 2183eqtrd 2477 . . . . . 6  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( G  -  N ) )
220207, 219syl5eq 2485 . . . . 5  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  ( G  -  N ) )
221220eqeq2d 2452 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  ( G  -  N ) ) )
22279oveq2i 6101 . . . . . 6  |-  ( (
-u Q  -  (
2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  -  ( 2  x.  G ) )  /  2 )
223212, 213oveq12d 6108 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  -  ( ( 2  x.  G )  /  2
) )  =  (
-u N  -  G
) )
224208, 170, 181, 183divsubdird 10142 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  -  (
( 2  x.  G
)  /  2 ) ) )
22556, 60addcomd 9567 . . . . . . . . 9  |-  ( ph  ->  ( G  +  N
)  =  ( N  +  G ) )
226225negeqd 9600 . . . . . . . 8  |-  ( ph  -> 
-u ( G  +  N )  =  -u ( N  +  G
) )
22760, 56negdi2d 9729 . . . . . . . 8  |-  ( ph  -> 
-u ( N  +  G )  =  (
-u N  -  G
) )
228226, 227eqtrd 2473 . . . . . . 7  |-  ( ph  -> 
-u ( G  +  N )  =  (
-u N  -  G
) )
229223, 224, 2283eqtr4d 2483 . . . . . 6  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  -u ( G  +  N
) )
230222, 229syl5eq 2485 . . . . 5  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  -u ( G  +  N
) )
231230eqeq2d 2452 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  -  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  -u ( G  +  N
) ) )
232221, 231orbi12d 704 . . 3  |-  ( ph  ->  ( ( ( U ^ 3 )  =  ( ( -u Q  +  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  \/  ( U ^ 3 )  =  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) ) )  <->  ( ( U ^ 3 )  =  ( G  -  N
)  \/  ( U ^ 3 )  = 
-u ( G  +  N ) ) ) )
233206, 232mpbid 210 . 2  |-  ( ph  ->  ( ( U ^
3 )  =  ( G  -  N )  \/  ( U ^
3 )  =  -u ( G  +  N
) ) )
23436, 152, 233mpjaodan 779 1  |-  ( ph  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714   class class class wbr 4289  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   2c2 10367   3c3 10368   4c4 10369   NN0cn0 10575   ZZcz 10642   ^cexp 11861    || cdivides 13531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532
This theorem is referenced by:  dcubic  22200
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