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Theorem dcubic2 22251
Description: Reverse direction of dcubic 22253. 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 10119 . . . 4  |-  ( ph  ->  ( U  /  T
)  e.  CC )
54adantr 465 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  ( U  /  T )  e.  CC )
6 3nn0 10609 . . . . . . 7  |-  3  e.  NN0
76a1i 11 . . . . . 6  |-  ( ph  ->  3  e.  NN0 )
81, 2, 3, 7expdivd 12034 . . . . 5  |-  ( ph  ->  ( ( U  /  T ) ^ 3 )  =  ( ( U ^ 3 )  /  ( T ^
3 ) ) )
98adantr 465 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  ( ( U ^ 3 )  / 
( T ^ 3 ) ) )
10 oveq1 6110 . . . . 5  |-  ( ( U ^ 3 )  =  ( G  -  N )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
11 dcubic.3 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
1211oveq1d 6118 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
13 expcl 11895 . . . . . . . 8  |-  ( ( T  e.  CC  /\  3  e.  NN0 )  -> 
( T ^ 3 )  e.  CC )
142, 6, 13sylancl 662 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  e.  CC )
15 3z 10691 . . . . . . . . 9  |-  3  e.  ZZ
1615a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  ZZ )
172, 3, 16expne0d 12026 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =/=  0 )
1814, 17dividd 10117 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  1 )
1912, 18eqtr3d 2477 . . . . 5  |-  ( ph  ->  ( ( G  -  N )  /  ( T ^ 3 ) )  =  1 )
2010, 19sylan9eqr 2497 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  1 )
219, 20eqtrd 2475 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  1 )
22 dcubic2.2 . . . . 5  |-  ( ph  ->  X  =  ( U  -  ( M  /  U ) ) )
231, 2, 3divcan1d 10120 . . . . . 6  |-  ( ph  ->  ( ( U  /  T )  x.  T
)  =  U )
2423oveq2d 6119 . . . . . 6  |-  ( ph  ->  ( M  /  (
( U  /  T
)  x.  T ) )  =  ( M  /  U ) )
2523, 24oveq12d 6121 . . . . 5  |-  ( ph  ->  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) )  =  ( U  -  ( M  /  U
) ) )
2622, 25eqtr4d 2478 . . . 4  |-  ( ph  ->  X  =  ( ( ( U  /  T
)  x.  T )  -  ( M  / 
( ( U  /  T )  x.  T
) ) ) )
2726adantr 465 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
28 oveq1 6110 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
r ^ 3 )  =  ( ( U  /  T ) ^
3 ) )
2928eqeq1d 2451 . . . . 5  |-  ( r  =  ( U  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( U  /  T
) ^ 3 )  =  1 ) )
30 oveq1 6110 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  (
r  x.  T )  =  ( ( U  /  T )  x.  T ) )
3130oveq2d 6119 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( U  /  T
)  x.  T ) ) )
3230, 31oveq12d 6121 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) )  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
3332eqeq2d 2454 . . . . 5  |-  ( r  =  ( U  /  T )  ->  ( X  =  ( (
r  x.  T )  -  ( M  / 
( r  x.  T
) ) )  <->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) ) ) )
3429, 33anbi12d 710 . . . 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 3085 . . 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 1216 . 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 10408 . . . . . . . . . 10  |-  3  e.  CC
4039a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  e.  CC )
41 3ne0 10428 . . . . . . . . . 10  |-  3  =/=  0
4241a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  =/=  0 )
4338, 40, 42divcld 10119 . . . . . . . 8  |-  ( ph  ->  ( P  /  3
)  e.  CC )
4437, 43eqeltrd 2517 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
45 dcubic2.z . . . . . . 7  |-  ( ph  ->  U  =/=  0 )
4644, 1, 45divcld 10119 . . . . . 6  |-  ( ph  ->  ( M  /  U
)  e.  CC )
4746negcld 9718 . . . . 5  |-  ( ph  -> 
-u ( M  /  U )  e.  CC )
4847, 2, 3divcld 10119 . . . 4  |-  ( ph  ->  ( -u ( M  /  U )  /  T )  e.  CC )
4948adantr 465 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( M  /  U )  /  T
)  e.  CC )
5047, 2, 3, 7expdivd 12034 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( (
-u ( M  /  U ) ^ 3 )  /  ( T ^ 3 ) ) )
5144, 1, 45divnegd 10132 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  (
-u M  /  U
) )
5251oveq1d 6118 . . . . . . . 8  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u M  /  U ) ^ 3 ) )
5344negcld 9718 . . . . . . . . 9  |-  ( ph  -> 
-u M  e.  CC )
5453, 1, 45, 7expdivd 12034 . . . . . . . 8  |-  ( ph  ->  ( ( -u M  /  U ) ^ 3 )  =  ( (
-u M ^ 3 )  /  ( U ^ 3 ) ) )
5511oveq2d 6119 . . . . . . . . . . . . . 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 10581 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
6057, 59eqeltrd 2517 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  CC )
61 subsq 11985 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6256, 60, 61syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6355, 62eqtr4d 2478 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( ( G ^ 2 )  -  ( N ^ 2 ) ) )
64 dcubic.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
6564oveq1d 6118 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( M ^
3 ) )  -  ( N ^ 2 ) ) )
6660sqcld 12018 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
67 expcl 11895 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
6844, 6, 67sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
6966, 68pncan2d 9733 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( N ^ 2 ) )  =  ( M ^
3 ) )
7063, 65, 693eqtrd 2479 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( M ^
3 ) )
7170negeqd 9616 . . . . . . . . . . 11  |-  ( ph  -> 
-u ( ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
7256, 60addcld 9417 . . . . . . . . . . . 12  |-  ( ph  ->  ( G  +  N
)  e.  CC )
7372, 14mulneg1d 9809 . . . . . . . . . . 11  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( ( G  +  N )  x.  ( T ^ 3 ) ) )
74 3nn 10492 . . . . . . . . . . . . 13  |-  3  e.  NN
7574a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  3  e.  NN )
76 2nn 10491 . . . . . . . . . . . . . 14  |-  2  e.  NN
77 1nn0 10607 . . . . . . . . . . . . . 14  |-  1  e.  NN0
78 1nn 10345 . . . . . . . . . . . . . 14  |-  1  e.  NN
79 2t1e2 10482 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  1 )  =  2
8079oveq1i 6113 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
81 2p1e3 10457 . . . . . . . . . . . . . . 15  |-  ( 2  +  1 )  =  3
8280, 81eqtri 2463 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  1 )  +  1 )  =  3
83 1lt2 10500 . . . . . . . . . . . . . 14  |-  1  <  2
8476, 77, 78, 82, 83ndvdsi 13626 . . . . . . . . . . . . 13  |-  -.  2  ||  3
8584a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  -.  2  ||  3
)
86 oexpneg 13607 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN  /\  -.  2  ||  3 )  -> 
( -u M ^ 3 )  =  -u ( M ^ 3 ) )
8744, 75, 85, 86syl3anc 1218 . . . . . . . . . . 11  |-  ( ph  ->  ( -u M ^
3 )  =  -u ( M ^ 3 ) )
8871, 73, 873eqtr4d 2485 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  (
-u M ^ 3 ) )
8988oveq1d 6118 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u M ^ 3 )  / 
( U ^ 3 ) ) )
9072negcld 9718 . . . . . . . . . 10  |-  ( ph  -> 
-u ( G  +  N )  e.  CC )
91 expcl 11895 . . . . . . . . . . 11  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
921, 6, 91sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
931, 45, 16expne0d 12026 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  =/=  0 )
9490, 14, 92, 93div23d 10156 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) ) )
9589, 94eqtr3d 2477 . . . . . . . 8  |-  ( ph  ->  ( ( -u M ^ 3 )  / 
( U ^ 3 ) )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9652, 54, 953eqtrd 2479 . . . . . . 7  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9796oveq1d 6118 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U ) ^
3 )  /  ( T ^ 3 ) )  =  ( ( (
-u ( G  +  N )  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) ) )
9890, 92, 93divcld 10119 . . . . . . 7  |-  ( ph  ->  ( -u ( G  +  N )  / 
( U ^ 3 ) )  e.  CC )
9998, 14, 17divcan4d 10125 . . . . . 6  |-  ( ph  ->  ( ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) )  =  (
-u ( G  +  N )  /  ( U ^ 3 ) ) )
10050, 97, 993eqtrd 2479 . . . . 5  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
101100adantr 465 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
102 oveq1 6110 . . . . . 6  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  (
( U ^ 3 )  /  ( U ^ 3 ) )  =  ( -u ( G  +  N )  /  ( U ^
3 ) ) )
103102eqcomd 2448 . . . . 5  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  ( -u ( G  +  N
)  /  ( U ^ 3 ) )  =  ( ( U ^ 3 )  / 
( U ^ 3 ) ) )
10492, 93dividd 10117 . . . . 5  |-  ( ph  ->  ( ( U ^
3 )  /  ( U ^ 3 ) )  =  1 )
105103, 104sylan9eqr 2497 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  /  ( U ^ 3 ) )  =  1 )
106101, 105eqtrd 2475 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  1 )
10746, 1neg2subd 9748 . . . . . 6  |-  ( ph  ->  ( -u ( M  /  U )  -  -u U )  =  ( U  -  ( M  /  U ) ) )
10822, 107eqtr4d 2478 . . . . 5  |-  ( ph  ->  X  =  ( -u ( M  /  U
)  -  -u U
) )
109108adantr 465 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( -u ( M  /  U )  -  -u U ) )
11047, 2, 3divcan1d 10120 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
111110adantr 465 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
11244, 1, 45divneg2d 10133 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  ( M  /  -u U
) )
113110, 112eqtrd 2475 . . . . . . . 8  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
114113adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
115114oveq2d 6119 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  ( M  /  ( M  /  -u U ) ) )
11644adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  e.  CC )
1171negcld 9718 . . . . . . . 8  |-  ( ph  -> 
-u U  e.  CC )
118117adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  e.  CC )
11973, 71eqtrd 2475 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
120119adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^
3 ) )
12190adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  e.  CC )
12214adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  e.  CC )
123 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =  -u ( G  +  N )
)
12493adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =/=  0 )
125123, 124eqnetrrd 2640 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  =/=  0 )
12617adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  =/=  0 )
127121, 122, 125, 126mulne0d 10000 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =/=  0 )
128120, 127eqnetrrd 2640 . . . . . . . 8  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( M ^ 3 )  =/=  0 )
129 oveq1 6110 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ 3 )  =  ( 0 ^ 3 ) )
130 0exp 11911 . . . . . . . . . . . . 13  |-  ( 3  e.  NN  ->  (
0 ^ 3 )  =  0 )
13174, 130ax-mp 5 . . . . . . . . . . . 12  |-  ( 0 ^ 3 )  =  0
132129, 131syl6eq 2491 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( M ^ 3 )  =  0 )
133132negeqd 9616 . . . . . . . . . 10  |-  ( M  =  0  ->  -u ( M ^ 3 )  = 
-u 0 )
134 neg0 9667 . . . . . . . . . 10  |-  -u 0  =  0
135133, 134syl6eq 2491 . . . . . . . . 9  |-  ( M  =  0  ->  -u ( M ^ 3 )  =  0 )
136135necon3i 2662 . . . . . . . 8  |-  ( -u ( M ^ 3 )  =/=  0  ->  M  =/=  0 )
137128, 136syl 16 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  =/=  0 )
1381, 45negne0d 9729 . . . . . . . 8  |-  ( ph  -> 
-u U  =/=  0
)
139138adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  =/=  0 )
140116, 118, 137, 139ddcand 10139 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  ( M  /  -u U ) )  =  -u U )
141115, 140eqtrd 2475 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  -u U
)
142111, 141oveq12d 6121 . . . 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 2478 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( (
( -u ( M  /  U )  /  T
)  x.  T )  -  ( M  / 
( ( -u ( M  /  U )  /  T )  x.  T
) ) ) )
144 oveq1 6110 . . . . . 6  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r ^ 3 )  =  ( ( -u ( M  /  U
)  /  T ) ^ 3 ) )
145144eqeq1d 2451 . . . . 5  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( -u ( M  /  U )  /  T
) ^ 3 )  =  1 ) )
146 oveq1 6110 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r  x.  T )  =  ( ( -u ( M  /  U
)  /  T )  x.  T ) )
147146oveq2d 6119 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) )
148146, 147oveq12d 6121 . . . . . 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 2454 . . . . 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 710 . . . 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 3085 . . 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 1216 . 2  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
15392sqcld 12018 . . . . . . 7  |-  ( ph  ->  ( ( U ^
3 ) ^ 2 )  e.  CC )
154153mulid2d 9416 . . . . . 6  |-  ( ph  ->  ( 1  x.  (
( U ^ 3 ) ^ 2 ) )  =  ( ( U ^ 3 ) ^ 2 ) )
15558, 92mulcld 9418 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( U ^ 3 ) )  e.  CC )
156155, 68negsubd 9737 . . . . . 6  |-  ( ph  ->  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^ 3 ) )  =  ( ( Q  x.  ( U ^
3 ) )  -  ( M ^ 3 ) ) )
157154, 156oveq12d 6121 . . . . 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 22250 . . . . . 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 2475 . . . 4  |-  ( ph  ->  ( ( 1  x.  ( ( U ^
3 ) ^ 2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^
3 ) ) )  =  0 )
163 ax-1cn 9352 . . . . . 6  |-  1  e.  CC
164163a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
165 ax-1ne0 9363 . . . . . 6  |-  1  =/=  0
166165a1i 11 . . . . 5  |-  ( ph  ->  1  =/=  0 )
16768negcld 9718 . . . . 5  |-  ( ph  -> 
-u ( M ^
3 )  e.  CC )
168 2cn 10404 . . . . . 6  |-  2  e.  CC
169 mulcl 9378 . . . . . 6  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( 2  x.  G
)  e.  CC )
170168, 56, 169sylancr 663 . . . . 5  |-  ( ph  ->  ( 2  x.  G
)  e.  CC )
171 sqmul 11941 . . . . . . 7  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
172168, 56, 171sylancr 663 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
17364oveq2d 6119 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( G ^ 2 ) )  =  ( ( 2 ^ 2 )  x.  ( ( N ^
2 )  +  ( M ^ 3 ) ) ) )
174168sqcli 11958 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
175 mulcl 9378 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
176174, 66, 175sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
177 mulcl 9378 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
178174, 68, 177sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
179176, 178subnegd 9738 . . . . . . 7  |-  ( ph  ->  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  -  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
18057oveq2d 6119 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
181168a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  CC )
182 2ne0 10426 . . . . . . . . . . . . 13  |-  2  =/=  0
183182a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
18458, 181, 183divcan2d 10121 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
185180, 184eqtrd 2475 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
186185oveq1d 6118 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( Q ^ 2 ) )
187 sqmul 11941 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  N  e.  CC )  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
188168, 60, 187sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
189186, 188eqtr3d 2477 . . . . . . . 8  |-  ( ph  ->  ( Q ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
190167mulid2d 9416 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  x.  -u ( M ^ 3 ) )  =  -u ( M ^
3 ) )
191190oveq2d 6119 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  ( 4  x.  -u ( M ^
3 ) ) )
192 4cn 10411 . . . . . . . . . . 11  |-  4  e.  CC
193 mulneg2 9794 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
194192, 68, 193sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
195191, 194eqtrd 2475 . . . . . . . . 9  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
4  x.  ( M ^ 3 ) ) )
196 sq2 11974 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
197196oveq1i 6113 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( M ^
3 ) )  =  ( 4  x.  ( M ^ 3 ) )
198197negeqi 9615 . . . . . . . . 9  |-  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) )
199195, 198syl6eqr 2493 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )
200189, 199oveq12d 6121 . . . . . . 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 9422 . . . . . . 7  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
203179, 200, 2023eqtr4rd 2486 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
204172, 173, 2033eqtrd 2479 . . . . 5  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
205164, 166, 58, 167, 92, 170, 204quad2 22246 . . . 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 6114 . . . . . 6  |-  ( (
-u Q  +  ( 2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  +  ( 2  x.  G ) )  /  2 )
20858negcld 9718 . . . . . . . 8  |-  ( ph  -> 
-u Q  e.  CC )
209208, 170, 181, 183divdird 10157 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  +  ( ( 2  x.  G
)  /  2 ) ) )
21057negeqd 9616 . . . . . . . . 9  |-  ( ph  -> 
-u N  =  -u ( Q  /  2
) )
21158, 181, 183divnegd 10132 . . . . . . . . 9  |-  ( ph  -> 
-u ( Q  / 
2 )  =  (
-u Q  /  2
) )
212210, 211eqtr2d 2476 . . . . . . . 8  |-  ( ph  ->  ( -u Q  / 
2 )  =  -u N )
21356, 181, 183divcan3d 10124 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  G )  /  2
)  =  G )
214212, 213oveq12d 6121 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  +  ( ( 2  x.  G )  /  2
) )  =  (
-u N  +  G
) )
21560negcld 9718 . . . . . . . . 9  |-  ( ph  -> 
-u N  e.  CC )
216215, 56addcomd 9583 . . . . . . . 8  |-  ( ph  ->  ( -u N  +  G )  =  ( G  +  -u N
) )
21756, 60negsubd 9737 . . . . . . . 8  |-  ( ph  ->  ( G  +  -u N )  =  ( G  -  N ) )
218216, 217eqtrd 2475 . . . . . . 7  |-  ( ph  ->  ( -u N  +  G )  =  ( G  -  N ) )
219209, 214, 2183eqtrd 2479 . . . . . 6  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( G  -  N ) )
220207, 219syl5eq 2487 . . . . 5  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  ( G  -  N ) )
221220eqeq2d 2454 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  ( G  -  N ) ) )
22279oveq2i 6114 . . . . . 6  |-  ( (
-u Q  -  (
2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  -  ( 2  x.  G ) )  /  2 )
223212, 213oveq12d 6121 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  -  ( ( 2  x.  G )  /  2
) )  =  (
-u N  -  G
) )
224208, 170, 181, 183divsubdird 10158 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  -  (
( 2  x.  G
)  /  2 ) ) )
22556, 60addcomd 9583 . . . . . . . . 9  |-  ( ph  ->  ( G  +  N
)  =  ( N  +  G ) )
226225negeqd 9616 . . . . . . . 8  |-  ( ph  -> 
-u ( G  +  N )  =  -u ( N  +  G
) )
22760, 56negdi2d 9745 . . . . . . . 8  |-  ( ph  -> 
-u ( N  +  G )  =  (
-u N  -  G
) )
228226, 227eqtrd 2475 . . . . . . 7  |-  ( ph  -> 
-u ( G  +  N )  =  (
-u N  -  G
) )
229223, 224, 2283eqtr4d 2485 . . . . . 6  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  -u ( G  +  N
) )
230222, 229syl5eq 2487 . . . . 5  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  -u ( G  +  N
) )
231230eqeq2d 2454 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  -  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  -u ( G  +  N
) ) )
232221, 231orbi12d 709 . . 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 784 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 1369    e. wcel 1756    =/= wne 2618   E.wrex 2728   class class class wbr 4304  (class class class)co 6103   CCcc 9292   0cc0 9294   1c1 9295    + caddc 9297    x. cmul 9299    - cmin 9607   -ucneg 9608    / cdiv 10005   NNcn 10334   2c2 10383   3c3 10384   4c4 10385   NN0cn0 10591   ZZcz 10658   ^cexp 11877    || cdivides 13547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548
This theorem is referenced by:  dcubic  22253
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