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Theorem dcubic2 23291
Description: Reverse direction of dcubic 23293. 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 10237 . . . 4  |-  ( ph  ->  ( U  /  T
)  e.  CC )
54adantr 463 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  ( U  /  T )  e.  CC )
6 3nn0 10730 . . . . . . 7  |-  3  e.  NN0
76a1i 11 . . . . . 6  |-  ( ph  ->  3  e.  NN0 )
81, 2, 3, 7expdivd 12226 . . . . 5  |-  ( ph  ->  ( ( U  /  T ) ^ 3 )  =  ( ( U ^ 3 )  /  ( T ^
3 ) ) )
98adantr 463 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  ( ( U ^ 3 )  / 
( T ^ 3 ) ) )
10 oveq1 6203 . . . . 5  |-  ( ( U ^ 3 )  =  ( G  -  N )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
11 dcubic.3 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
1211oveq1d 6211 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
13 expcl 12087 . . . . . . . 8  |-  ( ( T  e.  CC  /\  3  e.  NN0 )  -> 
( T ^ 3 )  e.  CC )
142, 6, 13sylancl 660 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  e.  CC )
15 3z 10814 . . . . . . . . 9  |-  3  e.  ZZ
1615a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  ZZ )
172, 3, 16expne0d 12218 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =/=  0 )
1814, 17dividd 10235 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  1 )
1912, 18eqtr3d 2425 . . . . 5  |-  ( ph  ->  ( ( G  -  N )  /  ( T ^ 3 ) )  =  1 )
2010, 19sylan9eqr 2445 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  1 )
219, 20eqtrd 2423 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  1 )
22 dcubic2.2 . . . . 5  |-  ( ph  ->  X  =  ( U  -  ( M  /  U ) ) )
231, 2, 3divcan1d 10238 . . . . . 6  |-  ( ph  ->  ( ( U  /  T )  x.  T
)  =  U )
2423oveq2d 6212 . . . . . 6  |-  ( ph  ->  ( M  /  (
( U  /  T
)  x.  T ) )  =  ( M  /  U ) )
2523, 24oveq12d 6214 . . . . 5  |-  ( ph  ->  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) )  =  ( U  -  ( M  /  U
) ) )
2622, 25eqtr4d 2426 . . . 4  |-  ( ph  ->  X  =  ( ( ( U  /  T
)  x.  T )  -  ( M  / 
( ( U  /  T )  x.  T
) ) ) )
2726adantr 463 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
28 oveq1 6203 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
r ^ 3 )  =  ( ( U  /  T ) ^
3 ) )
2928eqeq1d 2384 . . . . 5  |-  ( r  =  ( U  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( U  /  T
) ^ 3 )  =  1 ) )
30 oveq1 6203 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  (
r  x.  T )  =  ( ( U  /  T )  x.  T ) )
3130oveq2d 6212 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( U  /  T
)  x.  T ) ) )
3230, 31oveq12d 6214 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) )  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
3332eqeq2d 2396 . . . . 5  |-  ( r  =  ( U  /  T )  ->  ( X  =  ( (
r  x.  T )  -  ( M  / 
( r  x.  T
) ) )  <->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) ) ) )
3429, 33anbi12d 708 . . . 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 3135 . . 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 1224 . 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 10527 . . . . . . . . . 10  |-  3  e.  CC
4039a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  e.  CC )
41 3ne0 10547 . . . . . . . . . 10  |-  3  =/=  0
4241a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  =/=  0 )
4338, 40, 42divcld 10237 . . . . . . . 8  |-  ( ph  ->  ( P  /  3
)  e.  CC )
4437, 43eqeltrd 2470 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
45 dcubic2.z . . . . . . 7  |-  ( ph  ->  U  =/=  0 )
4644, 1, 45divcld 10237 . . . . . 6  |-  ( ph  ->  ( M  /  U
)  e.  CC )
4746negcld 9831 . . . . 5  |-  ( ph  -> 
-u ( M  /  U )  e.  CC )
4847, 2, 3divcld 10237 . . . 4  |-  ( ph  ->  ( -u ( M  /  U )  /  T )  e.  CC )
4948adantr 463 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( M  /  U )  /  T
)  e.  CC )
5047, 2, 3, 7expdivd 12226 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( (
-u ( M  /  U ) ^ 3 )  /  ( T ^ 3 ) ) )
5144, 1, 45divnegd 10250 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  (
-u M  /  U
) )
5251oveq1d 6211 . . . . . . . 8  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u M  /  U ) ^ 3 ) )
5344negcld 9831 . . . . . . . . 9  |-  ( ph  -> 
-u M  e.  CC )
5453, 1, 45, 7expdivd 12226 . . . . . . . 8  |-  ( ph  ->  ( ( -u M  /  U ) ^ 3 )  =  ( (
-u M ^ 3 )  /  ( U ^ 3 ) ) )
5511oveq2d 6212 . . . . . . . . . . . . . 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 10700 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
6057, 59eqeltrd 2470 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  CC )
61 subsq 12178 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6256, 60, 61syl2anc 659 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6355, 62eqtr4d 2426 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( ( G ^ 2 )  -  ( N ^ 2 ) ) )
64 dcubic.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
6564oveq1d 6211 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( M ^
3 ) )  -  ( N ^ 2 ) ) )
6660sqcld 12210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
67 expcl 12087 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
6844, 6, 67sylancl 660 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
6966, 68pncan2d 9846 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( N ^ 2 ) )  =  ( M ^
3 ) )
7063, 65, 693eqtrd 2427 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( M ^
3 ) )
7170negeqd 9727 . . . . . . . . . . 11  |-  ( ph  -> 
-u ( ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
7256, 60addcld 9526 . . . . . . . . . . . 12  |-  ( ph  ->  ( G  +  N
)  e.  CC )
7372, 14mulneg1d 9927 . . . . . . . . . . 11  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( ( G  +  N )  x.  ( T ^ 3 ) ) )
74 3nn 10611 . . . . . . . . . . . . 13  |-  3  e.  NN
7574a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  3  e.  NN )
76 2nn 10610 . . . . . . . . . . . . . 14  |-  2  e.  NN
77 1nn0 10728 . . . . . . . . . . . . . 14  |-  1  e.  NN0
78 1nn 10463 . . . . . . . . . . . . . 14  |-  1  e.  NN
79 2t1e2 10601 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  1 )  =  2
8079oveq1i 6206 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
81 2p1e3 10576 . . . . . . . . . . . . . . 15  |-  ( 2  +  1 )  =  3
8280, 81eqtri 2411 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  1 )  +  1 )  =  3
83 1lt2 10619 . . . . . . . . . . . . . 14  |-  1  <  2
8476, 77, 78, 82, 83ndvdsi 14070 . . . . . . . . . . . . 13  |-  -.  2  ||  3
8584a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  -.  2  ||  3
)
86 oexpneg 14051 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN  /\  -.  2  ||  3 )  -> 
( -u M ^ 3 )  =  -u ( M ^ 3 ) )
8744, 75, 85, 86syl3anc 1226 . . . . . . . . . . 11  |-  ( ph  ->  ( -u M ^
3 )  =  -u ( M ^ 3 ) )
8871, 73, 873eqtr4d 2433 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  (
-u M ^ 3 ) )
8988oveq1d 6211 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u M ^ 3 )  / 
( U ^ 3 ) ) )
9072negcld 9831 . . . . . . . . . 10  |-  ( ph  -> 
-u ( G  +  N )  e.  CC )
91 expcl 12087 . . . . . . . . . . 11  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
921, 6, 91sylancl 660 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
931, 45, 16expne0d 12218 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  =/=  0 )
9490, 14, 92, 93div23d 10274 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) ) )
9589, 94eqtr3d 2425 . . . . . . . 8  |-  ( ph  ->  ( ( -u M ^ 3 )  / 
( U ^ 3 ) )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9652, 54, 953eqtrd 2427 . . . . . . 7  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9796oveq1d 6211 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U ) ^
3 )  /  ( T ^ 3 ) )  =  ( ( (
-u ( G  +  N )  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) ) )
9890, 92, 93divcld 10237 . . . . . . 7  |-  ( ph  ->  ( -u ( G  +  N )  / 
( U ^ 3 ) )  e.  CC )
9998, 14, 17divcan4d 10243 . . . . . 6  |-  ( ph  ->  ( ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) )  =  (
-u ( G  +  N )  /  ( U ^ 3 ) ) )
10050, 97, 993eqtrd 2427 . . . . 5  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
101100adantr 463 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
102 oveq1 6203 . . . . . 6  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  (
( U ^ 3 )  /  ( U ^ 3 ) )  =  ( -u ( G  +  N )  /  ( U ^
3 ) ) )
103102eqcomd 2390 . . . . 5  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  ( -u ( G  +  N
)  /  ( U ^ 3 ) )  =  ( ( U ^ 3 )  / 
( U ^ 3 ) ) )
10492, 93dividd 10235 . . . . 5  |-  ( ph  ->  ( ( U ^
3 )  /  ( U ^ 3 ) )  =  1 )
105103, 104sylan9eqr 2445 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  /  ( U ^ 3 ) )  =  1 )
106101, 105eqtrd 2423 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  1 )
10746, 1neg2subd 9861 . . . . . 6  |-  ( ph  ->  ( -u ( M  /  U )  -  -u U )  =  ( U  -  ( M  /  U ) ) )
10822, 107eqtr4d 2426 . . . . 5  |-  ( ph  ->  X  =  ( -u ( M  /  U
)  -  -u U
) )
109108adantr 463 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( -u ( M  /  U )  -  -u U ) )
11047, 2, 3divcan1d 10238 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
111110adantr 463 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
11244, 1, 45divneg2d 10251 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  ( M  /  -u U
) )
113110, 112eqtrd 2423 . . . . . . . 8  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
114113adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
115114oveq2d 6212 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  ( M  /  ( M  /  -u U ) ) )
11644adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  e.  CC )
1171negcld 9831 . . . . . . . 8  |-  ( ph  -> 
-u U  e.  CC )
118117adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  e.  CC )
11973, 71eqtrd 2423 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
120119adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^
3 ) )
12190adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  e.  CC )
12214adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  e.  CC )
123 simpr 459 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =  -u ( G  +  N )
)
12493adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =/=  0 )
125123, 124eqnetrrd 2676 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  =/=  0 )
12617adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  =/=  0 )
127121, 122, 125, 126mulne0d 10118 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =/=  0 )
128120, 127eqnetrrd 2676 . . . . . . . 8  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( M ^ 3 )  =/=  0 )
129 oveq1 6203 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ 3 )  =  ( 0 ^ 3 ) )
130 0exp 12104 . . . . . . . . . . . . 13  |-  ( 3  e.  NN  ->  (
0 ^ 3 )  =  0 )
13174, 130ax-mp 5 . . . . . . . . . . . 12  |-  ( 0 ^ 3 )  =  0
132129, 131syl6eq 2439 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( M ^ 3 )  =  0 )
133132negeqd 9727 . . . . . . . . . 10  |-  ( M  =  0  ->  -u ( M ^ 3 )  = 
-u 0 )
134 neg0 9778 . . . . . . . . . 10  |-  -u 0  =  0
135133, 134syl6eq 2439 . . . . . . . . 9  |-  ( M  =  0  ->  -u ( M ^ 3 )  =  0 )
136135necon3i 2622 . . . . . . . 8  |-  ( -u ( M ^ 3 )  =/=  0  ->  M  =/=  0 )
137128, 136syl 16 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  =/=  0 )
1381, 45negne0d 9842 . . . . . . . 8  |-  ( ph  -> 
-u U  =/=  0
)
139138adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  =/=  0 )
140116, 118, 137, 139ddcand 10257 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  ( M  /  -u U ) )  =  -u U )
141115, 140eqtrd 2423 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  -u U
)
142111, 141oveq12d 6214 . . . 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 2426 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( (
( -u ( M  /  U )  /  T
)  x.  T )  -  ( M  / 
( ( -u ( M  /  U )  /  T )  x.  T
) ) ) )
144 oveq1 6203 . . . . . 6  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r ^ 3 )  =  ( ( -u ( M  /  U
)  /  T ) ^ 3 ) )
145144eqeq1d 2384 . . . . 5  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( -u ( M  /  U )  /  T
) ^ 3 )  =  1 ) )
146 oveq1 6203 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r  x.  T )  =  ( ( -u ( M  /  U
)  /  T )  x.  T ) )
147146oveq2d 6212 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) )
148146, 147oveq12d 6214 . . . . . 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 2396 . . . . 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 708 . . . 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 3135 . . 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 1224 . 2  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
15392sqcld 12210 . . . . . . 7  |-  ( ph  ->  ( ( U ^
3 ) ^ 2 )  e.  CC )
154153mulid2d 9525 . . . . . 6  |-  ( ph  ->  ( 1  x.  (
( U ^ 3 ) ^ 2 ) )  =  ( ( U ^ 3 ) ^ 2 ) )
15558, 92mulcld 9527 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( U ^ 3 ) )  e.  CC )
156155, 68negsubd 9850 . . . . . 6  |-  ( ph  ->  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^ 3 ) )  =  ( ( Q  x.  ( U ^
3 ) )  -  ( M ^ 3 ) ) )
157154, 156oveq12d 6214 . . . . 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 23290 . . . . . 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 2423 . . . 4  |-  ( ph  ->  ( ( 1  x.  ( ( U ^
3 ) ^ 2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^
3 ) ) )  =  0 )
163 1cnd 9523 . . . . 5  |-  ( ph  ->  1  e.  CC )
164 ax-1ne0 9472 . . . . . 6  |-  1  =/=  0
165164a1i 11 . . . . 5  |-  ( ph  ->  1  =/=  0 )
16668negcld 9831 . . . . 5  |-  ( ph  -> 
-u ( M ^
3 )  e.  CC )
167 2cn 10523 . . . . . 6  |-  2  e.  CC
168 mulcl 9487 . . . . . 6  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( 2  x.  G
)  e.  CC )
169167, 56, 168sylancr 661 . . . . 5  |-  ( ph  ->  ( 2  x.  G
)  e.  CC )
170 sqmul 12134 . . . . . . 7  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
171167, 56, 170sylancr 661 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
17264oveq2d 6212 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( G ^ 2 ) )  =  ( ( 2 ^ 2 )  x.  ( ( N ^
2 )  +  ( M ^ 3 ) ) ) )
173167sqcli 12151 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
174 mulcl 9487 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
175173, 66, 174sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
176 mulcl 9487 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
177173, 68, 176sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
178175, 177subnegd 9851 . . . . . . 7  |-  ( ph  ->  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  -  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
17957oveq2d 6212 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
180167a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  CC )
181 2ne0 10545 . . . . . . . . . . . . 13  |-  2  =/=  0
182181a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
18358, 180, 182divcan2d 10239 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
184179, 183eqtrd 2423 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
185184oveq1d 6211 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( Q ^ 2 ) )
186 sqmul 12134 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  N  e.  CC )  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
187167, 60, 186sylancr 661 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
188185, 187eqtr3d 2425 . . . . . . . 8  |-  ( ph  ->  ( Q ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
189166mulid2d 9525 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  x.  -u ( M ^ 3 ) )  =  -u ( M ^
3 ) )
190189oveq2d 6212 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  ( 4  x.  -u ( M ^
3 ) ) )
191 4cn 10530 . . . . . . . . . . 11  |-  4  e.  CC
192 mulneg2 9912 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
193191, 68, 192sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
194190, 193eqtrd 2423 . . . . . . . . 9  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
4  x.  ( M ^ 3 ) ) )
195 sq2 12167 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
196195oveq1i 6206 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( M ^
3 ) )  =  ( 4  x.  ( M ^ 3 ) )
197196negeqi 9726 . . . . . . . . 9  |-  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) )
198194, 197syl6eqr 2441 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )
199188, 198oveq12d 6214 . . . . . . 7  |-  ( ph  ->  ( ( Q ^
2 )  -  (
4  x.  ( 1  x.  -u ( M ^
3 ) ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  -  -u ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
200173a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ 2 )  e.  CC )
201200, 66, 68adddid 9531 . . . . . . 7  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
202178, 199, 2013eqtr4rd 2434 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
203171, 172, 2023eqtrd 2427 . . . . 5  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
204163, 165, 58, 166, 92, 169, 203quad2 23286 . . . 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 ) ) ) ) )
205162, 204mpbid 210 . . 3  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  \/  ( U ^ 3 )  =  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) ) ) )
20679oveq2i 6207 . . . . . 6  |-  ( (
-u Q  +  ( 2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  +  ( 2  x.  G ) )  /  2 )
20758negcld 9831 . . . . . . . 8  |-  ( ph  -> 
-u Q  e.  CC )
208207, 169, 180, 182divdird 10275 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  +  ( ( 2  x.  G
)  /  2 ) ) )
20957negeqd 9727 . . . . . . . . 9  |-  ( ph  -> 
-u N  =  -u ( Q  /  2
) )
21058, 180, 182divnegd 10250 . . . . . . . . 9  |-  ( ph  -> 
-u ( Q  / 
2 )  =  (
-u Q  /  2
) )
211209, 210eqtr2d 2424 . . . . . . . 8  |-  ( ph  ->  ( -u Q  / 
2 )  =  -u N )
21256, 180, 182divcan3d 10242 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  G )  /  2
)  =  G )
213211, 212oveq12d 6214 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  +  ( ( 2  x.  G )  /  2
) )  =  (
-u N  +  G
) )
21460negcld 9831 . . . . . . . . 9  |-  ( ph  -> 
-u N  e.  CC )
215214, 56addcomd 9693 . . . . . . . 8  |-  ( ph  ->  ( -u N  +  G )  =  ( G  +  -u N
) )
21656, 60negsubd 9850 . . . . . . . 8  |-  ( ph  ->  ( G  +  -u N )  =  ( G  -  N ) )
217215, 216eqtrd 2423 . . . . . . 7  |-  ( ph  ->  ( -u N  +  G )  =  ( G  -  N ) )
218208, 213, 2173eqtrd 2427 . . . . . 6  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( G  -  N ) )
219206, 218syl5eq 2435 . . . . 5  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  ( G  -  N ) )
220219eqeq2d 2396 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  ( G  -  N ) ) )
22179oveq2i 6207 . . . . . 6  |-  ( (
-u Q  -  (
2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  -  ( 2  x.  G ) )  /  2 )
222211, 212oveq12d 6214 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  -  ( ( 2  x.  G )  /  2
) )  =  (
-u N  -  G
) )
223207, 169, 180, 182divsubdird 10276 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  -  (
( 2  x.  G
)  /  2 ) ) )
22456, 60addcomd 9693 . . . . . . . . 9  |-  ( ph  ->  ( G  +  N
)  =  ( N  +  G ) )
225224negeqd 9727 . . . . . . . 8  |-  ( ph  -> 
-u ( G  +  N )  =  -u ( N  +  G
) )
22660, 56negdi2d 9858 . . . . . . . 8  |-  ( ph  -> 
-u ( N  +  G )  =  (
-u N  -  G
) )
227225, 226eqtrd 2423 . . . . . . 7  |-  ( ph  -> 
-u ( G  +  N )  =  (
-u N  -  G
) )
228222, 223, 2273eqtr4d 2433 . . . . . 6  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  -u ( G  +  N
) )
229221, 228syl5eq 2435 . . . . 5  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  -u ( G  +  N
) )
230229eqeq2d 2396 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  -  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  -u ( G  +  N
) ) )
231220, 230orbi12d 707 . . 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 ) ) ) )
232205, 231mpbid 210 . 2  |-  ( ph  ->  ( ( U ^
3 )  =  ( G  -  N )  \/  ( U ^
3 )  =  -u ( G  +  N
) ) )
23336, 152, 232mpjaodan 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 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733   class class class wbr 4367  (class class class)co 6196   CCcc 9401   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408    - cmin 9718   -ucneg 9719    / cdiv 10123   NNcn 10452   2c2 10502   3c3 10503   4c4 10504   NN0cn0 10712   ZZcz 10781   ^cexp 12069    || cdvds 13988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989
This theorem is referenced by:  dcubic  23293
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