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Theorem dcubic2 20553
Description: Reverse direction of dcubic 20555. 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 9724 . . . 4  |-  ( ph  ->  ( U  /  T
)  e.  CC )
54adantr 452 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  ( U  /  T )  e.  CC )
6 3nn0 10173 . . . . . . 7  |-  3  e.  NN0
76a1i 11 . . . . . 6  |-  ( ph  ->  3  e.  NN0 )
81, 2, 3, 7expdivd 11466 . . . . 5  |-  ( ph  ->  ( ( U  /  T ) ^ 3 )  =  ( ( U ^ 3 )  /  ( T ^
3 ) ) )
98adantr 452 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  ( ( U ^ 3 )  / 
( T ^ 3 ) ) )
10 oveq1 6029 . . . . 5  |-  ( ( U ^ 3 )  =  ( G  -  N )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
11 dcubic.3 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =  ( G  -  N ) )
1211oveq1d 6037 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  ( ( G  -  N )  / 
( T ^ 3 ) ) )
13 expcl 11328 . . . . . . . 8  |-  ( ( T  e.  CC  /\  3  e.  NN0 )  -> 
( T ^ 3 )  e.  CC )
142, 6, 13sylancl 644 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  e.  CC )
156nn0zi 10240 . . . . . . . . 9  |-  3  e.  ZZ
1615a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  ZZ )
172, 3, 16expne0d 11458 . . . . . . 7  |-  ( ph  ->  ( T ^ 3 )  =/=  0 )
1814, 17dividd 9722 . . . . . 6  |-  ( ph  ->  ( ( T ^
3 )  /  ( T ^ 3 ) )  =  1 )
1912, 18eqtr3d 2423 . . . . 5  |-  ( ph  ->  ( ( G  -  N )  /  ( T ^ 3 ) )  =  1 )
2010, 19sylan9eqr 2443 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U ^ 3 )  /  ( T ^ 3 ) )  =  1 )
219, 20eqtrd 2421 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  (
( U  /  T
) ^ 3 )  =  1 )
22 dcubic2.2 . . . . 5  |-  ( ph  ->  X  =  ( U  -  ( M  /  U ) ) )
231, 2, 3divcan1d 9725 . . . . . 6  |-  ( ph  ->  ( ( U  /  T )  x.  T
)  =  U )
2423oveq2d 6038 . . . . . 6  |-  ( ph  ->  ( M  /  (
( U  /  T
)  x.  T ) )  =  ( M  /  U ) )
2523, 24oveq12d 6040 . . . . 5  |-  ( ph  ->  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) )  =  ( U  -  ( M  /  U
) ) )
2622, 25eqtr4d 2424 . . . 4  |-  ( ph  ->  X  =  ( ( ( U  /  T
)  x.  T )  -  ( M  / 
( ( U  /  T )  x.  T
) ) ) )
2726adantr 452 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  =  ( G  -  N
) )  ->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
28 oveq1 6029 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
r ^ 3 )  =  ( ( U  /  T ) ^
3 ) )
2928eqeq1d 2397 . . . . 5  |-  ( r  =  ( U  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( U  /  T
) ^ 3 )  =  1 ) )
30 oveq1 6029 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  (
r  x.  T )  =  ( ( U  /  T )  x.  T ) )
3130oveq2d 6038 . . . . . . 7  |-  ( r  =  ( U  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( U  /  T
)  x.  T ) ) )
3230, 31oveq12d 6040 . . . . . 6  |-  ( r  =  ( U  /  T )  ->  (
( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) )  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  (
( U  /  T
)  x.  T ) ) ) )
3332eqeq2d 2400 . . . . 5  |-  ( r  =  ( U  /  T )  ->  ( X  =  ( (
r  x.  T )  -  ( M  / 
( r  x.  T
) ) )  <->  X  =  ( ( ( U  /  T )  x.  T )  -  ( M  /  ( ( U  /  T )  x.  T ) ) ) ) )
3429, 33anbi12d 692 . . . 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 2997 . . 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 1182 . 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 10006 . . . . . . . . . 10  |-  3  e.  CC
4039a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  e.  CC )
41 3ne0 10019 . . . . . . . . . 10  |-  3  =/=  0
4241a1i 11 . . . . . . . . 9  |-  ( ph  ->  3  =/=  0 )
4338, 40, 42divcld 9724 . . . . . . . 8  |-  ( ph  ->  ( P  /  3
)  e.  CC )
4437, 43eqeltrd 2463 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
45 dcubic2.z . . . . . . 7  |-  ( ph  ->  U  =/=  0 )
4644, 1, 45divcld 9724 . . . . . 6  |-  ( ph  ->  ( M  /  U
)  e.  CC )
4746negcld 9332 . . . . 5  |-  ( ph  -> 
-u ( M  /  U )  e.  CC )
4847, 2, 3divcld 9724 . . . 4  |-  ( ph  ->  ( -u ( M  /  U )  /  T )  e.  CC )
4948adantr 452 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( M  /  U )  /  T
)  e.  CC )
5047, 2, 3, 7expdivd 11466 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( (
-u ( M  /  U ) ^ 3 )  /  ( T ^ 3 ) ) )
5144, 1, 45divnegd 9737 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  (
-u M  /  U
) )
5251oveq1d 6037 . . . . . . . 8  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u M  /  U ) ^ 3 ) )
5344negcld 9332 . . . . . . . . 9  |-  ( ph  -> 
-u M  e.  CC )
5453, 1, 45, 7expdivd 11466 . . . . . . . 8  |-  ( ph  ->  ( ( -u M  /  U ) ^ 3 )  =  ( (
-u M ^ 3 )  /  ( U ^ 3 ) ) )
5511oveq2d 6038 . . . . . . . . . . . . . 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 10146 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( Q  /  2
)  e.  CC )
6057, 59eqeltrd 2463 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  CC )
61 subsq 11417 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  CC  /\  N  e.  CC )  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6256, 60, 61syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( G  +  N )  x.  ( G  -  N
) ) )
6355, 62eqtr4d 2424 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( ( G ^ 2 )  -  ( N ^ 2 ) ) )
64 dcubic.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( G ^ 2 )  =  ( ( N ^ 2 )  +  ( M ^
3 ) ) )
6564oveq1d 6037 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G ^
2 )  -  ( N ^ 2 ) )  =  ( ( ( N ^ 2 )  +  ( M ^
3 ) )  -  ( N ^ 2 ) ) )
6660sqcld 11450 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( N ^ 2 )  e.  CC )
67 expcl 11328 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  CC  /\  3  e.  NN0 )  -> 
( M ^ 3 )  e.  CC )
6844, 6, 67sylancl 644 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M ^ 3 )  e.  CC )
6966, 68pncan2d 9347 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( N ^ 2 )  +  ( M ^ 3 ) )  -  ( N ^ 2 ) )  =  ( M ^
3 ) )
7063, 65, 693eqtrd 2425 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G  +  N )  x.  ( T ^ 3 ) )  =  ( M ^
3 ) )
7170negeqd 9234 . . . . . . . . . . 11  |-  ( ph  -> 
-u ( ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
7256, 60addcld 9042 . . . . . . . . . . . 12  |-  ( ph  ->  ( G  +  N
)  e.  CC )
7372, 14mulneg1d 9420 . . . . . . . . . . 11  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( ( G  +  N )  x.  ( T ^ 3 ) ) )
74 3nn 10068 . . . . . . . . . . . . 13  |-  3  e.  NN
7574a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  3  e.  NN )
76 2nn 10067 . . . . . . . . . . . . . 14  |-  2  e.  NN
77 1nn0 10171 . . . . . . . . . . . . . 14  |-  1  e.  NN0
78 1nn 9945 . . . . . . . . . . . . . 14  |-  1  e.  NN
79 2cn 10004 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
8079mulid1i 9027 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  1 )  =  2
8180oveq1i 6032 . . . . . . . . . . . . . . 15  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
82 2p1e3 10037 . . . . . . . . . . . . . . 15  |-  ( 2  +  1 )  =  3
8381, 82eqtri 2409 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  1 )  +  1 )  =  3
84 1lt2 10076 . . . . . . . . . . . . . 14  |-  1  <  2
8576, 77, 78, 83, 84ndvdsi 12859 . . . . . . . . . . . . 13  |-  -.  2  ||  3
8685a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  -.  2  ||  3
)
87 oexpneg 12840 . . . . . . . . . . . 12  |-  ( ( M  e.  CC  /\  3  e.  NN  /\  -.  2  ||  3 )  -> 
( -u M ^ 3 )  =  -u ( M ^ 3 ) )
8844, 75, 86, 87syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( -u M ^
3 )  =  -u ( M ^ 3 ) )
8971, 73, 883eqtr4d 2431 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  (
-u M ^ 3 ) )
9089oveq1d 6037 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u M ^ 3 )  / 
( U ^ 3 ) ) )
9172negcld 9332 . . . . . . . . . 10  |-  ( ph  -> 
-u ( G  +  N )  e.  CC )
92 expcl 11328 . . . . . . . . . . 11  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
931, 6, 92sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
941, 45, 16expne0d 11458 . . . . . . . . . 10  |-  ( ph  ->  ( U ^ 3 )  =/=  0 )
9591, 14, 93, 94div23d 9761 . . . . . . . . 9  |-  ( ph  ->  ( ( -u ( G  +  N )  x.  ( T ^ 3 ) )  /  ( U ^ 3 ) )  =  ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) ) )
9690, 95eqtr3d 2423 . . . . . . . 8  |-  ( ph  ->  ( ( -u M ^ 3 )  / 
( U ^ 3 ) )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9752, 54, 963eqtrd 2425 . . . . . . 7  |-  ( ph  ->  ( -u ( M  /  U ) ^
3 )  =  ( ( -u ( G  +  N )  / 
( U ^ 3 ) )  x.  ( T ^ 3 ) ) )
9897oveq1d 6037 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U ) ^
3 )  /  ( T ^ 3 ) )  =  ( ( (
-u ( G  +  N )  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) ) )
9991, 93, 94divcld 9724 . . . . . . 7  |-  ( ph  ->  ( -u ( G  +  N )  / 
( U ^ 3 ) )  e.  CC )
10099, 14, 17divcan4d 9730 . . . . . 6  |-  ( ph  ->  ( ( ( -u ( G  +  N
)  /  ( U ^ 3 ) )  x.  ( T ^
3 ) )  / 
( T ^ 3 ) )  =  (
-u ( G  +  N )  /  ( U ^ 3 ) ) )
10150, 98, 1003eqtrd 2425 . . . . 5  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
102101adantr 452 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  ( -u ( G  +  N
)  /  ( U ^ 3 ) ) )
103 oveq1 6029 . . . . . 6  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  (
( U ^ 3 )  /  ( U ^ 3 ) )  =  ( -u ( G  +  N )  /  ( U ^
3 ) ) )
104103eqcomd 2394 . . . . 5  |-  ( ( U ^ 3 )  =  -u ( G  +  N )  ->  ( -u ( G  +  N
)  /  ( U ^ 3 ) )  =  ( ( U ^ 3 )  / 
( U ^ 3 ) ) )
10593, 94dividd 9722 . . . . 5  |-  ( ph  ->  ( ( U ^
3 )  /  ( U ^ 3 ) )  =  1 )
106104, 105sylan9eqr 2443 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  /  ( U ^ 3 ) )  =  1 )
107102, 106eqtrd 2421 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T ) ^ 3 )  =  1 )
10846, 1neg2subd 9362 . . . . . 6  |-  ( ph  ->  ( -u ( M  /  U )  -  -u U )  =  ( U  -  ( M  /  U ) ) )
10922, 108eqtr4d 2424 . . . . 5  |-  ( ph  ->  X  =  ( -u ( M  /  U
)  -  -u U
) )
110109adantr 452 . . . 4  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( -u ( M  /  U )  -  -u U ) )
11147, 2, 3divcan1d 9725 . . . . . 6  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
112111adantr 452 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  -u ( M  /  U ) )
11344, 1, 45divneg2d 9738 . . . . . . . . 9  |-  ( ph  -> 
-u ( M  /  U )  =  ( M  /  -u U
) )
114111, 113eqtrd 2421 . . . . . . . 8  |-  ( ph  ->  ( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
115114adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( ( -u ( M  /  U )  /  T )  x.  T
)  =  ( M  /  -u U ) )
116115oveq2d 6038 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  ( M  /  ( M  /  -u U ) ) )
11744adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  e.  CC )
1181negcld 9332 . . . . . . . 8  |-  ( ph  -> 
-u U  e.  CC )
119118adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  e.  CC )
12073, 71eqtrd 2421 . . . . . . . . . 10  |-  ( ph  ->  ( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^ 3 ) )
121120adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =  -u ( M ^
3 ) )
12291adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  e.  CC )
12314adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  e.  CC )
124 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =  -u ( G  +  N )
)
12594adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( U ^ 3 )  =/=  0 )
126124, 125eqnetrrd 2572 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( G  +  N
)  =/=  0 )
12717adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( T ^ 3 )  =/=  0 )
128122, 123, 126, 127mulne0d 9608 . . . . . . . . 9  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( -u ( G  +  N )  x.  ( T ^ 3 ) )  =/=  0 )
129121, 128eqnetrrd 2572 . . . . . . . 8  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u ( M ^ 3 )  =/=  0 )
130 oveq1 6029 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ 3 )  =  ( 0 ^ 3 ) )
131 0exp 11344 . . . . . . . . . . . . 13  |-  ( 3  e.  NN  ->  (
0 ^ 3 )  =  0 )
13274, 131ax-mp 8 . . . . . . . . . . . 12  |-  ( 0 ^ 3 )  =  0
133130, 132syl6eq 2437 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( M ^ 3 )  =  0 )
134133negeqd 9234 . . . . . . . . . 10  |-  ( M  =  0  ->  -u ( M ^ 3 )  = 
-u 0 )
135 neg0 9281 . . . . . . . . . 10  |-  -u 0  =  0
136134, 135syl6eq 2437 . . . . . . . . 9  |-  ( M  =  0  ->  -u ( M ^ 3 )  =  0 )
137136necon3i 2591 . . . . . . . 8  |-  ( -u ( M ^ 3 )  =/=  0  ->  M  =/=  0 )
138129, 137syl 16 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  M  =/=  0 )
1391, 45negne0d 9343 . . . . . . . 8  |-  ( ph  -> 
-u U  =/=  0
)
140139adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  -u U  =/=  0 )
141117, 119, 138, 140ddcand 9744 . . . . . 6  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  ( M  /  -u U ) )  =  -u U )
142116, 141eqtrd 2421 . . . . 5  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  -> 
( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) )  =  -u U
)
143112, 142oveq12d 6040 . . . 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
) )
144110, 143eqtr4d 2424 . . 3  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  X  =  ( (
( -u ( M  /  U )  /  T
)  x.  T )  -  ( M  / 
( ( -u ( M  /  U )  /  T )  x.  T
) ) ) )
145 oveq1 6029 . . . . . 6  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r ^ 3 )  =  ( ( -u ( M  /  U
)  /  T ) ^ 3 ) )
146145eqeq1d 2397 . . . . 5  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
( r ^ 3 )  =  1  <->  (
( -u ( M  /  U )  /  T
) ^ 3 )  =  1 ) )
147 oveq1 6029 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  (
r  x.  T )  =  ( ( -u ( M  /  U
)  /  T )  x.  T ) )
148147oveq2d 6038 . . . . . . 7  |-  ( r  =  ( -u ( M  /  U )  /  T )  ->  ( M  /  ( r  x.  T ) )  =  ( M  /  (
( -u ( M  /  U )  /  T
)  x.  T ) ) )
149147, 148oveq12d 6040 . . . . . 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
) ) ) )
150149eqeq2d 2400 . . . . 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 ) ) ) ) )
151146, 150anbi12d 692 . . . 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
) ) ) ) ) )
152151rspcev 2997 . . 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 ) ) ) ) )
15349, 107, 144, 152syl12anc 1182 . 2  |-  ( (
ph  /\  ( U ^ 3 )  = 
-u ( G  +  N ) )  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
15493sqcld 11450 . . . . . . 7  |-  ( ph  ->  ( ( U ^
3 ) ^ 2 )  e.  CC )
155154mulid2d 9041 . . . . . 6  |-  ( ph  ->  ( 1  x.  (
( U ^ 3 ) ^ 2 ) )  =  ( ( U ^ 3 ) ^ 2 ) )
15658, 93mulcld 9043 . . . . . . 7  |-  ( ph  ->  ( Q  x.  ( U ^ 3 ) )  e.  CC )
157156, 68negsubd 9351 . . . . . 6  |-  ( ph  ->  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^ 3 ) )  =  ( ( Q  x.  ( U ^
3 ) )  -  ( M ^ 3 ) ) )
158155, 157oveq12d 6040 . . . . 5  |-  ( ph  ->  ( ( 1  x.  ( ( U ^
3 ) ^ 2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^
3 ) ) )  =  ( ( ( U ^ 3 ) ^ 2 )  +  ( ( Q  x.  ( U ^ 3 ) )  -  ( M ^ 3 ) ) ) )
159 dcubic2.x . . . . . 6  |-  ( ph  ->  ( ( X ^
3 )  +  ( ( P  x.  X
)  +  Q ) )  =  0 )
160 dcubic.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
16138, 58, 160, 2, 11, 56, 64, 37, 57, 3, 1, 45, 22dcubic1lem 20552 . . . . . 6  |-  ( ph  ->  ( ( ( X ^ 3 )  +  ( ( P  x.  X )  +  Q
) )  =  0  <-> 
( ( ( U ^ 3 ) ^
2 )  +  ( ( Q  x.  ( U ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 ) )
162159, 161mpbid 202 . . . . 5  |-  ( ph  ->  ( ( ( U ^ 3 ) ^
2 )  +  ( ( Q  x.  ( U ^ 3 ) )  -  ( M ^
3 ) ) )  =  0 )
163158, 162eqtrd 2421 . . . 4  |-  ( ph  ->  ( ( 1  x.  ( ( U ^
3 ) ^ 2 ) )  +  ( ( Q  x.  ( U ^ 3 ) )  +  -u ( M ^
3 ) ) )  =  0 )
164 ax-1cn 8983 . . . . . 6  |-  1  e.  CC
165164a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
166 ax-1ne0 8994 . . . . . 6  |-  1  =/=  0
167166a1i 11 . . . . 5  |-  ( ph  ->  1  =/=  0 )
16868negcld 9332 . . . . 5  |-  ( ph  -> 
-u ( M ^
3 )  e.  CC )
169 mulcl 9009 . . . . . 6  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( 2  x.  G
)  e.  CC )
17079, 56, 169sylancr 645 . . . . 5  |-  ( ph  ->  ( 2  x.  G
)  e.  CC )
171 sqmul 11374 . . . . . . 7  |-  ( ( 2  e.  CC  /\  G  e.  CC )  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
17279, 56, 171sylancr 645 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( G ^
2 ) ) )
17364oveq2d 6038 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( G ^ 2 ) )  =  ( ( 2 ^ 2 )  x.  ( ( N ^
2 )  +  ( M ^ 3 ) ) ) )
17479sqcli 11391 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
175 mulcl 9009 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( N ^ 2 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
176174, 66, 175sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  e.  CC )
177 mulcl 9009 . . . . . . . . 9  |-  ( ( ( 2 ^ 2 )  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
178174, 68, 177sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) )  e.  CC )
179176, 178subnegd 9352 . . . . . . 7  |-  ( ph  ->  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  -  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
18057oveq2d 6038 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  N
)  =  ( 2  x.  ( Q  / 
2 ) ) )
18179a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  CC )
182 2ne0 10017 . . . . . . . . . . . . 13  |-  2  =/=  0
183182a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  2  =/=  0 )
18458, 181, 183divcan2d 9726 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( Q  /  2 ) )  =  Q )
185180, 184eqtrd 2421 . . . . . . . . . 10  |-  ( ph  ->  ( 2  x.  N
)  =  Q )
186185oveq1d 6037 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( Q ^ 2 ) )
187 sqmul 11374 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  N  e.  CC )  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
18879, 60, 187sylancr 645 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  N ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
189186, 188eqtr3d 2423 . . . . . . . 8  |-  ( ph  ->  ( Q ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^
2 ) ) )
190168mulid2d 9041 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  x.  -u ( M ^ 3 ) )  =  -u ( M ^
3 ) )
191190oveq2d 6038 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  ( 4  x.  -u ( M ^
3 ) ) )
192 4cn 10008 . . . . . . . . . . 11  |-  4  e.  CC
193 mulneg2 9405 . . . . . . . . . . 11  |-  ( ( 4  e.  CC  /\  ( M ^ 3 )  e.  CC )  -> 
( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
194192, 68, 193sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 4  x.  -u ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) ) )
195191, 194eqtrd 2421 . . . . . . . . 9  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
4  x.  ( M ^ 3 ) ) )
196 sq2 11406 . . . . . . . . . . 11  |-  ( 2 ^ 2 )  =  4
197196oveq1i 6032 . . . . . . . . . 10  |-  ( ( 2 ^ 2 )  x.  ( M ^
3 ) )  =  ( 4  x.  ( M ^ 3 ) )
198197negeqi 9233 . . . . . . . . 9  |-  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) )  =  -u ( 4  x.  ( M ^ 3 ) )
199195, 198syl6eqr 2439 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  (
1  x.  -u ( M ^ 3 ) ) )  =  -u (
( 2 ^ 2 )  x.  ( M ^ 3 ) ) )
200189, 199oveq12d 6040 . . . . . . 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 9047 . . . . . . 7  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( ( 2 ^ 2 )  x.  ( N ^ 2 ) )  +  ( ( 2 ^ 2 )  x.  ( M ^ 3 ) ) ) )
203179, 200, 2023eqtr4rd 2432 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  (
( N ^ 2 )  +  ( M ^ 3 ) ) )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
204172, 173, 2033eqtrd 2425 . . . . 5  |-  ( ph  ->  ( ( 2  x.  G ) ^ 2 )  =  ( ( Q ^ 2 )  -  ( 4  x.  ( 1  x.  -u ( M ^ 3 ) ) ) ) )
205165, 167, 58, 168, 93, 170, 204quad2 20548 . . . 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 ) ) ) ) )
206163, 205mpbid 202 . . 3  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  \/  ( U ^ 3 )  =  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) ) ) )
20780oveq2i 6033 . . . . . 6  |-  ( (
-u Q  +  ( 2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  +  ( 2  x.  G ) )  /  2 )
20858negcld 9332 . . . . . . . 8  |-  ( ph  -> 
-u Q  e.  CC )
209208, 170, 181, 183divdird 9762 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  +  ( ( 2  x.  G
)  /  2 ) ) )
21057negeqd 9234 . . . . . . . . 9  |-  ( ph  -> 
-u N  =  -u ( Q  /  2
) )
21158, 181, 183divnegd 9737 . . . . . . . . 9  |-  ( ph  -> 
-u ( Q  / 
2 )  =  (
-u Q  /  2
) )
212210, 211eqtr2d 2422 . . . . . . . 8  |-  ( ph  ->  ( -u Q  / 
2 )  =  -u N )
21356, 181, 183divcan3d 9729 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  G )  /  2
)  =  G )
214212, 213oveq12d 6040 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  +  ( ( 2  x.  G )  /  2
) )  =  (
-u N  +  G
) )
21560negcld 9332 . . . . . . . . 9  |-  ( ph  -> 
-u N  e.  CC )
216215, 56addcomd 9202 . . . . . . . 8  |-  ( ph  ->  ( -u N  +  G )  =  ( G  +  -u N
) )
21756, 60negsubd 9351 . . . . . . . 8  |-  ( ph  ->  ( G  +  -u N )  =  ( G  -  N ) )
218216, 217eqtrd 2421 . . . . . . 7  |-  ( ph  ->  ( -u N  +  G )  =  ( G  -  N ) )
219209, 214, 2183eqtrd 2425 . . . . . 6  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
2 )  =  ( G  -  N ) )
220207, 219syl5eq 2433 . . . . 5  |-  ( ph  ->  ( ( -u Q  +  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  ( G  -  N ) )
221220eqeq2d 2400 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  +  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  ( G  -  N ) ) )
22280oveq2i 6033 . . . . . 6  |-  ( (
-u Q  -  (
2  x.  G ) )  /  ( 2  x.  1 ) )  =  ( ( -u Q  -  ( 2  x.  G ) )  /  2 )
223212, 213oveq12d 6040 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  /  2 )  -  ( ( 2  x.  G )  /  2
) )  =  (
-u N  -  G
) )
224208, 170, 181, 183divsubdird 9763 . . . . . . 7  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  ( ( -u Q  / 
2 )  -  (
( 2  x.  G
)  /  2 ) ) )
22556, 60addcomd 9202 . . . . . . . . 9  |-  ( ph  ->  ( G  +  N
)  =  ( N  +  G ) )
226225negeqd 9234 . . . . . . . 8  |-  ( ph  -> 
-u ( G  +  N )  =  -u ( N  +  G
) )
22760, 56negdi2d 9359 . . . . . . . 8  |-  ( ph  -> 
-u ( N  +  G )  =  (
-u N  -  G
) )
228226, 227eqtrd 2421 . . . . . . 7  |-  ( ph  -> 
-u ( G  +  N )  =  (
-u N  -  G
) )
229223, 224, 2283eqtr4d 2431 . . . . . 6  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
2 )  =  -u ( G  +  N
) )
230222, 229syl5eq 2433 . . . . 5  |-  ( ph  ->  ( ( -u Q  -  ( 2  x.  G ) )  / 
( 2  x.  1 ) )  =  -u ( G  +  N
) )
231230eqeq2d 2400 . . . 4  |-  ( ph  ->  ( ( U ^
3 )  =  ( ( -u Q  -  ( 2  x.  G
) )  /  (
2  x.  1 ) )  <->  ( U ^
3 )  =  -u ( G  +  N
) ) )
232221, 231orbi12d 691 . . 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 202 . 2  |-  ( ph  ->  ( ( U ^
3 )  =  ( G  -  N )  \/  ( U ^
3 )  =  -u ( G  +  N
) ) )
23436, 153, 233mpjaodan 762 1  |-  ( ph  ->  E. r  e.  CC  ( ( r ^
3 )  =  1  /\  X  =  ( ( r  x.  T
)  -  ( M  /  ( r  x.  T ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652   class class class wbr 4155  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    - cmin 9225   -ucneg 9226    / cdiv 9611   NNcn 9934   2c2 9983   3c3 9984   4c4 9985   NN0cn0 10155   ZZcz 10216   ^cexp 11311    || cdivides 12781
This theorem is referenced by:  dcubic  20555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-dvds 12782
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