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Theorem quad2 20151
Description: The quadratic equation, without specifying the particular branch  D to the square root. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
quad.a  |-  ( ph  ->  A  e.  CC )
quad.z  |-  ( ph  ->  A  =/=  0 )
quad.b  |-  ( ph  ->  B  e.  CC )
quad.c  |-  ( ph  ->  C  e.  CC )
quad.x  |-  ( ph  ->  X  e.  CC )
quad2.d  |-  ( ph  ->  D  e.  CC )
quad2.2  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
Assertion
Ref Expression
quad2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  \/  X  =  ( ( -u B  -  D )  /  (
2  x.  A ) ) ) ) )

Proof of Theorem quad2
StepHypRef Expression
1 2cn 9832 . . . . . . . 8  |-  2  e.  CC
2 quad.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 mulcl 8837 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
41, 2, 3sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 2  x.  A
)  e.  CC )
5 quad.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64, 5mulcld 8871 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  A )  x.  X
)  e.  CC )
7 quad.b . . . . . 6  |-  ( ph  ->  B  e.  CC )
86, 7addcld 8870 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  +  B
)  e.  CC )
98sqcld 11259 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  e.  CC )
10 quad2.d . . . . 5  |-  ( ph  ->  D  e.  CC )
1110sqcld 11259 . . . 4  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
12 subeq0 9089 . . . 4  |-  ( ( ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  e.  CC  /\  ( D ^ 2 )  e.  CC )  -> 
( ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) )  =  0  <->  (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  =  ( D ^
2 ) ) )
139, 11, 12syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) )  =  0  <->  (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  =  ( D ^
2 ) ) )
145sqcld 11259 . . . . . . 7  |-  ( ph  ->  ( X ^ 2 )  e.  CC )
152, 14mulcld 8871 . . . . . 6  |-  ( ph  ->  ( A  x.  ( X ^ 2 ) )  e.  CC )
167, 5mulcld 8871 . . . . . . 7  |-  ( ph  ->  ( B  x.  X
)  e.  CC )
17 quad.c . . . . . . 7  |-  ( ph  ->  C  e.  CC )
1816, 17addcld 8870 . . . . . 6  |-  ( ph  ->  ( ( B  x.  X )  +  C
)  e.  CC )
1915, 18addcld 8870 . . . . 5  |-  ( ph  ->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  e.  CC )
20 0cn 8847 . . . . . 6  |-  0  e.  CC
2120a1i 10 . . . . 5  |-  ( ph  ->  0  e.  CC )
22 4cn 9836 . . . . . 6  |-  4  e.  CC
23 mulcl 8837 . . . . . 6  |-  ( ( 4  e.  CC  /\  A  e.  CC )  ->  ( 4  x.  A
)  e.  CC )
2422, 2, 23sylancr 644 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  e.  CC )
2522a1i 10 . . . . . 6  |-  ( ph  ->  4  e.  CC )
26 4nn 9895 . . . . . . . 8  |-  4  e.  NN
2726nnne0i 9796 . . . . . . 7  |-  4  =/=  0
2827a1i 10 . . . . . 6  |-  ( ph  ->  4  =/=  0 )
29 quad.z . . . . . 6  |-  ( ph  ->  A  =/=  0 )
3025, 2, 28, 29mulne0d 9436 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  =/=  0 )
3119, 21, 24, 30mulcand 9417 . . . 4  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( 4  x.  A )  x.  0 )  <->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  0 ) )
326sqcld 11259 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  e.  CC )
336, 7mulcld 8871 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )
34 mulcl 8837 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )  ->  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  e.  CC )
351, 33, 34sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) )  e.  CC )
362, 17mulcld 8871 . . . . . . . . 9  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
37 mulcl 8837 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 4  x.  ( A  x.  C
) )  e.  CC )
3822, 36, 37sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( A  x.  C )
)  e.  CC )
3932, 35, 38addassd 8873 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) )  +  ( 4  x.  ( A  x.  C ) ) )  =  ( ( ( ( 2  x.  A )  x.  X
) ^ 2 )  +  ( ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C )
) ) ) )
407sqcld 11259 . . . . . . . 8  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
4132, 35addcld 8870 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) )  e.  CC )
4240, 41, 38pnncand 9212 . . . . . . 7  |-  ( ph  ->  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) ) )  -  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
434, 5sqmuld 11273 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  =  ( ( ( 2  x.  A
) ^ 2 )  x.  ( X ^
2 ) ) )
44 sq2 11215 . . . . . . . . . . . . 13  |-  ( 2 ^ 2 )  =  4
4544a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2 ^ 2 )  =  4 )
462sqvald 11258 . . . . . . . . . . . 12  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
4745, 46oveq12d 5892 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( A ^ 2 ) )  =  ( 4  x.  ( A  x.  A
) ) )
48 sqmul 11183 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
491, 2, 48sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
5025, 2, 2mulassd 8874 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 4  x.  A )  x.  A
)  =  ( 4  x.  ( A  x.  A ) ) )
5147, 49, 503eqtr4d 2338 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A ) )
5251oveq1d 5889 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A ) ^
2 )  x.  ( X ^ 2 ) )  =  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) ) )
5324, 2, 14mulassd 8874 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) )  =  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) ) )
5443, 52, 533eqtrrd 2333 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  =  ( ( ( 2  x.  A
)  x.  X ) ^ 2 ) )
5524, 16, 17adddid 8875 . . . . . . . . 9  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( ( 4  x.  A
)  x.  ( B  x.  X ) )  +  ( ( 4  x.  A )  x.  C ) ) )
56 2t2e4 9887 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  2 )  =  4
5756oveq1i 5884 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
581a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  e.  CC )
5958, 58, 2mulassd 8874 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2  x.  2 )  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
6057, 59syl5eqr 2342 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 4  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
6160oveq1d 5889 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( ( 2  x.  ( 2  x.  A ) )  x.  B ) )
6258, 4, 7mulassd 8874 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2  x.  ( 2  x.  A
) )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6361, 62eqtrd 2328 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6463oveq1d 5889 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 2  x.  ( ( 2  x.  A )  x.  B ) )  x.  X ) )
654, 7mulcld 8871 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2  x.  A )  x.  B
)  e.  CC )
6658, 65, 5mulassd 8874 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  A )  x.  B
) )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6764, 66eqtrd 2328 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6824, 7, 5mulassd 8874 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 4  x.  A )  x.  ( B  x.  X ) ) )
694, 7, 5mul32d 9038 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  B )  x.  X
)  =  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )
7069oveq2d 5890 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  B
)  x.  X ) )  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
7167, 68, 703eqtr3d 2336 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( B  x.  X )
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
7225, 2, 17mulassd 8874 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  C
)  =  ( 4  x.  ( A  x.  C ) ) )
7371, 72oveq12d 5892 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( B  x.  X
) )  +  ( ( 4  x.  A
)  x.  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7455, 73eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7554, 74oveq12d 5892 . . . . . . 7  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A )  x.  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  +  ( 4  x.  ( A  x.  C ) ) ) ) )
7639, 42, 753eqtr4rd 2339 . . . . . 6  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A )  x.  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) )  -  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )
7724, 15, 18adddid 8875 . . . . . 6  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A
)  x.  ( ( B  x.  X )  +  C ) ) ) )
78 binom2 11234 . . . . . . . . 9  |-  ( ( ( ( 2  x.  A )  x.  X
)  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( B ^
2 ) ) )
796, 7, 78syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( B ^
2 ) ) )
8041, 40addcomd 9030 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) ) )
8179, 80eqtrd 2328 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) ) )
82 quad2.2 . . . . . . 7  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
8381, 82oveq12d 5892 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) )  -  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )
8476, 77, 833eqtr4d 2338 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) ) )
8524mul01d 9027 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  0 )  =  0 )
8684, 85eqeq12d 2310 . . . 4  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( 4  x.  A )  x.  0 )  <->  ( (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  -  ( D ^
2 ) )  =  0 ) )
8731, 86bitr3d 246 . . 3  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  0 ) )
886, 7subnegd 9180 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  =  ( ( ( 2  x.  A
)  x.  X )  +  B ) )
8988oveq1d 5889 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 ) )
9089eqeq1d 2304 . . 3  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( D ^ 2 ) ) )
9113, 87, 903bitr4d 276 . 2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( D ^ 2 ) ) )
927negcld 9160 . . . 4  |-  ( ph  -> 
-u B  e.  CC )
936, 92subcld 9173 . . 3  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC )
94 sqeqor 11233 . . 3  |-  ( ( ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC  /\  D  e.  CC )  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D ) ) )
9593, 10, 94syl2anc 642 . 2  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D ) ) )
966, 92, 10subaddd 9191 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
( -u B  +  D
)  =  ( ( 2  x.  A )  x.  X ) ) )
9792, 10addcld 8870 . . . . . 6  |-  ( ph  ->  ( -u B  +  D )  e.  CC )
98 2ne0 9845 . . . . . . . 8  |-  2  =/=  0
9998a1i 10 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
10058, 2, 99, 29mulne0d 9436 . . . . . 6  |-  ( ph  ->  ( 2  x.  A
)  =/=  0 )
10197, 4, 5, 100divmuld 9574 . . . . 5  |-  ( ph  ->  ( ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  +  D
) ) )
102 eqcom 2298 . . . . 5  |-  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X )
103 eqcom 2298 . . . . 5  |-  ( (
-u B  +  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  +  D ) )
104101, 102, 1033bitr4g 279 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  <->  ( -u B  +  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10596, 104bitr4d 247 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
X  =  ( (
-u B  +  D
)  /  ( 2  x.  A ) ) ) )
10692, 10negsubd 9179 . . . . 5  |-  ( ph  ->  ( -u B  +  -u D )  =  (
-u B  -  D
) )
107106eqeq1d 2304 . . . 4  |-  ( ph  ->  ( ( -u B  +  -u D )  =  ( ( 2  x.  A )  x.  X
)  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10810negcld 9160 . . . . 5  |-  ( ph  -> 
-u D  e.  CC )
1096, 92, 108subaddd 9191 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  ( -u B  +  -u D )  =  ( ( 2  x.  A
)  x.  X ) ) )
11092, 10subcld 9173 . . . . . 6  |-  ( ph  ->  ( -u B  -  D )  e.  CC )
111110, 4, 5, 100divmuld 9574 . . . . 5  |-  ( ph  ->  ( ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  -  D
) ) )
112 eqcom 2298 . . . . 5  |-  ( X  =  ( ( -u B  -  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X )
113 eqcom 2298 . . . . 5  |-  ( (
-u B  -  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  -  D ) )
114111, 112, 1133bitr4g 279 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  -  D )  /  (
2  x.  A ) )  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
115107, 109, 1143bitr4d 276 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) )
116105, 115orbi12d 690 . 2  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A
)  x.  X )  -  -u B )  = 
-u D )  <->  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  \/  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) ) )
11791, 95, 1163bitrd 270 1  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  \/  X  =  ( ( -u B  -  D )  /  (
2  x.  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1632    e. wcel 1696    =/= wne 2459  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   4c4 9813   ^cexp 11120
This theorem is referenced by:  quad  20152  dcubic2  20156  dquartlem1  20163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
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