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Theorem quad2 20097
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 9784 . . . . . . . 8  |-  2  e.  CC
2 quad.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 mulcl 8789 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
41, 2, 3sylancr 647 . . . . . . 7  |-  ( ph  ->  ( 2  x.  A
)  e.  CC )
5 quad.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64, 5mulcld 8823 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  A )  x.  X
)  e.  CC )
7 quad.b . . . . . 6  |-  ( ph  ->  B  e.  CC )
86, 7addcld 8822 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  +  B
)  e.  CC )
98sqcld 11209 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  e.  CC )
10 quad2.d . . . . 5  |-  ( ph  ->  D  e.  CC )
1110sqcld 11209 . . . 4  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
12 subeq0 9041 . . . 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 645 . . 3  |-  ( ph  ->  ( ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) )  =  0  <->  (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  =  ( D ^
2 ) ) )
145sqcld 11209 . . . . . . 7  |-  ( ph  ->  ( X ^ 2 )  e.  CC )
152, 14mulcld 8823 . . . . . 6  |-  ( ph  ->  ( A  x.  ( X ^ 2 ) )  e.  CC )
167, 5mulcld 8823 . . . . . . 7  |-  ( ph  ->  ( B  x.  X
)  e.  CC )
17 quad.c . . . . . . 7  |-  ( ph  ->  C  e.  CC )
1816, 17addcld 8822 . . . . . 6  |-  ( ph  ->  ( ( B  x.  X )  +  C
)  e.  CC )
1915, 18addcld 8822 . . . . 5  |-  ( ph  ->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  e.  CC )
20 0cn 8799 . . . . . 6  |-  0  e.  CC
2120a1i 12 . . . . 5  |-  ( ph  ->  0  e.  CC )
22 4cn 9788 . . . . . 6  |-  4  e.  CC
23 mulcl 8789 . . . . . 6  |-  ( ( 4  e.  CC  /\  A  e.  CC )  ->  ( 4  x.  A
)  e.  CC )
2422, 2, 23sylancr 647 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  e.  CC )
2522a1i 12 . . . . . 6  |-  ( ph  ->  4  e.  CC )
26 4nn 9846 . . . . . . . 8  |-  4  e.  NN
2726nnne0i 9748 . . . . . . 7  |-  4  =/=  0
2827a1i 12 . . . . . 6  |-  ( ph  ->  4  =/=  0 )
29 quad.z . . . . . 6  |-  ( ph  ->  A  =/=  0 )
3025, 2, 28, 29mulne0d 9388 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  =/=  0 )
3119, 21, 24, 30mulcand 9369 . . . 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 11209 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  e.  CC )
336, 7mulcld 8823 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )
34 mulcl 8789 . . . . . . . . 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 647 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) )  e.  CC )
362, 17mulcld 8823 . . . . . . . . 9  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
37 mulcl 8789 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 4  x.  ( A  x.  C
) )  e.  CC )
3822, 36, 37sylancr 647 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( A  x.  C )
)  e.  CC )
3932, 35, 38addassd 8825 . . . . . . 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 11209 . . . . . . . 8  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
4132, 35addcld 8822 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) )  e.  CC )
4240, 41, 38pnncand 9164 . . . . . . 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 11223 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  =  ( ( ( 2  x.  A
) ^ 2 )  x.  ( X ^
2 ) ) )
44 sq2 11165 . . . . . . . . . . . . 13  |-  ( 2 ^ 2 )  =  4
4544a1i 12 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2 ^ 2 )  =  4 )
462sqvald 11208 . . . . . . . . . . . 12  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
4745, 46oveq12d 5810 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( A ^ 2 ) )  =  ( 4  x.  ( A  x.  A
) ) )
48 sqmul 11133 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
491, 2, 48sylancr 647 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
5025, 2, 2mulassd 8826 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 4  x.  A )  x.  A
)  =  ( 4  x.  ( A  x.  A ) ) )
5147, 49, 503eqtr4d 2300 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A ) )
5251oveq1d 5807 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A ) ^
2 )  x.  ( X ^ 2 ) )  =  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) ) )
5324, 2, 14mulassd 8826 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) )  =  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) ) )
5443, 52, 533eqtrrd 2295 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  =  ( ( ( 2  x.  A
)  x.  X ) ^ 2 ) )
5524, 16, 17adddid 8827 . . . . . . . . 9  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( ( 4  x.  A
)  x.  ( B  x.  X ) )  +  ( ( 4  x.  A )  x.  C ) ) )
56 2t2e4 9838 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  2 )  =  4
5756oveq1i 5802 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
581a1i 12 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  e.  CC )
5958, 58, 2mulassd 8826 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2  x.  2 )  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
6057, 59syl5eqr 2304 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 4  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
6160oveq1d 5807 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( ( 2  x.  ( 2  x.  A ) )  x.  B ) )
6258, 4, 7mulassd 8826 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2  x.  ( 2  x.  A
) )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6361, 62eqtrd 2290 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6463oveq1d 5807 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 2  x.  ( ( 2  x.  A )  x.  B ) )  x.  X ) )
654, 7mulcld 8823 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2  x.  A )  x.  B
)  e.  CC )
6658, 65, 5mulassd 8826 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  A )  x.  B
) )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6764, 66eqtrd 2290 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6824, 7, 5mulassd 8826 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 4  x.  A )  x.  ( B  x.  X ) ) )
694, 7, 5mul32d 8990 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  B )  x.  X
)  =  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )
7069oveq2d 5808 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  B
)  x.  X ) )  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
7167, 68, 703eqtr3d 2298 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( B  x.  X )
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
7225, 2, 17mulassd 8826 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  C
)  =  ( 4  x.  ( A  x.  C ) ) )
7371, 72oveq12d 5810 . . . . . . . . 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 2290 . . . . . . . 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 5810 . . . . . . 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 2301 . . . . . 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 8827 . . . . . 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 11184 . . . . . . . . 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 645 . . . . . . . 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 8982 . . . . . . . 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 2290 . . . . . . 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 5810 . . . . . 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 2300 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) ) )
8524mul01d 8979 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  0 )  =  0 )
8684, 85eqeq12d 2272 . . . 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 248 . . 3  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  0 ) )
886, 7subnegd 9132 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  =  ( ( ( 2  x.  A
)  x.  X )  +  B ) )
8988oveq1d 5807 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 ) )
9089eqeq1d 2266 . . 3  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( D ^ 2 ) ) )
9113, 87, 903bitr4d 278 . 2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( D ^ 2 ) ) )
927negcld 9112 . . . 4  |-  ( ph  -> 
-u B  e.  CC )
936, 92subcld 9125 . . 3  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC )
94 sqeqor 11183 . . 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 645 . 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 9143 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
( -u B  +  D
)  =  ( ( 2  x.  A )  x.  X ) ) )
9792, 10addcld 8822 . . . . . 6  |-  ( ph  ->  ( -u B  +  D )  e.  CC )
98 2ne0 9797 . . . . . . . 8  |-  2  =/=  0
9998a1i 12 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
10058, 2, 99, 29mulne0d 9388 . . . . . 6  |-  ( ph  ->  ( 2  x.  A
)  =/=  0 )
10197, 4, 5, 100divmuld 9526 . . . . 5  |-  ( ph  ->  ( ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  +  D
) ) )
102 eqcom 2260 . . . . 5  |-  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X )
103 eqcom 2260 . . . . 5  |-  ( (
-u B  +  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  +  D ) )
104101, 102, 1033bitr4g 281 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  <->  ( -u B  +  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10596, 104bitr4d 249 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
X  =  ( (
-u B  +  D
)  /  ( 2  x.  A ) ) ) )
10692, 10negsubd 9131 . . . . 5  |-  ( ph  ->  ( -u B  +  -u D )  =  (
-u B  -  D
) )
107106eqeq1d 2266 . . . 4  |-  ( ph  ->  ( ( -u B  +  -u D )  =  ( ( 2  x.  A )  x.  X
)  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10810negcld 9112 . . . . 5  |-  ( ph  -> 
-u D  e.  CC )
1096, 92, 108subaddd 9143 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  ( -u B  +  -u D )  =  ( ( 2  x.  A
)  x.  X ) ) )
11092, 10subcld 9125 . . . . . 6  |-  ( ph  ->  ( -u B  -  D )  e.  CC )
111110, 4, 5, 100divmuld 9526 . . . . 5  |-  ( ph  ->  ( ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  -  D
) ) )
112 eqcom 2260 . . . . 5  |-  ( X  =  ( ( -u B  -  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X )
113 eqcom 2260 . . . . 5  |-  ( (
-u B  -  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  -  D ) )
114111, 112, 1133bitr4g 281 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  -  D )  /  (
2  x.  A ) )  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
115107, 109, 1143bitr4d 278 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) )
116105, 115orbi12d 693 . 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 272 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 6    <-> wb 178    \/ wo 359    = wceq 1619    e. wcel 1621    =/= wne 2421  (class class class)co 5792   CCcc 8703   0cc0 8705    + caddc 8708    x. cmul 8710    - cmin 9005   -ucneg 9006    / cdiv 9391   2c2 9763   4c4 9765   ^cexp 11070
This theorem is referenced by:  quad  20098  dcubic2  20102  dquartlem1  20109
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-n0 9933  df-z 9992  df-uz 10198  df-seq 11013  df-exp 11071
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