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Theorem quad2 20679
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 10070 . . . . . . . 8  |-  2  e.  CC
2 quad.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 mulcl 9074 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
41, 2, 3sylancr 645 . . . . . . 7  |-  ( ph  ->  ( 2  x.  A
)  e.  CC )
5 quad.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64, 5mulcld 9108 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  A )  x.  X
)  e.  CC )
7 quad.b . . . . . 6  |-  ( ph  ->  B  e.  CC )
86, 7addcld 9107 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  +  B
)  e.  CC )
98sqcld 11521 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  e.  CC )
10 quad2.d . . . . 5  |-  ( ph  ->  D  e.  CC )
1110sqcld 11521 . . . 4  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
129, 11subeq0ad 9421 . . 3  |-  ( ph  ->  ( ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) )  =  0  <->  (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  =  ( D ^
2 ) ) )
135sqcld 11521 . . . . . . 7  |-  ( ph  ->  ( X ^ 2 )  e.  CC )
142, 13mulcld 9108 . . . . . 6  |-  ( ph  ->  ( A  x.  ( X ^ 2 ) )  e.  CC )
157, 5mulcld 9108 . . . . . . 7  |-  ( ph  ->  ( B  x.  X
)  e.  CC )
16 quad.c . . . . . . 7  |-  ( ph  ->  C  e.  CC )
1715, 16addcld 9107 . . . . . 6  |-  ( ph  ->  ( ( B  x.  X )  +  C
)  e.  CC )
1814, 17addcld 9107 . . . . 5  |-  ( ph  ->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  e.  CC )
19 0cn 9084 . . . . . 6  |-  0  e.  CC
2019a1i 11 . . . . 5  |-  ( ph  ->  0  e.  CC )
21 4cn 10074 . . . . . 6  |-  4  e.  CC
22 mulcl 9074 . . . . . 6  |-  ( ( 4  e.  CC  /\  A  e.  CC )  ->  ( 4  x.  A
)  e.  CC )
2321, 2, 22sylancr 645 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  e.  CC )
2421a1i 11 . . . . . 6  |-  ( ph  ->  4  e.  CC )
25 4nn 10135 . . . . . . . 8  |-  4  e.  NN
2625nnne0i 10034 . . . . . . 7  |-  4  =/=  0
2726a1i 11 . . . . . 6  |-  ( ph  ->  4  =/=  0 )
28 quad.z . . . . . 6  |-  ( ph  ->  A  =/=  0 )
2924, 2, 27, 28mulne0d 9674 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  =/=  0 )
3018, 20, 23, 29mulcand 9655 . . . 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 ) )
316sqcld 11521 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  e.  CC )
326, 7mulcld 9108 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )
33 mulcl 9074 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )  ->  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  e.  CC )
341, 32, 33sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) )  e.  CC )
352, 16mulcld 9108 . . . . . . . . 9  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
36 mulcl 9074 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 4  x.  ( A  x.  C
) )  e.  CC )
3721, 35, 36sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( A  x.  C )
)  e.  CC )
3831, 34, 37addassd 9110 . . . . . . 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 )
) ) ) )
397sqcld 11521 . . . . . . . 8  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
4031, 34addcld 9107 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) )  e.  CC )
4139, 40, 37pnncand 9450 . . . . . . 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
) ) ) )
424, 5sqmuld 11535 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  =  ( ( ( 2  x.  A
) ^ 2 )  x.  ( X ^
2 ) ) )
43 sq2 11477 . . . . . . . . . . . . 13  |-  ( 2 ^ 2 )  =  4
4443a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2 ^ 2 )  =  4 )
452sqvald 11520 . . . . . . . . . . . 12  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
4644, 45oveq12d 6099 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( A ^ 2 ) )  =  ( 4  x.  ( A  x.  A
) ) )
47 sqmul 11445 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
481, 2, 47sylancr 645 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
4924, 2, 2mulassd 9111 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 4  x.  A )  x.  A
)  =  ( 4  x.  ( A  x.  A ) ) )
5046, 48, 493eqtr4d 2478 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A ) )
5150oveq1d 6096 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A ) ^
2 )  x.  ( X ^ 2 ) )  =  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) ) )
5223, 2, 13mulassd 9111 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) )  =  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) ) )
5342, 51, 523eqtrrd 2473 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  =  ( ( ( 2  x.  A
)  x.  X ) ^ 2 ) )
5423, 15, 16adddid 9112 . . . . . . . . 9  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( ( 4  x.  A
)  x.  ( B  x.  X ) )  +  ( ( 4  x.  A )  x.  C ) ) )
55 2t2e4 10127 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  2 )  =  4
5655oveq1i 6091 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
571a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  e.  CC )
5857, 57, 2mulassd 9111 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2  x.  2 )  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
5956, 58syl5eqr 2482 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 4  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
6059oveq1d 6096 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( ( 2  x.  ( 2  x.  A ) )  x.  B ) )
6157, 4, 7mulassd 9111 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2  x.  ( 2  x.  A
) )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6260, 61eqtrd 2468 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6362oveq1d 6096 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 2  x.  ( ( 2  x.  A )  x.  B ) )  x.  X ) )
644, 7mulcld 9108 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2  x.  A )  x.  B
)  e.  CC )
6557, 64, 5mulassd 9111 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  A )  x.  B
) )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6663, 65eqtrd 2468 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6723, 7, 5mulassd 9111 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 4  x.  A )  x.  ( B  x.  X ) ) )
684, 7, 5mul32d 9276 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  B )  x.  X
)  =  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )
6968oveq2d 6097 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  B
)  x.  X ) )  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
7066, 67, 693eqtr3d 2476 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( B  x.  X )
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
7124, 2, 16mulassd 9111 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  C
)  =  ( 4  x.  ( A  x.  C ) ) )
7270, 71oveq12d 6099 . . . . . . . . 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
) ) ) )
7354, 72eqtrd 2468 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7453, 73oveq12d 6099 . . . . . . 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 ) ) ) ) )
7538, 41, 743eqtr4rd 2479 . . . . . 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 ) ) ) ) )
7623, 14, 17adddid 9112 . . . . . 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 ) ) ) )
77 binom2 11496 . . . . . . . . 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 ) ) )
786, 7, 77syl2anc 643 . . . . . . . 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 ) ) )
7940, 39addcomd 9268 . . . . . . . 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 ) ) ) ) )
8078, 79eqtrd 2468 . . . . . . 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 ) ) ) ) )
81 quad2.2 . . . . . . 7  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
8280, 81oveq12d 6099 . . . . . 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 ) ) ) ) )
8375, 76, 823eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) ) )
8423mul01d 9265 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  0 )  =  0 )
8583, 84eqeq12d 2450 . . . 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 ) )
8630, 85bitr3d 247 . . 3  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  0 ) )
876, 7subnegd 9418 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  =  ( ( ( 2  x.  A
)  x.  X )  +  B ) )
8887oveq1d 6096 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 ) )
8988eqeq1d 2444 . . 3  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( D ^ 2 ) ) )
9012, 86, 893bitr4d 277 . 2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( D ^ 2 ) ) )
917negcld 9398 . . . 4  |-  ( ph  -> 
-u B  e.  CC )
926, 91subcld 9411 . . 3  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC )
93 sqeqor 11495 . . 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 ) ) )
9492, 10, 93syl2anc 643 . 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 ) ) )
956, 91, 10subaddd 9429 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
( -u B  +  D
)  =  ( ( 2  x.  A )  x.  X ) ) )
9691, 10addcld 9107 . . . . . 6  |-  ( ph  ->  ( -u B  +  D )  e.  CC )
97 2ne0 10083 . . . . . . . 8  |-  2  =/=  0
9897a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
9957, 2, 98, 28mulne0d 9674 . . . . . 6  |-  ( ph  ->  ( 2  x.  A
)  =/=  0 )
10096, 4, 5, 99divmuld 9812 . . . . 5  |-  ( ph  ->  ( ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  +  D
) ) )
101 eqcom 2438 . . . . 5  |-  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X )
102 eqcom 2438 . . . . 5  |-  ( (
-u B  +  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  +  D ) )
103100, 101, 1023bitr4g 280 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  <->  ( -u B  +  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10495, 103bitr4d 248 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
X  =  ( (
-u B  +  D
)  /  ( 2  x.  A ) ) ) )
10591, 10negsubd 9417 . . . . 5  |-  ( ph  ->  ( -u B  +  -u D )  =  (
-u B  -  D
) )
106105eqeq1d 2444 . . . 4  |-  ( ph  ->  ( ( -u B  +  -u D )  =  ( ( 2  x.  A )  x.  X
)  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10710negcld 9398 . . . . 5  |-  ( ph  -> 
-u D  e.  CC )
1086, 91, 107subaddd 9429 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  ( -u B  +  -u D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10991, 10subcld 9411 . . . . . 6  |-  ( ph  ->  ( -u B  -  D )  e.  CC )
110109, 4, 5, 99divmuld 9812 . . . . 5  |-  ( ph  ->  ( ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  -  D
) ) )
111 eqcom 2438 . . . . 5  |-  ( X  =  ( ( -u B  -  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X )
112 eqcom 2438 . . . . 5  |-  ( (
-u B  -  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  -  D ) )
113110, 111, 1123bitr4g 280 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  -  D )  /  (
2  x.  A ) )  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
114106, 108, 1133bitr4d 277 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) )
115104, 114orbi12d 691 . 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 ) ) ) ) )
11690, 94, 1153bitrd 271 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 177    \/ wo 358    = wceq 1652    e. wcel 1725    =/= wne 2599  (class class class)co 6081   CCcc 8988   0cc0 8990    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   2c2 10049   4c4 10051   ^cexp 11382
This theorem is referenced by:  quad  20680  dcubic2  20684  dquartlem1  20691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-exp 11383
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