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Theorem quad3 28897
Description: Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.)
Hypotheses
Ref Expression
quad3.1  |-  X  e.  CC
quad3.2  |-  A  e.  CC
quad3.3  |-  A  =/=  0
quad3.4  |-  B  e.  CC
quad3.5  |-  C  e.  CC
quad3.6  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  0
Assertion
Ref Expression
quad3  |-  ( X  =  ( ( -u B  +  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) )  \/  X  =  ( ( -u B  -  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) ) )

Proof of Theorem quad3
StepHypRef Expression
1 2cn 10612 . . . . . 6  |-  2  e.  CC
2 quad3.2 . . . . . 6  |-  A  e.  CC
31, 2mulcli 9604 . . . . 5  |-  ( 2  x.  A )  e.  CC
4 quad3.1 . . . . . 6  |-  X  e.  CC
5 quad3.4 . . . . . . 7  |-  B  e.  CC
6 2ne0 10634 . . . . . . . 8  |-  2  =/=  0
7 quad3.3 . . . . . . . 8  |-  A  =/=  0
81, 2, 6, 7mulne0i 10198 . . . . . . 7  |-  ( 2  x.  A )  =/=  0
95, 3, 8divcli 10292 . . . . . 6  |-  ( B  /  ( 2  x.  A ) )  e.  CC
104, 9addcli 9603 . . . . 5  |-  ( X  +  ( B  / 
( 2  x.  A
) ) )  e.  CC
113, 10sqmuli 12230 . . . 4  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) ) ^ 2 )  =  ( ( ( 2  x.  A ) ^
2 )  x.  (
( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 ) )
124, 9binom2i 12256 . . . . . . 7  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 )  =  ( ( ( X ^ 2 )  +  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) ) )  +  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
134sqcli 12227 . . . . . . . . . . . 12  |-  ( X ^ 2 )  e.  CC
142, 13mulcli 9604 . . . . . . . . . . 11  |-  ( A  x.  ( X ^
2 ) )  e.  CC
155, 4mulcli 9604 . . . . . . . . . . 11  |-  ( B  x.  X )  e.  CC
1614, 15, 2, 7divdiri 10307 . . . . . . . . . 10  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  /  A )  =  ( ( ( A  x.  ( X ^ 2 ) )  /  A )  +  ( ( B  x.  X )  /  A ) )
1713, 2, 7divcan3i 10296 . . . . . . . . . . 11  |-  ( ( A  x.  ( X ^ 2 ) )  /  A )  =  ( X ^ 2 )
185, 4, 2, 7div23i 10308 . . . . . . . . . . 11  |-  ( ( B  x.  X )  /  A )  =  ( ( B  /  A )  x.  X
)
1917, 18oveq12i 6293 . . . . . . . . . 10  |-  ( ( ( A  x.  ( X ^ 2 ) )  /  A )  +  ( ( B  x.  X )  /  A
) )  =  ( ( X ^ 2 )  +  ( ( B  /  A )  x.  X ) )
2016, 19eqtr2i 2473 . . . . . . . . 9  |-  ( ( X ^ 2 )  +  ( ( B  /  A )  x.  X ) )  =  ( ( ( A  x.  ( X ^
2 ) )  +  ( B  x.  X
) )  /  A
)
215, 2, 7divcli 10292 . . . . . . . . . . . 12  |-  ( B  /  A )  e.  CC
2221, 4mulcomi 9605 . . . . . . . . . . 11  |-  ( ( B  /  A )  x.  X )  =  ( X  x.  ( B  /  A ) )
234, 21mulcli 9604 . . . . . . . . . . . 12  |-  ( X  x.  ( B  /  A ) )  e.  CC
2423, 1, 6divcan2i 10293 . . . . . . . . . . 11  |-  ( 2  x.  ( ( X  x.  ( B  /  A ) )  / 
2 ) )  =  ( X  x.  ( B  /  A ) )
254, 21, 1, 6divassi 10306 . . . . . . . . . . . . 13  |-  ( ( X  x.  ( B  /  A ) )  /  2 )  =  ( X  x.  (
( B  /  A
)  /  2 ) )
262, 7pm3.2i 455 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  /\  A  =/=  0 )
271, 6pm3.2i 455 . . . . . . . . . . . . . . . 16  |-  ( 2  e.  CC  /\  2  =/=  0 )
28 divdiv1 10261 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 )  /\  ( 2  e.  CC  /\  2  =/=  0 ) )  -> 
( ( B  /  A )  /  2
)  =  ( B  /  ( A  x.  2 ) ) )
295, 26, 27, 28mp3an 1325 . . . . . . . . . . . . . . 15  |-  ( ( B  /  A )  /  2 )  =  ( B  /  ( A  x.  2 ) )
302, 1mulcomi 9605 . . . . . . . . . . . . . . . 16  |-  ( A  x.  2 )  =  ( 2  x.  A
)
3130oveq2i 6292 . . . . . . . . . . . . . . 15  |-  ( B  /  ( A  x.  2 ) )  =  ( B  /  (
2  x.  A ) )
3229, 31eqtri 2472 . . . . . . . . . . . . . 14  |-  ( ( B  /  A )  /  2 )  =  ( B  /  (
2  x.  A ) )
3332oveq2i 6292 . . . . . . . . . . . . 13  |-  ( X  x.  ( ( B  /  A )  / 
2 ) )  =  ( X  x.  ( B  /  ( 2  x.  A ) ) )
3425, 33eqtri 2472 . . . . . . . . . . . 12  |-  ( ( X  x.  ( B  /  A ) )  /  2 )  =  ( X  x.  ( B  /  ( 2  x.  A ) ) )
3534oveq2i 6292 . . . . . . . . . . 11  |-  ( 2  x.  ( ( X  x.  ( B  /  A ) )  / 
2 ) )  =  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) )
3622, 24, 353eqtr2i 2478 . . . . . . . . . 10  |-  ( ( B  /  A )  x.  X )  =  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) )
3736oveq2i 6292 . . . . . . . . 9  |-  ( ( X ^ 2 )  +  ( ( B  /  A )  x.  X ) )  =  ( ( X ^
2 )  +  ( 2  x.  ( X  x.  ( B  / 
( 2  x.  A
) ) ) ) )
38 quad3.5 . . . . . . . . . . . . . . 15  |-  C  e.  CC
3914, 15, 38addassi 9607 . . . . . . . . . . . . . 14  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  +  C )  =  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )
4039eqcomi 2456 . . . . . . . . . . . . 13  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  ( ( ( A  x.  ( X ^
2 ) )  +  ( B  x.  X
) )  +  C
)
4140oveq1i 6291 . . . . . . . . . . . 12  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  -  C )  =  ( ( ( ( A  x.  ( X ^
2 ) )  +  ( B  x.  X
) )  +  C
)  -  C )
4214, 15addcli 9603 . . . . . . . . . . . . 13  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  e.  CC
4342, 38pncan3oi 9841 . . . . . . . . . . . 12  |-  ( ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  +  C )  -  C )  =  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )
4441, 43eqtr2i 2473 . . . . . . . . . . 11  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  =  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  -  C
)
45 quad3.6 . . . . . . . . . . . . 13  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  0
4645oveq1i 6291 . . . . . . . . . . . 12  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  -  C )  =  ( 0  -  C )
47 df-neg 9813 . . . . . . . . . . . 12  |-  -u C  =  ( 0  -  C )
4846, 47eqtr4i 2475 . . . . . . . . . . 11  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  -  C )  =  -u C
4944, 48eqtri 2472 . . . . . . . . . 10  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  = 
-u C
5049oveq1i 6291 . . . . . . . . 9  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  /  A )  =  (
-u C  /  A
)
5120, 37, 503eqtr3i 2480 . . . . . . . 8  |-  ( ( X ^ 2 )  +  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) ) )  =  (
-u C  /  A
)
5251oveq1i 6291 . . . . . . 7  |-  ( ( ( X ^ 2 )  +  ( 2  x.  ( X  x.  ( B  /  (
2  x.  A ) ) ) ) )  +  ( ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( ( -u C  /  A )  +  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
5312, 52eqtri 2472 . . . . . 6  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 )  =  ( ( -u C  /  A )  +  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
5438negcli 9892 . . . . . . . 8  |-  -u C  e.  CC
5554, 2, 7divcli 10292 . . . . . . 7  |-  ( -u C  /  A )  e.  CC
569sqcli 12227 . . . . . . 7  |-  ( ( B  /  ( 2  x.  A ) ) ^ 2 )  e.  CC
5755, 56addcomi 9774 . . . . . 6  |-  ( (
-u C  /  A
)  +  ( ( B  /  ( 2  x.  A ) ) ^ 2 ) )  =  ( ( ( B  /  ( 2  x.  A ) ) ^ 2 )  +  ( -u C  /  A ) )
585, 3, 8sqdivi 12231 . . . . . . . 8  |-  ( ( B  /  ( 2  x.  A ) ) ^ 2 )  =  ( ( B ^
2 )  /  (
( 2  x.  A
) ^ 2 ) )
59 4cn 10619 . . . . . . . . . . 11  |-  4  e.  CC
6059, 2mulcli 9604 . . . . . . . . . 10  |-  ( 4  x.  A )  e.  CC
61 4ne0 10638 . . . . . . . . . . 11  |-  4  =/=  0
6259, 2, 61, 7mulne0i 10198 . . . . . . . . . 10  |-  ( 4  x.  A )  =/=  0
6360, 60, 54, 2, 62, 7divmuldivi 10310 . . . . . . . . 9  |-  ( ( ( 4  x.  A
)  /  ( 4  x.  A ) )  x.  ( -u C  /  A ) )  =  ( ( ( 4  x.  A )  x.  -u C )  /  (
( 4  x.  A
)  x.  A ) )
6460, 62dividi 10283 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  /  ( 4  x.  A ) )  =  1
6564eqcomi 2456 . . . . . . . . . . 11  |-  1  =  ( ( 4  x.  A )  / 
( 4  x.  A
) )
6665oveq1i 6291 . . . . . . . . . 10  |-  ( 1  x.  ( -u C  /  A ) )  =  ( ( ( 4  x.  A )  / 
( 4  x.  A
) )  x.  ( -u C  /  A ) )
6755mulid2i 9602 . . . . . . . . . 10  |-  ( 1  x.  ( -u C  /  A ) )  =  ( -u C  /  A )
6866, 67eqtr3i 2474 . . . . . . . . 9  |-  ( ( ( 4  x.  A
)  /  ( 4  x.  A ) )  x.  ( -u C  /  A ) )  =  ( -u C  /  A )
6938mulm1i 10007 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  C )  =  -u C
7069eqcomi 2456 . . . . . . . . . . . . . 14  |-  -u C  =  ( -u 1  x.  C )
7170oveq2i 6292 . . . . . . . . . . . . 13  |-  ( ( 4  x.  A )  x.  -u C )  =  ( ( 4  x.  A )  x.  ( -u 1  x.  C ) )
72 neg1cn 10645 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
7360, 72, 38mulassi 9608 . . . . . . . . . . . . 13  |-  ( ( ( 4  x.  A
)  x.  -u 1
)  x.  C )  =  ( ( 4  x.  A )  x.  ( -u 1  x.  C ) )
7471, 73eqtr4i 2475 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  x.  -u C )  =  ( ( ( 4  x.  A )  x.  -u 1 )  x.  C )
7560, 72mulcomi 9605 . . . . . . . . . . . . 13  |-  ( ( 4  x.  A )  x.  -u 1 )  =  ( -u 1  x.  ( 4  x.  A
) )
7675oveq1i 6291 . . . . . . . . . . . 12  |-  ( ( ( 4  x.  A
)  x.  -u 1
)  x.  C )  =  ( ( -u
1  x.  ( 4  x.  A ) )  x.  C )
7772, 60, 38mulassi 9608 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  (
4  x.  A ) )  x.  C )  =  ( -u 1  x.  ( ( 4  x.  A )  x.  C
) )
7874, 76, 773eqtri 2476 . . . . . . . . . . 11  |-  ( ( 4  x.  A )  x.  -u C )  =  ( -u 1  x.  ( ( 4  x.  A )  x.  C
) )
7959, 2, 38mulassi 9608 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  x.  C )  =  ( 4  x.  ( A  x.  C )
)
8079oveq2i 6292 . . . . . . . . . . 11  |-  ( -u
1  x.  ( ( 4  x.  A )  x.  C ) )  =  ( -u 1  x.  ( 4  x.  ( A  x.  C )
) )
812, 38mulcli 9604 . . . . . . . . . . . . 13  |-  ( A  x.  C )  e.  CC
8259, 81mulcli 9604 . . . . . . . . . . . 12  |-  ( 4  x.  ( A  x.  C ) )  e.  CC
8382mulm1i 10007 . . . . . . . . . . 11  |-  ( -u
1  x.  ( 4  x.  ( A  x.  C ) ) )  =  -u ( 4  x.  ( A  x.  C
) )
8478, 80, 833eqtri 2476 . . . . . . . . . 10  |-  ( ( 4  x.  A )  x.  -u C )  = 
-u ( 4  x.  ( A  x.  C
) )
85 2t2e4 10691 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  2 )  =  4
8685eqcomi 2456 . . . . . . . . . . . . . . 15  |-  4  =  ( 2  x.  2 )
8786oveq1i 6291 . . . . . . . . . . . . . 14  |-  ( 4  x.  A )  =  ( ( 2  x.  2 )  x.  A
)
8887oveq1i 6291 . . . . . . . . . . . . 13  |-  ( ( 4  x.  A )  x.  A )  =  ( ( ( 2  x.  2 )  x.  A )  x.  A
)
891, 1, 2mulassi 9608 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) )
9089oveq1i 6291 . . . . . . . . . . . . 13  |-  ( ( ( 2  x.  2 )  x.  A )  x.  A )  =  ( ( 2  x.  ( 2  x.  A
) )  x.  A
)
9188, 90eqtri 2472 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  x.  A )  =  ( ( 2  x.  ( 2  x.  A
) )  x.  A
)
921, 3mulcomi 9605 . . . . . . . . . . . . 13  |-  ( 2  x.  ( 2  x.  A ) )  =  ( ( 2  x.  A )  x.  2 )
9392oveq1i 6291 . . . . . . . . . . . 12  |-  ( ( 2  x.  ( 2  x.  A ) )  x.  A )  =  ( ( ( 2  x.  A )  x.  2 )  x.  A
)
943, 1, 2mulassi 9608 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) )
9591, 93, 943eqtri 2476 . . . . . . . . . . 11  |-  ( ( 4  x.  A )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) )
963sqvali 12226 . . . . . . . . . . 11  |-  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) )
9795, 96eqtr4i 2475 . . . . . . . . . 10  |-  ( ( 4  x.  A )  x.  A )  =  ( ( 2  x.  A ) ^ 2 )
9884, 97oveq12i 6293 . . . . . . . . 9  |-  ( ( ( 4  x.  A
)  x.  -u C
)  /  ( ( 4  x.  A )  x.  A ) )  =  ( -u (
4  x.  ( A  x.  C ) )  /  ( ( 2  x.  A ) ^
2 ) )
9963, 68, 983eqtr3i 2480 . . . . . . . 8  |-  ( -u C  /  A )  =  ( -u ( 4  x.  ( A  x.  C ) )  / 
( ( 2  x.  A ) ^ 2 ) )
10058, 99oveq12i 6293 . . . . . . 7  |-  ( ( ( B  /  (
2  x.  A ) ) ^ 2 )  +  ( -u C  /  A ) )  =  ( ( ( B ^ 2 )  / 
( ( 2  x.  A ) ^ 2 ) )  +  (
-u ( 4  x.  ( A  x.  C
) )  /  (
( 2  x.  A
) ^ 2 ) ) )
1015sqcli 12227 . . . . . . . 8  |-  ( B ^ 2 )  e.  CC
10282negcli 9892 . . . . . . . 8  |-  -u (
4  x.  ( A  x.  C ) )  e.  CC
1033sqcli 12227 . . . . . . . 8  |-  ( ( 2  x.  A ) ^ 2 )  e.  CC
1043, 3, 8, 8mulne0i 10198 . . . . . . . . 9  |-  ( ( 2  x.  A )  x.  ( 2  x.  A ) )  =/=  0
10596, 104eqnetri 2739 . . . . . . . 8  |-  ( ( 2  x.  A ) ^ 2 )  =/=  0
106101, 102, 103, 105divdiri 10307 . . . . . . 7  |-  ( ( ( B ^ 2 )  +  -u (
4  x.  ( A  x.  C ) ) )  /  ( ( 2  x.  A ) ^ 2 ) )  =  ( ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^
2 ) )  +  ( -u ( 4  x.  ( A  x.  C ) )  / 
( ( 2  x.  A ) ^ 2 ) ) )
107101, 82negsubi 9902 . . . . . . . 8  |-  ( ( B ^ 2 )  +  -u ( 4  x.  ( A  x.  C
) ) )  =  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )
108107oveq1i 6291 . . . . . . 7  |-  ( ( ( B ^ 2 )  +  -u (
4  x.  ( A  x.  C ) ) )  /  ( ( 2  x.  A ) ^ 2 ) )  =  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  / 
( ( 2  x.  A ) ^ 2 ) )
109100, 106, 1083eqtr2i 2478 . . . . . 6  |-  ( ( ( B  /  (
2  x.  A ) ) ^ 2 )  +  ( -u C  /  A ) )  =  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  /  (
( 2  x.  A
) ^ 2 ) )
11053, 57, 1093eqtri 2476 . . . . 5  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 )  =  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  /  (
( 2  x.  A
) ^ 2 ) )
111110oveq2i 6292 . . . 4  |-  ( ( ( 2  x.  A
) ^ 2 )  x.  ( ( X  +  ( B  / 
( 2  x.  A
) ) ) ^
2 ) )  =  ( ( ( 2  x.  A ) ^
2 )  x.  (
( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  /  ( ( 2  x.  A ) ^ 2 ) ) )
112101, 82subcli 9900 . . . . 5  |-  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  e.  CC
113112, 103, 105divcan2i 10293 . . . 4  |-  ( ( ( 2  x.  A
) ^ 2 )  x.  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  / 
( ( 2  x.  A ) ^ 2 ) ) )  =  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )
11411, 111, 1133eqtri 2476 . . 3  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) ) ^ 2 )  =  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )
1153, 10mulcli 9604 . . . . 5  |-  ( ( 2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  e.  CC
116115, 112pm3.2i 455 . . . 4  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  e.  CC  /\  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  e.  CC )
117 eqsqrtor 13178 . . . 4  |-  ( ( ( ( 2  x.  A )  x.  ( X  +  ( B  /  ( 2  x.  A ) ) ) )  e.  CC  /\  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  e.  CC )  ->  ( ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) ) ^ 2 )  =  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <->  ( ( ( 2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  =  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  \/  ( ( 2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  = 
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) ) ) )
118116, 117ax-mp 5 . . 3  |-  ( ( ( ( 2  x.  A )  x.  ( X  +  ( B  /  ( 2  x.  A ) ) ) ) ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <->  ( (
( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  \/  ( ( 2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  = 
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) ) )
119114, 118mpbi 208 . 2  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  \/  ( ( 2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  = 
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )
120 sqrtcl 13173 . . . . . . 7  |-  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  e.  CC  ->  ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  e.  CC )
121112, 120ax-mp 5 . . . . . 6  |-  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  e.  CC
122121, 3, 10, 8divmuli 10304 . . . . 5  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  =  ( X  +  ( B  /  (
2  x.  A ) ) )  <->  ( (
2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  =  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )
123 eqcom 2452 . . . . 5  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  =  ( X  +  ( B  /  (
2  x.  A ) ) )  <->  ( X  +  ( B  / 
( 2  x.  A
) ) )  =  ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
124122, 123bitr3i 251 . . . 4  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  <->  ( X  +  ( B  /  (
2  x.  A ) ) )  =  ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
125121, 3, 8divcli 10292 . . . . . 6  |-  ( ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )  /  ( 2  x.  A ) )  e.  CC
126125, 9, 4subadd2i 9913 . . . . 5  |-  ( ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) )  =  X  <->  ( X  +  ( B  /  (
2  x.  A ) ) )  =  ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
127 eqcom 2452 . . . . 5  |-  ( ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) )  =  X  <->  X  =  (
( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) ) )
128126, 127bitr3i 251 . . . 4  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) )  =  ( ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) )  <->  X  =  (
( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) ) )
129 divneg 10245 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  ( 2  x.  A
)  e.  CC  /\  ( 2  x.  A
)  =/=  0 )  ->  -u ( B  / 
( 2  x.  A
) )  =  (
-u B  /  (
2  x.  A ) ) )
1305, 3, 8, 129mp3an 1325 . . . . . . . 8  |-  -u ( B  /  ( 2  x.  A ) )  =  ( -u B  / 
( 2  x.  A
) )
131130oveq2i 6292 . . . . . . 7  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  +  -u ( B  / 
( 2  x.  A
) ) )  =  ( ( ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) )  +  ( -u B  /  ( 2  x.  A ) ) )
132125, 9negsubi 9902 . . . . . . 7  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  +  -u ( B  / 
( 2  x.  A
) ) )  =  ( ( ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) )  -  ( B  /  ( 2  x.  A ) ) )
1335negcli 9892 . . . . . . . . 9  |-  -u B  e.  CC
134133, 3, 8divcli 10292 . . . . . . . 8  |-  ( -u B  /  ( 2  x.  A ) )  e.  CC
135125, 134addcomi 9774 . . . . . . 7  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  +  ( -u B  /  ( 2  x.  A ) ) )  =  ( ( -u B  /  ( 2  x.  A ) )  +  ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
136131, 132, 1353eqtr3i 2480 . . . . . 6  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) )  =  ( ( -u B  /  ( 2  x.  A ) )  +  ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
137133, 121, 3, 8divdiri 10307 . . . . . 6  |-  ( (
-u B  +  ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) ) )  /  ( 2  x.  A ) )  =  ( ( -u B  /  ( 2  x.  A ) )  +  ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
138136, 137eqtr4i 2475 . . . . 5  |-  ( ( ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) )  =  ( ( -u B  +  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )  /  (
2  x.  A ) )
139138eqeq2i 2461 . . . 4  |-  ( X  =  ( ( ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )  /  ( 2  x.  A ) )  -  ( B  /  (
2  x.  A ) ) )  <->  X  =  ( ( -u B  +  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )  /  (
2  x.  A ) ) )
140124, 128, 1393bitri 271 . . 3  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  <->  X  =  (
( -u B  +  ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) ) )  /  ( 2  x.  A ) ) )
141121negcli 9892 . . . . . 6  |-  -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  e.  CC
142141, 3, 10, 8divmuli 10304 . . . . 5  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  =  ( X  +  ( B  /  (
2  x.  A ) ) )  <->  ( (
2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  = 
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )
143 eqcom 2452 . . . . 5  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  =  ( X  +  ( B  /  (
2  x.  A ) ) )  <->  ( X  +  ( B  / 
( 2  x.  A
) ) )  =  ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) ) )
144142, 143bitr3i 251 . . . 4  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  -u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  <->  ( X  +  ( B  /  (
2  x.  A ) ) )  =  (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) ) )
145141, 3, 8divcli 10292 . . . . . 6  |-  ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )  /  ( 2  x.  A ) )  e.  CC
146145, 9, 4subadd2i 9913 . . . . 5  |-  ( ( ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) )  -  ( B  /  ( 2  x.  A ) ) )  =  X  <->  ( X  +  ( B  / 
( 2  x.  A
) ) )  =  ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) ) )
147 eqcom 2452 . . . . 5  |-  ( ( ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) )  -  ( B  /  ( 2  x.  A ) ) )  =  X  <->  X  =  ( ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  / 
( 2  x.  A
) )  -  ( B  /  ( 2  x.  A ) ) ) )
148146, 147bitr3i 251 . . . 4  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) )  =  ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  / 
( 2  x.  A
) )  <->  X  =  ( ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  / 
( 2  x.  A
) )  -  ( B  /  ( 2  x.  A ) ) ) )
149130oveq2i 6292 . . . . . . 7  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  +  -u ( B  / 
( 2  x.  A
) ) )  =  ( ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  / 
( 2  x.  A
) )  +  (
-u B  /  (
2  x.  A ) ) )
150145, 9negsubi 9902 . . . . . . 7  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  +  -u ( B  / 
( 2  x.  A
) ) )  =  ( ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  / 
( 2  x.  A
) )  -  ( B  /  ( 2  x.  A ) ) )
151145, 134addcomi 9774 . . . . . . 7  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  +  ( -u B  /  ( 2  x.  A ) ) )  =  ( ( -u B  /  ( 2  x.  A ) )  +  ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) ) )
152149, 150, 1513eqtr3i 2480 . . . . . 6  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) )  =  ( ( -u B  /  ( 2  x.  A ) )  +  ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) ) )
153133, 141, 3, 8divdiri 10307 . . . . . 6  |-  ( (
-u B  +  -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) ) )  /  ( 2  x.  A ) )  =  ( ( -u B  /  ( 2  x.  A ) )  +  ( -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  /  (
2  x.  A ) ) )
154133, 121negsubi 9902 . . . . . . 7  |-  ( -u B  +  -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  =  ( -u B  -  ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) ) )
155154oveq1i 6291 . . . . . 6  |-  ( (
-u B  +  -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) ) )  /  ( 2  x.  A ) )  =  ( ( -u B  -  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) )
156152, 153, 1553eqtr2i 2478 . . . . 5  |-  ( (
-u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  /  ( 2  x.  A ) )  -  ( B  / 
( 2  x.  A
) ) )  =  ( ( -u B  -  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )  /  (
2  x.  A ) )
157156eqeq2i 2461 . . . 4  |-  ( X  =  ( ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )  /  ( 2  x.  A ) )  -  ( B  /  (
2  x.  A ) ) )  <->  X  =  ( ( -u B  -  ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )  /  (
2  x.  A ) ) )
158144, 148, 1573bitri 271 . . 3  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  -u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  <->  X  =  (
( -u B  -  ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) ) )  /  ( 2  x.  A ) ) )
159140, 158orbi12i 521 . 2  |-  ( ( ( ( 2  x.  A )  x.  ( X  +  ( B  /  ( 2  x.  A ) ) ) )  =  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) )  \/  (
( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  =  -u ( sqr `  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) ) )  <->  ( X  =  ( ( -u B  +  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) )  \/  X  =  ( ( -u B  -  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) ) ) )
160119, 159mpbi 208 1  |-  ( X  =  ( ( -u B  +  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) )  \/  X  =  ( ( -u B  -  ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  / 
( 2  x.  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    - cmin 9810   -ucneg 9811    / cdiv 10212   2c2 10591   4c4 10593   ^cexp 12145   sqrcsqrt 13045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048
This theorem is referenced by: (None)
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