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Theorem quad3 29288
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 10602 . . . . . 6  |-  2  e.  CC
2 quad3.2 . . . . . 6  |-  A  e.  CC
31, 2mulcli 9590 . . . . 5  |-  ( 2  x.  A )  e.  CC
4 quad3.1 . . . . . 6  |-  X  e.  CC
5 quad3.4 . . . . . . 7  |-  B  e.  CC
6 2ne0 10624 . . . . . . . 8  |-  2  =/=  0
7 quad3.3 . . . . . . . 8  |-  A  =/=  0
81, 2, 6, 7mulne0i 10188 . . . . . . 7  |-  ( 2  x.  A )  =/=  0
95, 3, 8divcli 10282 . . . . . 6  |-  ( B  /  ( 2  x.  A ) )  e.  CC
104, 9addcli 9589 . . . . 5  |-  ( X  +  ( B  / 
( 2  x.  A
) ) )  e.  CC
113, 10sqmuli 12233 . . . 4  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) ) ^ 2 )  =  ( ( ( 2  x.  A ) ^
2 )  x.  (
( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 ) )
124, 9binom2i 12259 . . . . . . 7  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 )  =  ( ( ( X ^ 2 )  +  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) ) )  +  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
134sqcli 12230 . . . . . . . . . . . 12  |-  ( X ^ 2 )  e.  CC
142, 13mulcli 9590 . . . . . . . . . . 11  |-  ( A  x.  ( X ^
2 ) )  e.  CC
155, 4mulcli 9590 . . . . . . . . . . 11  |-  ( B  x.  X )  e.  CC
1614, 15, 2, 7divdiri 10297 . . . . . . . . . 10  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  /  A )  =  ( ( ( A  x.  ( X ^ 2 ) )  /  A )  +  ( ( B  x.  X )  /  A ) )
1713, 2, 7divcan3i 10286 . . . . . . . . . . 11  |-  ( ( A  x.  ( X ^ 2 ) )  /  A )  =  ( X ^ 2 )
185, 4, 2, 7div23i 10298 . . . . . . . . . . 11  |-  ( ( B  x.  X )  /  A )  =  ( ( B  /  A )  x.  X
)
1917, 18oveq12i 6282 . . . . . . . . . 10  |-  ( ( ( A  x.  ( X ^ 2 ) )  /  A )  +  ( ( B  x.  X )  /  A
) )  =  ( ( X ^ 2 )  +  ( ( B  /  A )  x.  X ) )
2016, 19eqtr2i 2484 . . . . . . . . 9  |-  ( ( X ^ 2 )  +  ( ( B  /  A )  x.  X ) )  =  ( ( ( A  x.  ( X ^
2 ) )  +  ( B  x.  X
) )  /  A
)
215, 2, 7divcli 10282 . . . . . . . . . . . 12  |-  ( B  /  A )  e.  CC
2221, 4mulcomi 9591 . . . . . . . . . . 11  |-  ( ( B  /  A )  x.  X )  =  ( X  x.  ( B  /  A ) )
234, 21mulcli 9590 . . . . . . . . . . . 12  |-  ( X  x.  ( B  /  A ) )  e.  CC
2423, 1, 6divcan2i 10283 . . . . . . . . . . 11  |-  ( 2  x.  ( ( X  x.  ( B  /  A ) )  / 
2 ) )  =  ( X  x.  ( B  /  A ) )
254, 21, 1, 6divassi 10296 . . . . . . . . . . . . 13  |-  ( ( X  x.  ( B  /  A ) )  /  2 )  =  ( X  x.  (
( B  /  A
)  /  2 ) )
262, 7pm3.2i 453 . . . . . . . . . . . . . . . 16  |-  ( A  e.  CC  /\  A  =/=  0 )
271, 6pm3.2i 453 . . . . . . . . . . . . . . . 16  |-  ( 2  e.  CC  /\  2  =/=  0 )
28 divdiv1 10251 . . . . . . . . . . . . . . . 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 1322 . . . . . . . . . . . . . . 15  |-  ( ( B  /  A )  /  2 )  =  ( B  /  ( A  x.  2 ) )
302, 1mulcomi 9591 . . . . . . . . . . . . . . . 16  |-  ( A  x.  2 )  =  ( 2  x.  A
)
3130oveq2i 6281 . . . . . . . . . . . . . . 15  |-  ( B  /  ( A  x.  2 ) )  =  ( B  /  (
2  x.  A ) )
3229, 31eqtri 2483 . . . . . . . . . . . . . 14  |-  ( ( B  /  A )  /  2 )  =  ( B  /  (
2  x.  A ) )
3332oveq2i 6281 . . . . . . . . . . . . 13  |-  ( X  x.  ( ( B  /  A )  / 
2 ) )  =  ( X  x.  ( B  /  ( 2  x.  A ) ) )
3425, 33eqtri 2483 . . . . . . . . . . . 12  |-  ( ( X  x.  ( B  /  A ) )  /  2 )  =  ( X  x.  ( B  /  ( 2  x.  A ) ) )
3534oveq2i 6281 . . . . . . . . . . 11  |-  ( 2  x.  ( ( X  x.  ( B  /  A ) )  / 
2 ) )  =  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) )
3622, 24, 353eqtr2i 2489 . . . . . . . . . 10  |-  ( ( B  /  A )  x.  X )  =  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) )
3736oveq2i 6281 . . . . . . . . 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 9593 . . . . . . . . . . . . . 14  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  +  C )  =  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )
4039eqcomi 2467 . . . . . . . . . . . . 13  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  ( ( ( A  x.  ( X ^
2 ) )  +  ( B  x.  X
) )  +  C
)
4140oveq1i 6280 . . . . . . . . . . . 12  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  -  C )  =  ( ( ( ( A  x.  ( X ^
2 ) )  +  ( B  x.  X
) )  +  C
)  -  C )
4214, 15addcli 9589 . . . . . . . . . . . . 13  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  e.  CC
4342, 38pncan3oi 9827 . . . . . . . . . . . 12  |-  ( ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  +  C )  -  C )  =  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )
4441, 43eqtr2i 2484 . . . . . . . . . . 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 6280 . . . . . . . . . . . 12  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  -  C )  =  ( 0  -  C )
47 df-neg 9799 . . . . . . . . . . . 12  |-  -u C  =  ( 0  -  C )
4846, 47eqtr4i 2486 . . . . . . . . . . 11  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  -  C )  =  -u C
4944, 48eqtri 2483 . . . . . . . . . 10  |-  ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  = 
-u C
5049oveq1i 6280 . . . . . . . . 9  |-  ( ( ( A  x.  ( X ^ 2 ) )  +  ( B  x.  X ) )  /  A )  =  (
-u C  /  A
)
5120, 37, 503eqtr3i 2491 . . . . . . . 8  |-  ( ( X ^ 2 )  +  ( 2  x.  ( X  x.  ( B  /  ( 2  x.  A ) ) ) ) )  =  (
-u C  /  A
)
5251oveq1i 6280 . . . . . . 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 2483 . . . . . 6  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 )  =  ( ( -u C  /  A )  +  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
5438negcli 9878 . . . . . . . 8  |-  -u C  e.  CC
5554, 2, 7divcli 10282 . . . . . . 7  |-  ( -u C  /  A )  e.  CC
569sqcli 12230 . . . . . . 7  |-  ( ( B  /  ( 2  x.  A ) ) ^ 2 )  e.  CC
5755, 56addcomi 9760 . . . . . 6  |-  ( (
-u C  /  A
)  +  ( ( B  /  ( 2  x.  A ) ) ^ 2 ) )  =  ( ( ( B  /  ( 2  x.  A ) ) ^ 2 )  +  ( -u C  /  A ) )
585, 3, 8sqdivi 12234 . . . . . . . 8  |-  ( ( B  /  ( 2  x.  A ) ) ^ 2 )  =  ( ( B ^
2 )  /  (
( 2  x.  A
) ^ 2 ) )
59 4cn 10609 . . . . . . . . . . 11  |-  4  e.  CC
6059, 2mulcli 9590 . . . . . . . . . 10  |-  ( 4  x.  A )  e.  CC
61 4ne0 10628 . . . . . . . . . . 11  |-  4  =/=  0
6259, 2, 61, 7mulne0i 10188 . . . . . . . . . 10  |-  ( 4  x.  A )  =/=  0
6360, 60, 54, 2, 62, 7divmuldivi 10300 . . . . . . . . 9  |-  ( ( ( 4  x.  A
)  /  ( 4  x.  A ) )  x.  ( -u C  /  A ) )  =  ( ( ( 4  x.  A )  x.  -u C )  /  (
( 4  x.  A
)  x.  A ) )
6460, 62dividi 10273 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  /  ( 4  x.  A ) )  =  1
6564eqcomi 2467 . . . . . . . . . . 11  |-  1  =  ( ( 4  x.  A )  / 
( 4  x.  A
) )
6665oveq1i 6280 . . . . . . . . . 10  |-  ( 1  x.  ( -u C  /  A ) )  =  ( ( ( 4  x.  A )  / 
( 4  x.  A
) )  x.  ( -u C  /  A ) )
6755mulid2i 9588 . . . . . . . . . 10  |-  ( 1  x.  ( -u C  /  A ) )  =  ( -u C  /  A )
6866, 67eqtr3i 2485 . . . . . . . . 9  |-  ( ( ( 4  x.  A
)  /  ( 4  x.  A ) )  x.  ( -u C  /  A ) )  =  ( -u C  /  A )
6938mulm1i 9997 . . . . . . . . . . . . . . 15  |-  ( -u
1  x.  C )  =  -u C
7069eqcomi 2467 . . . . . . . . . . . . . 14  |-  -u C  =  ( -u 1  x.  C )
7170oveq2i 6281 . . . . . . . . . . . . 13  |-  ( ( 4  x.  A )  x.  -u C )  =  ( ( 4  x.  A )  x.  ( -u 1  x.  C ) )
72 neg1cn 10635 . . . . . . . . . . . . . 14  |-  -u 1  e.  CC
7360, 72, 38mulassi 9594 . . . . . . . . . . . . 13  |-  ( ( ( 4  x.  A
)  x.  -u 1
)  x.  C )  =  ( ( 4  x.  A )  x.  ( -u 1  x.  C ) )
7471, 73eqtr4i 2486 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  x.  -u C )  =  ( ( ( 4  x.  A )  x.  -u 1 )  x.  C )
7560, 72mulcomi 9591 . . . . . . . . . . . . 13  |-  ( ( 4  x.  A )  x.  -u 1 )  =  ( -u 1  x.  ( 4  x.  A
) )
7675oveq1i 6280 . . . . . . . . . . . 12  |-  ( ( ( 4  x.  A
)  x.  -u 1
)  x.  C )  =  ( ( -u
1  x.  ( 4  x.  A ) )  x.  C )
7772, 60, 38mulassi 9594 . . . . . . . . . . . 12  |-  ( (
-u 1  x.  (
4  x.  A ) )  x.  C )  =  ( -u 1  x.  ( ( 4  x.  A )  x.  C
) )
7874, 76, 773eqtri 2487 . . . . . . . . . . 11  |-  ( ( 4  x.  A )  x.  -u C )  =  ( -u 1  x.  ( ( 4  x.  A )  x.  C
) )
7959, 2, 38mulassi 9594 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  x.  C )  =  ( 4  x.  ( A  x.  C )
)
8079oveq2i 6281 . . . . . . . . . . 11  |-  ( -u
1  x.  ( ( 4  x.  A )  x.  C ) )  =  ( -u 1  x.  ( 4  x.  ( A  x.  C )
) )
812, 38mulcli 9590 . . . . . . . . . . . . 13  |-  ( A  x.  C )  e.  CC
8259, 81mulcli 9590 . . . . . . . . . . . 12  |-  ( 4  x.  ( A  x.  C ) )  e.  CC
8382mulm1i 9997 . . . . . . . . . . 11  |-  ( -u
1  x.  ( 4  x.  ( A  x.  C ) ) )  =  -u ( 4  x.  ( A  x.  C
) )
8478, 80, 833eqtri 2487 . . . . . . . . . 10  |-  ( ( 4  x.  A )  x.  -u C )  = 
-u ( 4  x.  ( A  x.  C
) )
85 2t2e4 10681 . . . . . . . . . . . . . . . 16  |-  ( 2  x.  2 )  =  4
8685eqcomi 2467 . . . . . . . . . . . . . . 15  |-  4  =  ( 2  x.  2 )
8786oveq1i 6280 . . . . . . . . . . . . . 14  |-  ( 4  x.  A )  =  ( ( 2  x.  2 )  x.  A
)
8887oveq1i 6280 . . . . . . . . . . . . 13  |-  ( ( 4  x.  A )  x.  A )  =  ( ( ( 2  x.  2 )  x.  A )  x.  A
)
891, 1, 2mulassi 9594 . . . . . . . . . . . . . 14  |-  ( ( 2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) )
9089oveq1i 6280 . . . . . . . . . . . . 13  |-  ( ( ( 2  x.  2 )  x.  A )  x.  A )  =  ( ( 2  x.  ( 2  x.  A
) )  x.  A
)
9188, 90eqtri 2483 . . . . . . . . . . . 12  |-  ( ( 4  x.  A )  x.  A )  =  ( ( 2  x.  ( 2  x.  A
) )  x.  A
)
921, 3mulcomi 9591 . . . . . . . . . . . . 13  |-  ( 2  x.  ( 2  x.  A ) )  =  ( ( 2  x.  A )  x.  2 )
9392oveq1i 6280 . . . . . . . . . . . 12  |-  ( ( 2  x.  ( 2  x.  A ) )  x.  A )  =  ( ( ( 2  x.  A )  x.  2 )  x.  A
)
943, 1, 2mulassi 9594 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) )
9591, 93, 943eqtri 2487 . . . . . . . . . . 11  |-  ( ( 4  x.  A )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) )
963sqvali 12229 . . . . . . . . . . 11  |-  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) )
9795, 96eqtr4i 2486 . . . . . . . . . 10  |-  ( ( 4  x.  A )  x.  A )  =  ( ( 2  x.  A ) ^ 2 )
9884, 97oveq12i 6282 . . . . . . . . 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 2491 . . . . . . . 8  |-  ( -u C  /  A )  =  ( -u ( 4  x.  ( A  x.  C ) )  / 
( ( 2  x.  A ) ^ 2 ) )
10058, 99oveq12i 6282 . . . . . . 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 12230 . . . . . . . 8  |-  ( B ^ 2 )  e.  CC
10282negcli 9878 . . . . . . . 8  |-  -u (
4  x.  ( A  x.  C ) )  e.  CC
1033sqcli 12230 . . . . . . . 8  |-  ( ( 2  x.  A ) ^ 2 )  e.  CC
1043, 3, 8, 8mulne0i 10188 . . . . . . . . 9  |-  ( ( 2  x.  A )  x.  ( 2  x.  A ) )  =/=  0
10596, 104eqnetri 2750 . . . . . . . 8  |-  ( ( 2  x.  A ) ^ 2 )  =/=  0
106101, 102, 103, 105divdiri 10297 . . . . . . 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 9888 . . . . . . . 8  |-  ( ( B ^ 2 )  +  -u ( 4  x.  ( A  x.  C
) ) )  =  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )
108107oveq1i 6280 . . . . . . 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 2489 . . . . . 6  |-  ( ( ( B  /  (
2  x.  A ) ) ^ 2 )  +  ( -u C  /  A ) )  =  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  /  (
( 2  x.  A
) ^ 2 ) )
11053, 57, 1093eqtri 2487 . . . . 5  |-  ( ( X  +  ( B  /  ( 2  x.  A ) ) ) ^ 2 )  =  ( ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  /  (
( 2  x.  A
) ^ 2 ) )
111110oveq2i 6281 . . . 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 9886 . . . . 5  |-  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) )  e.  CC
113112, 103, 105divcan2i 10283 . . . 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 2487 . . 3  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) ) ^ 2 )  =  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )
1153, 10mulcli 9590 . . . . 5  |-  ( ( 2  x.  A )  x.  ( X  +  ( B  /  (
2  x.  A ) ) ) )  e.  CC
116115, 112pm3.2i 453 . . . 4  |-  ( ( ( 2  x.  A
)  x.  ( X  +  ( B  / 
( 2  x.  A
) ) ) )  e.  CC  /\  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  e.  CC )
117 eqsqrtor 13281 . . . 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 13276 . . . . . . 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 10294 . . . . 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 2463 . . . . 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 10282 . . . . . 6  |-  ( ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )  /  ( 2  x.  A ) )  e.  CC
126125, 9, 4subadd2i 9899 . . . . 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 2463 . . . . 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 10235 . . . . . . . . 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 1322 . . . . . . . 8  |-  -u ( B  /  ( 2  x.  A ) )  =  ( -u B  / 
( 2  x.  A
) )
131130oveq2i 6281 . . . . . . 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 9888 . . . . . . 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 9878 . . . . . . . . 9  |-  -u B  e.  CC
134133, 3, 8divcli 10282 . . . . . . . 8  |-  ( -u B  /  ( 2  x.  A ) )  e.  CC
135125, 134addcomi 9760 . . . . . . 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 2491 . . . . . 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 10297 . . . . . 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 2486 . . . . 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 2472 . . . 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 9878 . . . . . 6  |-  -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) ) )  e.  CC
142141, 3, 10, 8divmuli 10294 . . . . 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 2463 . . . . 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 10282 . . . . . 6  |-  ( -u ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )  /  ( 2  x.  A ) )  e.  CC
146145, 9, 4subadd2i 9899 . . . . 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 2463 . . . . 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 6281 . . . . . . 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 9888 . . . . . . 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 9760 . . . . . . 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 2491 . . . . . 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 10297 . . . . . 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 9888 . . . . . . 7  |-  ( -u B  +  -u ( sqr `  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )  =  ( -u B  -  ( sqr `  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) ) )
155154oveq1i 6280 . . . . . 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 2489 . . . . 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 2472 . . . 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 519 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   2c2 10581   4c4 10583   ^cexp 12148   sqrcsqrt 13148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151
This theorem is referenced by: (None)
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