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Theorem discr 12271
Description: If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is nonpositive. (Contributed by NM, 10-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.)
Hypotheses
Ref Expression
discr.1  |-  ( ph  ->  A  e.  RR )
discr.2  |-  ( ph  ->  B  e.  RR )
discr.3  |-  ( ph  ->  C  e.  RR )
discr.4  |-  ( (
ph  /\  x  e.  RR )  ->  0  <_ 
( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
Assertion
Ref Expression
discr  |-  ( ph  ->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
Distinct variable groups:    x, A    x, B    x, C    ph, x

Proof of Theorem discr
StepHypRef Expression
1 discr.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
21adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  B  e.  RR )
3 resqcl 12203 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
42, 3syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  RR )
54recnd 9622 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  CC )
6 4re 10612 . . . . . . . . 9  |-  4  e.  RR
7 discr.1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
87adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR )
9 discr.3 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
109adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  C  e.  RR )
118, 10remulcld 9624 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  RR )
12 remulcl 9577 . . . . . . . . 9  |-  ( ( 4  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( 4  x.  ( A  x.  C
) )  e.  RR )
136, 11, 12sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  RR )
1413recnd 9622 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  CC )
15 4pos 10631 . . . . . . . . . 10  |-  0  <  4
166, 15elrpii 11223 . . . . . . . . 9  |-  4  e.  RR+
17 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <  A )
188, 17elrpd 11254 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
19 rpmulcl 11241 . . . . . . . . 9  |-  ( ( 4  e.  RR+  /\  A  e.  RR+ )  ->  (
4  x.  A )  e.  RR+ )
2016, 18, 19sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  RR+ )
2120rpcnd 11258 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  CC )
2220rpne0d 11261 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =/=  0 )
235, 14, 21, 22divsubdird 10359 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  (
( 4  x.  ( A  x.  C )
)  /  ( 4  x.  A ) ) ) )
2411recnd 9622 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  CC )
258recnd 9622 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
26 4cn 10613 . . . . . . . . . 10  |-  4  e.  CC
2726a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  e.  CC )
2818rpne0d 11261 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
29 4ne0 10632 . . . . . . . . . 10  |-  4  =/=  0
3029a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  =/=  0 )
3124, 25, 27, 28, 30divcan5d 10346 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  ( ( A  x.  C )  /  A
) )
3210recnd 9622 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  C  e.  CC )
3332, 25, 28divcan3d 10325 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  C )  /  A )  =  C )
3431, 33eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  C )
3534oveq2d 6300 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  -  ( ( 4  x.  ( A  x.  C ) )  / 
( 4  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  C
) )
3623, 35eqtrd 2508 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  C
) )
374, 20rerpdivcld 11283 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  RR )
3837recnd 9622 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  CC )
39382timesd 10781 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  ( ( B ^ 2 )  /  ( 4  x.  A ) ) ) )
40 2t2e4 10685 . . . . . . . . . . . . 13  |-  ( 2  x.  2 )  =  4
4140oveq1i 6294 . . . . . . . . . . . 12  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
42 2cnd 10608 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  2  e.  CC )
4342, 42, 25mulassd 9619 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4441, 43syl5eqr 2522 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4544oveq2d 6300 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( ( 2  x.  ( B ^ 2 ) )  /  (
2  x.  ( 2  x.  A ) ) ) )
4642, 5, 21, 22divassd 10355 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( 2  x.  (
( B ^ 2 )  /  ( 4  x.  A ) ) ) )
47 2rp 11225 . . . . . . . . . . . . 13  |-  2  e.  RR+
48 rpmulcl 11241 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR+  /\  A  e.  RR+ )  ->  (
2  x.  A )  e.  RR+ )
4947, 18, 48sylancr 663 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  RR+ )
5049rpcnd 11258 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  CC )
5149rpne0d 11261 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  =/=  0 )
52 2ne0 10628 . . . . . . . . . . . 12  |-  2  =/=  0
5352a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  2  =/=  0 )
545, 50, 42, 51, 53divcan5d 10346 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 2  x.  ( 2  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5545, 46, 543eqtr3d 2516 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5639, 55eqtr3d 2510 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
572, 49rerpdivcld 11283 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  RR )
5857renegcld 9986 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  -u ( B  /  ( 2  x.  A ) )  e.  RR )
59 discr.4 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  0  <_ 
( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
6059ralrimiva 2878 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
6160adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A. x  e.  RR  0  <_  (
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
62 oveq1 6291 . . . . . . . . . . . . . . . 16  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
x ^ 2 )  =  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )
6362oveq2d 6300 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( A  x.  ( x ^ 2 ) )  =  ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) ) )
64 oveq2 6292 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( B  x.  x )  =  ( B  x.  -u ( B  /  (
2  x.  A ) ) ) )
6563, 64oveq12d 6302 . . . . . . . . . . . . . 14  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
( A  x.  (
x ^ 2 ) )  +  ( B  x.  x ) )  =  ( ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) ) )
6665oveq1d 6299 . . . . . . . . . . . . 13  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C )  =  ( ( ( A  x.  ( -u ( B  /  (
2  x.  A ) ) ^ 2 ) )  +  ( B  x.  -u ( B  / 
( 2  x.  A
) ) ) )  +  C ) )
6766breq2d 4459 . . . . . . . . . . . 12  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
0  <_  ( (
( A  x.  (
x ^ 2 ) )  +  ( B  x.  x ) )  +  C )  <->  0  <_  ( ( ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  +  C ) ) )
6867rspcv 3210 . . . . . . . . . . 11  |-  ( -u ( B  /  (
2  x.  A ) )  e.  RR  ->  ( A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C )  ->  0  <_  ( ( ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  +  C ) ) )
6958, 61, 68sylc 60 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  +  C ) )
7057recnd 9622 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  CC )
71 sqneg 12196 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  /  ( 2  x.  A ) )  e.  CC  ->  ( -u ( B  /  (
2  x.  A ) ) ^ 2 )  =  ( ( B  /  ( 2  x.  A ) ) ^
2 ) )
7270, 71syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
732recnd 9622 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  B  e.  CC )
74 sqdiv 12201 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  CC  /\  ( 2  x.  A
)  e.  CC  /\  ( 2  x.  A
)  =/=  0 )  ->  ( ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^ 2 ) ) )
7573, 50, 51, 74syl3anc 1228 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^ 2 ) ) )
76 sqval 12195 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 2  x.  A )  e.  CC  ->  (
( 2  x.  A
) ^ 2 )  =  ( ( 2  x.  A )  x.  ( 2  x.  A
) ) )
7750, 76syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7850, 42, 25mulassd 9619 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7942, 25, 42mul32d 9789 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( ( 2  x.  2 )  x.  A
) )
8079, 41syl6eq 2524 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( 4  x.  A
) )
8180oveq1d 6299 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 4  x.  A )  x.  A
) )
8277, 78, 813eqtr2d 2514 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A
) )
8382oveq2d 6300 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( ( 2  x.  A ) ^ 2 ) )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
8472, 75, 833eqtrd 2512 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
855, 21, 25, 22, 28divdiv1d 10351 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  /  A )  =  ( ( B ^
2 )  /  (
( 4  x.  A
)  x.  A ) ) )
8684, 85eqtr4d 2511 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) )
8786oveq2d 6300 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( A  x.  (
( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) ) )
8838, 25, 28divcan2d 10322 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  /  A
) )  =  ( ( B ^ 2 )  /  ( 4  x.  A ) ) )
8987, 88eqtrd 2508 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( ( B ^
2 )  /  (
4  x.  A ) ) )
9073, 70mulneg2d 10010 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
91 sqval 12195 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9273, 91syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9392oveq1d 6299 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( ( B  x.  B
)  /  ( 2  x.  A ) ) )
9473, 73, 50, 51divassd 10355 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  x.  B )  /  ( 2  x.  A ) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9593, 94eqtrd 2508 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9695negeqd 9814 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  -u ( ( B ^ 2 )  /  ( 2  x.  A ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
9790, 96eqtr4d 2511 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( ( B ^ 2 )  / 
( 2  x.  A
) ) )
9889, 97oveq12d 6302 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  =  ( ( ( B ^ 2 )  /  ( 4  x.  A ) )  +  -u ( ( B ^ 2 )  / 
( 2  x.  A
) ) ) )
994, 49rerpdivcld 11283 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  RR )
10099recnd 9622 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  CC )
10138, 100negsubd 9936 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  -u ( ( B ^ 2 )  / 
( 2  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  (
( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10298, 101eqtrd 2508 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  =  ( ( ( B ^ 2 )  /  ( 4  x.  A ) )  -  ( ( B ^ 2 )  / 
( 2  x.  A
) ) ) )
103102oveq1d 6299 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( (
( A  x.  ( -u ( B  /  (
2  x.  A ) ) ^ 2 ) )  +  ( B  x.  -u ( B  / 
( 2  x.  A
) ) ) )  +  C )  =  ( ( ( ( B ^ 2 )  /  ( 4  x.  A ) )  -  ( ( B ^
2 )  /  (
2  x.  A ) ) )  +  C
) )
10438, 32, 100addsubd 9951 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( (
( ( B ^
2 )  /  (
4  x.  A ) )  +  C )  -  ( ( B ^ 2 )  / 
( 2  x.  A
) ) )  =  ( ( ( ( B ^ 2 )  /  ( 4  x.  A ) )  -  ( ( B ^
2 )  /  (
2  x.  A ) ) )  +  C
) )
105103, 104eqtr4d 2511 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
( A  x.  ( -u ( B  /  (
2  x.  A ) ) ^ 2 ) )  +  ( B  x.  -u ( B  / 
( 2  x.  A
) ) ) )  +  C )  =  ( ( ( ( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  -  (
( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10669, 105breqtrd 4471 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
)  -  ( ( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10737, 10readdcld 9623 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  e.  RR )
108107, 99subge0d 10142 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 0  <_  ( ( ( ( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  -  ( ( B ^
2 )  /  (
2  x.  A ) ) )  <->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  <_  (
( ( B ^
2 )  /  (
4  x.  A ) )  +  C ) ) )
109106, 108mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  <_  (
( ( B ^
2 )  /  (
4  x.  A ) )  +  C ) )
11056, 109eqbrtrd 4467 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  <_ 
( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
) )
11137, 10, 37leadd2d 10147 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  <_  C  <->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  <_ 
( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
) ) )
112110, 111mpbird 232 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  <_  C
)
11337, 10suble0d 10143 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( (
( ( B ^
2 )  /  (
4  x.  A ) )  -  C )  <_  0  <->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  <_  C
) )
114112, 113mpbird 232 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  -  C )  <_ 
0 )
11536, 114eqbrtrd 4467 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  <_ 
0 )
1164, 13resubcld 9987 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  e.  RR )
117 0red 9597 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  0  e.  RR )
118116, 117, 20ledivmuld 11305 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( (
( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  <_  0  <->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  (
( 4  x.  A
)  x.  0 ) ) )
119115, 118mpbid 210 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  (
( 4  x.  A
)  x.  0 ) )
12021mul01d 9778 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  A )  x.  0 )  =  0 )
121119, 120breqtrd 4471 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  0
)
1229adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  e.  RR )
123122ltp1d 10476 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  <  ( C  + 
1 ) )
124 peano2re 9752 . . . . . . . . . . . . 13  |-  ( C  e.  RR  ->  ( C  +  1 )  e.  RR )
125122, 124syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  +  1 )  e.  RR )
126122, 125ltnegd 10130 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  <  ( C  +  1 )  <->  -u ( C  +  1 )  <  -u C
) )
127123, 126mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  -u C
)
128 df-neg 9808 . . . . . . . . . 10  |-  -u C  =  ( 0  -  C )
129127, 128syl6breq 4486 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  ( 0  -  C ) )
130125renegcld 9986 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  RR )
131 0red 9597 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  e.  RR )
132130, 122, 131ltaddsubd 10152 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( -u ( C  +  1 )  +  C )  <  0  <->  -u ( C  + 
1 )  <  (
0  -  C ) ) )
133129, 132mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  <  0 )
134133expr 615 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  ( B  =/=  0  ->  ( -u ( C  +  1 )  +  C )  <  0 ) )
1351adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  RR )
136 simprr 756 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  =/=  0 )
137130, 135, 136redivcld 10372 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  RR )
13860adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
139 oveq1 6291 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( x ^ 2 )  =  ( (
-u ( C  + 
1 )  /  B
) ^ 2 ) )
140139oveq2d 6300 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
141 oveq2 6292 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( B  x.  x
)  =  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )
142140, 141oveq12d 6302 . . . . . . . . . . . . . 14  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  =  ( ( A  x.  ( (
-u ( C  + 
1 )  /  B
) ^ 2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) ) )
143142oveq1d 6299 . . . . . . . . . . . . 13  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
)  =  ( ( ( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  +  C ) )
144143breq2d 4459 . . . . . . . . . . . 12  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( 0  <_  (
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C )  <->  0  <_  ( (
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  +  C ) ) )
145144rspcv 3210 . . . . . . . . . . 11  |-  ( (
-u ( C  + 
1 )  /  B
)  e.  RR  ->  ( A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C )  ->  0  <_  ( ( ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) )  +  ( B  x.  ( -u ( C  + 
1 )  /  B
) ) )  +  C ) ) )
146137, 138, 145sylc 60 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( (
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  +  C ) )
147 simprl 755 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  =  A )
148147oveq1d 6299 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
149137recnd 9622 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  CC )
150 sqcl 12198 . . . . . . . . . . . . . . . 16  |-  ( (
-u ( C  + 
1 )  /  B
)  e.  CC  ->  ( ( -u ( C  +  1 )  /  B ) ^ 2 )  e.  CC )
151149, 150syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( -u ( C  +  1 )  /  B ) ^
2 )  e.  CC )
152151mul02d 9777 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
153148, 152eqtr3d 2510 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
154130recnd 9622 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  CC )
155135recnd 9622 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  CC )
156154, 155, 136divcan2d 10322 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( B  x.  ( -u ( C  +  1 )  /  B ) )  =  -u ( C  +  1 ) )
157153, 156oveq12d 6302 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  ( 0  +  -u ( C  +  1 ) ) )
158154addid2d 9780 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  +  -u ( C  +  1
) )  =  -u ( C  +  1
) )
159157, 158eqtrd 2508 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  -u ( C  +  1
) )
160159oveq1d 6299 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) )  +  ( B  x.  ( -u ( C  + 
1 )  /  B
) ) )  +  C )  =  (
-u ( C  + 
1 )  +  C
) )
161146, 160breqtrd 4471 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( -u ( C  +  1 )  +  C ) )
162 0re 9596 . . . . . . . . . 10  |-  0  e.  RR
163130, 122readdcld 9623 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  e.  RR )
164 lenlt 9663 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( -u ( C  + 
1 )  +  C
)  e.  RR )  ->  ( 0  <_ 
( -u ( C  + 
1 )  +  C
)  <->  -.  ( -u ( C  +  1 )  +  C )  <  0 ) )
165162, 163, 164sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  <_  ( -u ( C  +  1 )  +  C )  <->  -.  ( -u ( C  +  1 )  +  C )  <  0
) )
166161, 165mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -.  ( -u ( C  +  1 )  +  C )  <  0
)
167166expr 615 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  ( B  =/=  0  ->  -.  ( -u ( C  + 
1 )  +  C
)  <  0 ) )
168134, 167pm2.65d 175 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  -.  B  =/=  0 )
169 nne 2668 . . . . . 6  |-  ( -.  B  =/=  0  <->  B  =  0 )
170168, 169sylib 196 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  B  =  0 )
171170sq0id 12229 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( B ^ 2 )  =  0 )
172 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
173172oveq1d 6299 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  ( A  x.  C ) )
1749recnd 9622 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
175174adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  C  e.  CC )
176175mul02d 9777 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  0 )
177173, 176eqtr3d 2510 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  ( A  x.  C )  =  0 )
178177oveq2d 6300 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  ( 4  x.  0 ) )
17926mul01i 9769 . . . . 5  |-  ( 4  x.  0 )  =  0
180178, 179syl6eq 2524 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  0 )
181171, 180oveq12d 6302 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  =  ( 0  -  0 ) )
182 0m0e0 10645 . . . 4  |-  ( 0  -  0 )  =  0
183 0le0 10625 . . . 4  |-  0  <_  0
184182, 183eqbrtri 4466 . . 3  |-  ( 0  -  0 )  <_ 
0
185181, 184syl6eqbr 4484 . 2  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0 )
186 eqid 2467 . . . 4  |-  if ( 1  <_  ( (
( B  +  if ( 0  <_  C ,  C ,  0 ) )  +  1 )  /  -u A ) ,  ( ( ( B  +  if ( 0  <_  C ,  C ,  0 ) )  +  1 )  /  -u A ) ,  1 )  =  if ( 1  <_  ( (
( B  +  if ( 0  <_  C ,  C ,  0 ) )  +  1 )  /  -u A ) ,  ( ( ( B  +  if ( 0  <_  C ,  C ,  0 ) )  +  1 )  /  -u A ) ,  1 )
1877, 1, 9, 59, 186discr1 12270 . . 3  |-  ( ph  ->  0  <_  A )
188 leloe 9671 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
189162, 7, 188sylancr 663 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
190187, 189mpbid 210 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
191121, 185, 190mpjaodan 784 1  |-  ( ph  ->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   ifcif 3939   class class class wbr 4447  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   -ucneg 9806    / cdiv 10206   2c2 10585   4c4 10587   RR+crp 11220   ^cexp 12134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-seq 12076  df-exp 12135
This theorem is referenced by:  csbren  21589  normlem6  25736
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