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Theorem discr 12306
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 12238 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
42, 3syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  RR )
54recnd 9639 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  CC )
6 4re 10633 . . . . . . . . 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 9641 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  RR )
12 remulcl 9594 . . . . . . . . 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 9639 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  CC )
15 4pos 10652 . . . . . . . . . 10  |-  0  <  4
166, 15elrpii 11248 . . . . . . . . 9  |-  4  e.  RR+
17 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <  A )
188, 17elrpd 11279 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
19 rpmulcl 11266 . . . . . . . . 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 11283 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  CC )
2220rpne0d 11286 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =/=  0 )
235, 14, 21, 22divsubdird 10380 . . . . . 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 9639 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  CC )
258recnd 9639 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
26 4cn 10634 . . . . . . . . . 10  |-  4  e.  CC
2726a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  e.  CC )
2818rpne0d 11286 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
29 4ne0 10653 . . . . . . . . . 10  |-  4  =/=  0
3029a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  =/=  0 )
3124, 25, 27, 28, 30divcan5d 10367 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  ( ( A  x.  C )  /  A
) )
3210recnd 9639 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  C  e.  CC )
3332, 25, 28divcan3d 10346 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  C )  /  A )  =  C )
3431, 33eqtrd 2498 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  C )
3534oveq2d 6312 . . . . . 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 2498 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  C
) )
374, 20rerpdivcld 11308 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  RR )
3837recnd 9639 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  CC )
39382timesd 10802 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  ( ( B ^ 2 )  /  ( 4  x.  A ) ) ) )
40 2t2e4 10706 . . . . . . . . . . . . 13  |-  ( 2  x.  2 )  =  4
4140oveq1i 6306 . . . . . . . . . . . 12  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
42 2cnd 10629 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  2  e.  CC )
4342, 42, 25mulassd 9636 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4441, 43syl5eqr 2512 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4544oveq2d 6312 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( ( 2  x.  ( B ^ 2 ) )  /  (
2  x.  ( 2  x.  A ) ) ) )
4642, 5, 21, 22divassd 10376 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( 2  x.  (
( B ^ 2 )  /  ( 4  x.  A ) ) ) )
47 2rp 11250 . . . . . . . . . . . . 13  |-  2  e.  RR+
48 rpmulcl 11266 . . . . . . . . . . . . 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 11283 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  CC )
5149rpne0d 11286 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  =/=  0 )
52 2ne0 10649 . . . . . . . . . . . 12  |-  2  =/=  0
5352a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  2  =/=  0 )
545, 50, 42, 51, 53divcan5d 10367 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 2  x.  ( 2  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5545, 46, 543eqtr3d 2506 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5639, 55eqtr3d 2500 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
572, 49rerpdivcld 11308 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  RR )
5857renegcld 10007 . . . . . . . . . . 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 2871 . . . . . . . . . . . 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 6303 . . . . . . . . . . . . . . . 16  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
x ^ 2 )  =  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )
6362oveq2d 6312 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( A  x.  ( x ^ 2 ) )  =  ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) ) )
64 oveq2 6304 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( B  x.  x )  =  ( B  x.  -u ( B  /  (
2  x.  A ) ) ) )
6563, 64oveq12d 6314 . . . . . . . . . . . . . 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 6311 . . . . . . . . . . . . 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 4468 . . . . . . . . . . . 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 3206 . . . . . . . . . . 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 9639 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  CC )
71 sqneg 12231 . . . . . . . . . . . . . . . . . . 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 9639 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  B  e.  CC )
74 sqdiv 12236 . . . . . . . . . . . . . . . . . . 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 12230 . . . . . . . . . . . . . . . . . . . . 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 9636 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7942, 25, 42mul32d 9807 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( ( 2  x.  2 )  x.  A
) )
8079, 41syl6eq 2514 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( 4  x.  A
) )
8180oveq1d 6311 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 4  x.  A )  x.  A
) )
8277, 78, 813eqtr2d 2504 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A
) )
8382oveq2d 6312 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( ( 2  x.  A ) ^ 2 ) )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
8472, 75, 833eqtrd 2502 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
855, 21, 25, 22, 28divdiv1d 10372 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  /  A )  =  ( ( B ^
2 )  /  (
( 4  x.  A
)  x.  A ) ) )
8684, 85eqtr4d 2501 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) )
8786oveq2d 6312 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( A  x.  (
( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) ) )
8838, 25, 28divcan2d 10343 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  /  A
) )  =  ( ( B ^ 2 )  /  ( 4  x.  A ) ) )
8987, 88eqtrd 2498 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( ( B ^
2 )  /  (
4  x.  A ) ) )
9073, 70mulneg2d 10031 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
91 sqval 12230 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9273, 91syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9392oveq1d 6311 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( ( B  x.  B
)  /  ( 2  x.  A ) ) )
9473, 73, 50, 51divassd 10376 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  x.  B )  /  ( 2  x.  A ) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9593, 94eqtrd 2498 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9695negeqd 9833 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  -u ( ( B ^ 2 )  /  ( 2  x.  A ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
9790, 96eqtr4d 2501 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( ( B ^ 2 )  / 
( 2  x.  A
) ) )
9889, 97oveq12d 6314 . . . . . . . . . . . . 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 11308 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  RR )
10099recnd 9639 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  CC )
10138, 100negsubd 9956 . . . . . . . . . . . . 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 2498 . . . . . . . . . . . 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 6311 . . . . . . . . . . 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 9971 . . . . . . . . . . 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 2501 . . . . . . . . . 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 4480 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
)  -  ( ( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10737, 10readdcld 9640 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  e.  RR )
108107, 99subge0d 10163 . . . . . . . . 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 4476 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  <_ 
( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
) )
11137, 10, 37leadd2d 10168 . . . . . . 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 10164 . . . . . 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 4476 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  <_ 
0 )
1164, 13resubcld 10008 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  e.  RR )
117 0red 9614 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  0  e.  RR )
118116, 117, 20ledivmuld 11330 . . . 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 9796 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  A )  x.  0 )  =  0 )
121119, 120breqtrd 4480 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  0
)
1229adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  e.  RR )
123122ltp1d 10496 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  <  ( C  + 
1 ) )
124 peano2re 9770 . . . . . . . . . . . . 13  |-  ( C  e.  RR  ->  ( C  +  1 )  e.  RR )
125122, 124syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  +  1 )  e.  RR )
126122, 125ltnegd 10151 . . . . . . . . . . 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 9827 . . . . . . . . . 10  |-  -u C  =  ( 0  -  C )
129127, 128syl6breq 4495 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  ( 0  -  C ) )
130125renegcld 10007 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  RR )
131 0red 9614 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  e.  RR )
132130, 122, 131ltaddsubd 10173 . . . . . . . . 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 757 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  =/=  0 )
137130, 135, 136redivcld 10393 . . . . . . . . . . 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 6303 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( x ^ 2 )  =  ( (
-u ( C  + 
1 )  /  B
) ^ 2 ) )
140139oveq2d 6312 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
141 oveq2 6304 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( B  x.  x
)  =  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )
142140, 141oveq12d 6314 . . . . . . . . . . . . . 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 6311 . . . . . . . . . . . . 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 4468 . . . . . . . . . . . 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 3206 . . . . . . . . . . 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 756 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  =  A )
148147oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
149137recnd 9639 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  CC )
150 sqcl 12233 . . . . . . . . . . . . . . . 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 9795 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
153148, 152eqtr3d 2500 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
154130recnd 9639 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  CC )
155135recnd 9639 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  CC )
156154, 155, 136divcan2d 10343 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( B  x.  ( -u ( C  +  1 )  /  B ) )  =  -u ( C  +  1 ) )
157153, 156oveq12d 6314 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  ( 0  +  -u ( C  +  1 ) ) )
158154addid2d 9798 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  +  -u ( C  +  1
) )  =  -u ( C  +  1
) )
159157, 158eqtrd 2498 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  -u ( C  +  1
) )
160159oveq1d 6311 . . . . . . . . . 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 4480 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( -u ( C  +  1 )  +  C ) )
162 0re 9613 . . . . . . . . . 10  |-  0  e.  RR
163130, 122readdcld 9640 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  e.  RR )
164 lenlt 9680 . . . . . . . . . 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 2658 . . . . . 6  |-  ( -.  B  =/=  0  <->  B  =  0 )
170168, 169sylib 196 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  B  =  0 )
171170sq0id 12264 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( B ^ 2 )  =  0 )
172 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
173172oveq1d 6311 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  ( A  x.  C ) )
1749recnd 9639 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
175174adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  C  e.  CC )
176175mul02d 9795 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  0 )
177173, 176eqtr3d 2500 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  ( A  x.  C )  =  0 )
178177oveq2d 6312 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  ( 4  x.  0 ) )
17926mul01i 9787 . . . . 5  |-  ( 4  x.  0 )  =  0
180178, 179syl6eq 2514 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  0 )
181171, 180oveq12d 6314 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  =  ( 0  -  0 ) )
182 0m0e0 10666 . . . 4  |-  ( 0  -  0 )  =  0
183 0le0 10646 . . . 4  |-  0  <_  0
184182, 183eqbrtri 4475 . . 3  |-  ( 0  -  0 )  <_ 
0
185181, 184syl6eqbr 4493 . 2  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0 )
186 eqid 2457 . . . 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 12305 . . 3  |-  ( ph  ->  0  <_  A )
188 leloe 9688 . . . 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 786 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ifcif 3944   class class class wbr 4456  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824   -ucneg 9825    / cdiv 10227   2c2 10606   4c4 10608   RR+crp 11245   ^cexp 12169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12111  df-exp 12170
This theorem is referenced by:  csbren  21952  normlem6  26159
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