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Theorem discr 12001
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 11933 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
42, 3syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  RR )
54recnd 9412 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  CC )
6 4re 10398 . . . . . . . . 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 9414 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  RR )
12 remulcl 9367 . . . . . . . . 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 9412 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  CC )
15 4pos 10417 . . . . . . . . . 10  |-  0  <  4
166, 15elrpii 10994 . . . . . . . . 9  |-  4  e.  RR+
17 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <  A )
188, 17elrpd 11025 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
19 rpmulcl 11012 . . . . . . . . 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 11029 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  CC )
2220rpne0d 11032 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =/=  0 )
235, 14, 21, 22divsubdird 10146 . . . . . 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 9412 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  CC )
258recnd 9412 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
26 4cn 10399 . . . . . . . . . 10  |-  4  e.  CC
2726a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  e.  CC )
2818rpne0d 11032 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
29 4ne0 10418 . . . . . . . . . 10  |-  4  =/=  0
3029a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  =/=  0 )
3124, 25, 27, 28, 30divcan5d 10133 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  ( ( A  x.  C )  /  A
) )
3210recnd 9412 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  C  e.  CC )
3332, 25, 28divcan3d 10112 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  C )  /  A )  =  C )
3431, 33eqtrd 2475 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  C )
3534oveq2d 6107 . . . . . 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 2475 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  C
) )
374, 20rerpdivcld 11054 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  RR )
3837recnd 9412 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  CC )
39382timesd 10567 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  ( ( B ^ 2 )  /  ( 4  x.  A ) ) ) )
40 2t2e4 10471 . . . . . . . . . . . . 13  |-  ( 2  x.  2 )  =  4
4140oveq1i 6101 . . . . . . . . . . . 12  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
42 2cnd 10394 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  2  e.  CC )
4342, 42, 25mulassd 9409 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4441, 43syl5eqr 2489 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4544oveq2d 6107 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( ( 2  x.  ( B ^ 2 ) )  /  (
2  x.  ( 2  x.  A ) ) ) )
4642, 5, 21, 22divassd 10142 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( 2  x.  (
( B ^ 2 )  /  ( 4  x.  A ) ) ) )
47 2rp 10996 . . . . . . . . . . . . 13  |-  2  e.  RR+
48 rpmulcl 11012 . . . . . . . . . . . . 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 11029 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  CC )
5149rpne0d 11032 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  =/=  0 )
52 2ne0 10414 . . . . . . . . . . . 12  |-  2  =/=  0
5352a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  2  =/=  0 )
545, 50, 42, 51, 53divcan5d 10133 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 2  x.  ( 2  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5545, 46, 543eqtr3d 2483 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5639, 55eqtr3d 2477 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
572, 49rerpdivcld 11054 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  RR )
5857renegcld 9775 . . . . . . . . . . 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 2799 . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . 16  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
x ^ 2 )  =  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )
6362oveq2d 6107 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( A  x.  ( x ^ 2 ) )  =  ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) ) )
64 oveq2 6099 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( B  x.  x )  =  ( B  x.  -u ( B  /  (
2  x.  A ) ) ) )
6563, 64oveq12d 6109 . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . 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 4304 . . . . . . . . . . . 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 3069 . . . . . . . . . . 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 9412 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  CC )
71 sqneg 11926 . . . . . . . . . . . . . . . . . . 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 9412 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  B  e.  CC )
74 sqdiv 11931 . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^ 2 ) ) )
76 sqval 11925 . . . . . . . . . . . . . . . . . . . . 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 9409 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7942, 25, 42mul32d 9579 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( ( 2  x.  2 )  x.  A
) )
8079, 41syl6eq 2491 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( 4  x.  A
) )
8180oveq1d 6106 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 4  x.  A )  x.  A
) )
8277, 78, 813eqtr2d 2481 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A
) )
8382oveq2d 6107 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( ( 2  x.  A ) ^ 2 ) )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
8472, 75, 833eqtrd 2479 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
855, 21, 25, 22, 28divdiv1d 10138 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  /  A )  =  ( ( B ^
2 )  /  (
( 4  x.  A
)  x.  A ) ) )
8684, 85eqtr4d 2478 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) )
8786oveq2d 6107 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( A  x.  (
( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) ) )
8838, 25, 28divcan2d 10109 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  /  A
) )  =  ( ( B ^ 2 )  /  ( 4  x.  A ) ) )
8987, 88eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( ( B ^
2 )  /  (
4  x.  A ) ) )
9073, 70mulneg2d 9798 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
91 sqval 11925 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9273, 91syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9392oveq1d 6106 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( ( B  x.  B
)  /  ( 2  x.  A ) ) )
9473, 73, 50, 51divassd 10142 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  x.  B )  /  ( 2  x.  A ) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9593, 94eqtrd 2475 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9695negeqd 9604 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  -u ( ( B ^ 2 )  /  ( 2  x.  A ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
9790, 96eqtr4d 2478 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( ( B ^ 2 )  / 
( 2  x.  A
) ) )
9889, 97oveq12d 6109 . . . . . . . . . . . . 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 11054 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  RR )
10099recnd 9412 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  CC )
10138, 100negsubd 9725 . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 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 6106 . . . . . . . . . . 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 9740 . . . . . . . . . . 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 2478 . . . . . . . . . 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 4316 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
)  -  ( ( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10737, 10readdcld 9413 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  e.  RR )
108107, 99subge0d 9929 . . . . . . . . 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 4312 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  <_ 
( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
) )
11137, 10, 37leadd2d 9934 . . . . . . 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 9930 . . . . . 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 4312 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  <_ 
0 )
1164, 13resubcld 9776 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  e.  RR )
117 0red 9387 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  0  e.  RR )
118116, 117, 20ledivmuld 11076 . . . 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 9568 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  A )  x.  0 )  =  0 )
121119, 120breqtrd 4316 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  0
)
1229adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  e.  RR )
123122ltp1d 10263 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  <  ( C  + 
1 ) )
124 peano2re 9542 . . . . . . . . . . . . 13  |-  ( C  e.  RR  ->  ( C  +  1 )  e.  RR )
125122, 124syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  +  1 )  e.  RR )
126122, 125ltnegd 9917 . . . . . . . . . . 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 9598 . . . . . . . . . 10  |-  -u C  =  ( 0  -  C )
129127, 128syl6breq 4331 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  ( 0  -  C ) )
130125renegcld 9775 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  RR )
131 0red 9387 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  e.  RR )
132130, 122, 131ltaddsubd 9939 . . . . . . . . 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 10159 . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( x ^ 2 )  =  ( (
-u ( C  + 
1 )  /  B
) ^ 2 ) )
140139oveq2d 6107 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
141 oveq2 6099 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( B  x.  x
)  =  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )
142140, 141oveq12d 6109 . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . 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 4304 . . . . . . . . . . . 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 3069 . . . . . . . . . . 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 6106 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
149137recnd 9412 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  CC )
150 sqcl 11928 . . . . . . . . . . . . . . . 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 9567 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
153148, 152eqtr3d 2477 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
154130recnd 9412 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  CC )
155135recnd 9412 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  CC )
156154, 155, 136divcan2d 10109 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( B  x.  ( -u ( C  +  1 )  /  B ) )  =  -u ( C  +  1 ) )
157153, 156oveq12d 6109 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  ( 0  +  -u ( C  +  1 ) ) )
158154addid2d 9570 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  +  -u ( C  +  1
) )  =  -u ( C  +  1
) )
159157, 158eqtrd 2475 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  -u ( C  +  1
) )
160159oveq1d 6106 . . . . . . . . . 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 4316 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( -u ( C  +  1 )  +  C ) )
162 0re 9386 . . . . . . . . . 10  |-  0  e.  RR
163130, 122readdcld 9413 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  e.  RR )
164 lenlt 9453 . . . . . . . . . 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 2612 . . . . . 6  |-  ( -.  B  =/=  0  <->  B  =  0 )
170168, 169sylib 196 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  B  =  0 )
171170sq0id 11959 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( B ^ 2 )  =  0 )
172 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
173172oveq1d 6106 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  ( A  x.  C ) )
1749recnd 9412 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
175174adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  C  e.  CC )
176175mul02d 9567 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  0 )
177173, 176eqtr3d 2477 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  ( A  x.  C )  =  0 )
178177oveq2d 6107 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  ( 4  x.  0 ) )
17926mul01i 9559 . . . . 5  |-  ( 4  x.  0 )  =  0
180178, 179syl6eq 2491 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  0 )
181171, 180oveq12d 6109 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  =  ( 0  -  0 ) )
182 0m0e0 10431 . . . 4  |-  ( 0  -  0 )  =  0
183 0le0 10411 . . . 4  |-  0  <_  0
184182, 183eqbrtri 4311 . . 3  |-  ( 0  -  0 )  <_ 
0
185181, 184syl6eqbr 4329 . 2  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0 )
186 eqid 2443 . . . 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 12000 . . 3  |-  ( ph  ->  0  <_  A )
188 leloe 9461 . . . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   ifcif 3791   class class class wbr 4292  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595   -ucneg 9596    / cdiv 9993   2c2 10371   4c4 10373   RR+crp 10991   ^cexp 11865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866
This theorem is referenced by:  csbren  20898  normlem6  24517
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