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Theorem discr 12408
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 466 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  B  e.  RR )
3 resqcl 12341 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
42, 3syl 17 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  RR )
54recnd 9669 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  CC )
6 4re 10686 . . . . . . . . 9  |-  4  e.  RR
7 discr.1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
87adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR )
9 discr.3 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
109adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  C  e.  RR )
118, 10remulcld 9671 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  RR )
12 remulcl 9624 . . . . . . . . 9  |-  ( ( 4  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( 4  x.  ( A  x.  C
) )  e.  RR )
136, 11, 12sylancr 667 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  RR )
1413recnd 9669 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  CC )
15 4pos 10705 . . . . . . . . . 10  |-  0  <  4
166, 15elrpii 11305 . . . . . . . . 9  |-  4  e.  RR+
17 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <  A )
188, 17elrpd 11338 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
19 rpmulcl 11324 . . . . . . . . 9  |-  ( ( 4  e.  RR+  /\  A  e.  RR+ )  ->  (
4  x.  A )  e.  RR+ )
2016, 18, 19sylancr 667 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  RR+ )
2120rpcnd 11343 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  CC )
2220rpne0d 11346 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =/=  0 )
235, 14, 21, 22divsubdird 10422 . . . . . 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 9669 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  CC )
258recnd 9669 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
26 4cn 10687 . . . . . . . . . 10  |-  4  e.  CC
2726a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  e.  CC )
2818rpne0d 11346 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
29 4ne0 10706 . . . . . . . . . 10  |-  4  =/=  0
3029a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  =/=  0 )
3124, 25, 27, 28, 30divcan5d 10409 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  ( ( A  x.  C )  /  A
) )
3210recnd 9669 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  C  e.  CC )
3332, 25, 28divcan3d 10388 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  C )  /  A )  =  C )
3431, 33eqtrd 2463 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  C )
3534oveq2d 6317 . . . . . 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 2463 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  C
) )
374, 20rerpdivcld 11369 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  RR )
3837recnd 9669 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  CC )
39382timesd 10855 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  ( ( B ^ 2 )  /  ( 4  x.  A ) ) ) )
40 2t2e4 10759 . . . . . . . . . . . . 13  |-  ( 2  x.  2 )  =  4
4140oveq1i 6311 . . . . . . . . . . . 12  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
42 2cnd 10682 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  2  e.  CC )
4342, 42, 25mulassd 9666 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4441, 43syl5eqr 2477 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4544oveq2d 6317 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( ( 2  x.  ( B ^ 2 ) )  /  (
2  x.  ( 2  x.  A ) ) ) )
4642, 5, 21, 22divassd 10418 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( 2  x.  (
( B ^ 2 )  /  ( 4  x.  A ) ) ) )
47 2rp 11307 . . . . . . . . . . . . 13  |-  2  e.  RR+
48 rpmulcl 11324 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR+  /\  A  e.  RR+ )  ->  (
2  x.  A )  e.  RR+ )
4947, 18, 48sylancr 667 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  RR+ )
5049rpcnd 11343 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  CC )
5149rpne0d 11346 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  =/=  0 )
52 2ne0 10702 . . . . . . . . . . . 12  |-  2  =/=  0
5352a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  2  =/=  0 )
545, 50, 42, 51, 53divcan5d 10409 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 2  x.  ( 2  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5545, 46, 543eqtr3d 2471 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5639, 55eqtr3d 2465 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
572, 49rerpdivcld 11369 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  RR )
5857renegcld 10046 . . . . . . . . . . 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 2839 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
6160adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A. x  e.  RR  0  <_  (
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
62 oveq1 6308 . . . . . . . . . . . . . . . 16  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
x ^ 2 )  =  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )
6362oveq2d 6317 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( A  x.  ( x ^ 2 ) )  =  ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) ) )
64 oveq2 6309 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( B  x.  x )  =  ( B  x.  -u ( B  /  (
2  x.  A ) ) ) )
6563, 64oveq12d 6319 . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . 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 4432 . . . . . . . . . . . 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 3178 . . . . . . . . . . 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 62 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  +  C ) )
7057recnd 9669 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  CC )
71 sqneg 12334 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  /  ( 2  x.  A ) )  e.  CC  ->  ( -u ( B  /  (
2  x.  A ) ) ^ 2 )  =  ( ( B  /  ( 2  x.  A ) ) ^
2 ) )
7270, 71syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
732recnd 9669 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  B  e.  CC )
74 sqdiv 12339 . . . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^ 2 ) ) )
76 sqval 12333 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 2  x.  A )  e.  CC  ->  (
( 2  x.  A
) ^ 2 )  =  ( ( 2  x.  A )  x.  ( 2  x.  A
) ) )
7750, 76syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7850, 42, 25mulassd 9666 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7942, 25, 42mul32d 9843 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( ( 2  x.  2 )  x.  A
) )
8079, 41syl6eq 2479 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( 4  x.  A
) )
8180oveq1d 6316 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 4  x.  A )  x.  A
) )
8277, 78, 813eqtr2d 2469 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A
) )
8382oveq2d 6317 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( ( 2  x.  A ) ^ 2 ) )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
8472, 75, 833eqtrd 2467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
855, 21, 25, 22, 28divdiv1d 10414 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  /  A )  =  ( ( B ^
2 )  /  (
( 4  x.  A
)  x.  A ) ) )
8684, 85eqtr4d 2466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) )
8786oveq2d 6317 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( A  x.  (
( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) ) )
8838, 25, 28divcan2d 10385 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  /  A
) )  =  ( ( B ^ 2 )  /  ( 4  x.  A ) ) )
8987, 88eqtrd 2463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( ( B ^
2 )  /  (
4  x.  A ) ) )
9073, 70mulneg2d 10072 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
91 sqval 12333 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9273, 91syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9392oveq1d 6316 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( ( B  x.  B
)  /  ( 2  x.  A ) ) )
9473, 73, 50, 51divassd 10418 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  x.  B )  /  ( 2  x.  A ) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9593, 94eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9695negeqd 9869 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  -u ( ( B ^ 2 )  /  ( 2  x.  A ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
9790, 96eqtr4d 2466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( ( B ^ 2 )  / 
( 2  x.  A
) ) )
9889, 97oveq12d 6319 . . . . . . . . . . . . 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 11369 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  RR )
10099recnd 9669 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  CC )
10138, 100negsubd 9992 . . . . . . . . . . . . 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 2463 . . . . . . . . . . . 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 6316 . . . . . . . . . . 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 10007 . . . . . . . . . . 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 2466 . . . . . . . . . 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 4445 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
)  -  ( ( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10737, 10readdcld 9670 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  e.  RR )
108107, 99subge0d 10203 . . . . . . . . 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 213 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  <_  (
( ( B ^
2 )  /  (
4  x.  A ) )  +  C ) )
11056, 109eqbrtrd 4441 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  <_ 
( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
) )
11137, 10, 37leadd2d 10208 . . . . . . 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 235 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  <_  C
)
11337, 10suble0d 10204 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( (
( ( B ^
2 )  /  (
4  x.  A ) )  -  C )  <_  0  <->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  <_  C
) )
114112, 113mpbird 235 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  -  C )  <_ 
0 )
11536, 114eqbrtrd 4441 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  <_ 
0 )
1164, 13resubcld 10047 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  e.  RR )
117 0red 9644 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  0  e.  RR )
118116, 117, 20ledivmuld 11391 . . . 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 213 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  (
( 4  x.  A
)  x.  0 ) )
12021mul01d 9832 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  A )  x.  0 )  =  0 )
121119, 120breqtrd 4445 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  0
)
1229adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  e.  RR )
123122ltp1d 10537 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  <  ( C  + 
1 ) )
124 peano2re 9806 . . . . . . . . . . . . 13  |-  ( C  e.  RR  ->  ( C  +  1 )  e.  RR )
125122, 124syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  +  1 )  e.  RR )
126122, 125ltnegd 10191 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  <  ( C  +  1 )  <->  -u ( C  +  1 )  <  -u C
) )
127123, 126mpbid 213 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  -u C
)
128 df-neg 9863 . . . . . . . . . 10  |-  -u C  =  ( 0  -  C )
129127, 128syl6breq 4460 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  ( 0  -  C ) )
130125renegcld 10046 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  RR )
131 0red 9644 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  e.  RR )
132130, 122, 131ltaddsubd 10213 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( -u ( C  +  1 )  +  C )  <  0  <->  -u ( C  + 
1 )  <  (
0  -  C ) ) )
133129, 132mpbird 235 . . . . . . . 8  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  <  0 )
134133expr 618 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  ( B  =/=  0  ->  ( -u ( C  +  1 )  +  C )  <  0 ) )
1351adantr 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  RR )
136 simprr 764 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  =/=  0 )
137130, 135, 136redivcld 10435 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  RR )
13860adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
139 oveq1 6308 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( x ^ 2 )  =  ( (
-u ( C  + 
1 )  /  B
) ^ 2 ) )
140139oveq2d 6317 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
141 oveq2 6309 . . . . . . . . . . . . . . 15  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( B  x.  x
)  =  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )
142140, 141oveq12d 6319 . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . 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 4432 . . . . . . . . . . . 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 3178 . . . . . . . . . . 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 62 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( (
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  +  C ) )
147 simprl 762 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  =  A )
148147oveq1d 6316 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
149137recnd 9669 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  CC )
150 sqcl 12336 . . . . . . . . . . . . . . . 16  |-  ( (
-u ( C  + 
1 )  /  B
)  e.  CC  ->  ( ( -u ( C  +  1 )  /  B ) ^ 2 )  e.  CC )
151149, 150syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( -u ( C  +  1 )  /  B ) ^
2 )  e.  CC )
152151mul02d 9831 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
153148, 152eqtr3d 2465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
154130recnd 9669 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  CC )
155135recnd 9669 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  CC )
156154, 155, 136divcan2d 10385 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( B  x.  ( -u ( C  +  1 )  /  B ) )  =  -u ( C  +  1 ) )
157153, 156oveq12d 6319 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  ( 0  +  -u ( C  +  1 ) ) )
158154addid2d 9834 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  +  -u ( C  +  1
) )  =  -u ( C  +  1
) )
159157, 158eqtrd 2463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  -u ( C  +  1
) )
160159oveq1d 6316 . . . . . . . . . 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 4445 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( -u ( C  +  1 )  +  C ) )
162 0re 9643 . . . . . . . . . 10  |-  0  e.  RR
163130, 122readdcld 9670 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  e.  RR )
164 lenlt 9712 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( -u ( C  + 
1 )  +  C
)  e.  RR )  ->  ( 0  <_ 
( -u ( C  + 
1 )  +  C
)  <->  -.  ( -u ( C  +  1 )  +  C )  <  0 ) )
165162, 163, 164sylancr 667 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  <_  ( -u ( C  +  1 )  +  C )  <->  -.  ( -u ( C  +  1 )  +  C )  <  0
) )
166161, 165mpbid 213 . . . . . . . 8  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -.  ( -u ( C  +  1 )  +  C )  <  0
)
167166expr 618 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  ( B  =/=  0  ->  -.  ( -u ( C  + 
1 )  +  C
)  <  0 ) )
168134, 167pm2.65d 178 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  -.  B  =/=  0 )
169 nne 2624 . . . . . 6  |-  ( -.  B  =/=  0  <->  B  =  0 )
170168, 169sylib 199 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  B  =  0 )
171170sq0id 12367 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( B ^ 2 )  =  0 )
172 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
173172oveq1d 6316 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  ( A  x.  C ) )
1749recnd 9669 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
175174adantr 466 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  C  e.  CC )
176175mul02d 9831 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  0 )
177173, 176eqtr3d 2465 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  ( A  x.  C )  =  0 )
178177oveq2d 6317 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  ( 4  x.  0 ) )
17926mul01i 9823 . . . . 5  |-  ( 4  x.  0 )  =  0
180178, 179syl6eq 2479 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  0 )
181171, 180oveq12d 6319 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  =  ( 0  -  0 ) )
182 0m0e0 10719 . . . 4  |-  ( 0  -  0 )  =  0
183 0le0 10699 . . . 4  |-  0  <_  0
184182, 183eqbrtri 4440 . . 3  |-  ( 0  -  0 )  <_ 
0
185181, 184syl6eqbr 4458 . 2  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0 )
186 eqid 2422 . . . 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 12407 . . 3  |-  ( ph  ->  0  <_  A )
188 leloe 9720 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
189162, 7, 188sylancr 667 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
190187, 189mpbid 213 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
191121, 185, 190mpjaodan 793 1  |-  ( ph  ->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   ifcif 3909   class class class wbr 4420  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860   -ucneg 9861    / cdiv 10269   2c2 10659   4c4 10661   RR+crp 11302   ^cexp 12271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12213  df-exp 12272
This theorem is referenced by:  csbren  22339  normlem6  26753
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