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Theorem discr 11254
Description: If a quadratic polynomial with real coefficients is nonnegative for all values, then its discriminant is non-positive. (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 451 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  B  e.  RR )
3 resqcl 11187 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B ^ 2 )  e.  RR )
42, 3syl 15 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  RR )
54recnd 8877 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  e.  CC )
6 4re 9835 . . . . . . . . 9  |-  4  e.  RR
7 discr.1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
87adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR )
9 discr.3 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  RR )
109adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  C  e.  RR )
118, 10remulcld 8879 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  RR )
12 remulcl 8838 . . . . . . . . 9  |-  ( ( 4  e.  RR  /\  ( A  x.  C
)  e.  RR )  ->  ( 4  x.  ( A  x.  C
) )  e.  RR )
136, 11, 12sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  RR )
1413recnd 8877 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  ( A  x.  C ) )  e.  CC )
15 4pos 9848 . . . . . . . . . 10  |-  0  <  4
166, 15elrpii 10373 . . . . . . . . 9  |-  4  e.  RR+
17 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <  A )
188, 17elrpd 10404 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  RR+ )
19 rpmulcl 10391 . . . . . . . . 9  |-  ( ( 4  e.  RR+  /\  A  e.  RR+ )  ->  (
4  x.  A )  e.  RR+ )
2016, 18, 19sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  RR+ )
2120rpcnd 10408 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  e.  CC )
2220rpne0d 10411 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =/=  0 )
235, 14, 21, 22divsubdird 9591 . . . . . 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 8877 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  C )  e.  CC )
258recnd 8877 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  e.  CC )
26 4cn 9836 . . . . . . . . . 10  |-  4  e.  CC
2726a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  e.  CC )
2818rpne0d 10411 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  A  =/=  0 )
296, 15gt0ne0ii 9325 . . . . . . . . . 10  |-  4  =/=  0
3029a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  4  =/=  0 )
3124, 25, 27, 28, 30divcan5d 9578 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  ( ( A  x.  C )  /  A
) )
3210recnd 8877 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  C  e.  CC )
3332, 25, 28divcan3d 9557 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( A  x.  C )  /  A )  =  C )
3431, 33eqtrd 2328 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  ( A  x.  C ) )  /  ( 4  x.  A ) )  =  C )
3534oveq2d 5890 . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  -  C
) )
374, 20rerpdivcld 10433 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  RR )
3837recnd 8877 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  e.  CC )
39382timesd 9970 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  ( ( B ^ 2 )  /  ( 4  x.  A ) ) ) )
40 2t2e4 9887 . . . . . . . . . . . . 13  |-  ( 2  x.  2 )  =  4
4140oveq1i 5884 . . . . . . . . . . . 12  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
42 2cn 9832 . . . . . . . . . . . . . 14  |-  2  e.  CC
4342a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  A )  ->  2  e.  CC )
4443, 43, 25mulassd 8874 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  2 )  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4541, 44syl5eqr 2342 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 4  x.  A )  =  ( 2  x.  (
2  x.  A ) ) )
4645oveq2d 5890 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( ( 2  x.  ( B ^ 2 ) )  /  (
2  x.  ( 2  x.  A ) ) ) )
4743, 5, 21, 22divassd 9587 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 4  x.  A ) )  =  ( 2  x.  (
( B ^ 2 )  /  ( 4  x.  A ) ) ) )
48 2rp 10375 . . . . . . . . . . . . 13  |-  2  e.  RR+
49 rpmulcl 10391 . . . . . . . . . . . . 13  |-  ( ( 2  e.  RR+  /\  A  e.  RR+ )  ->  (
2  x.  A )  e.  RR+ )
5048, 18, 49sylancr 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  RR+ )
5150rpcnd 10408 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  e.  CC )
5250rpne0d 10411 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  A )  =/=  0 )
53 2ne0 9845 . . . . . . . . . . . 12  |-  2  =/=  0
5453a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  2  =/=  0 )
555, 51, 43, 52, 54divcan5d 9578 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  ( B ^ 2 ) )  /  ( 2  x.  ( 2  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5646, 47, 553eqtr3d 2336 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  ( 2  x.  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
5739, 56eqtr3d 2330 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  =  ( ( B ^
2 )  /  (
2  x.  A ) ) )
582, 50rerpdivcld 10433 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  RR )
5958renegcld 9226 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  -u ( B  /  ( 2  x.  A ) )  e.  RR )
60 discr.4 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  0  <_ 
( ( ( A  x.  ( x ^
2 ) )  +  ( B  x.  x
) )  +  C
) )
6160ralrimiva 2639 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
6261adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  A )  ->  A. x  e.  RR  0  <_  (
( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
63 oveq1 5881 . . . . . . . . . . . . . . . 16  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  (
x ^ 2 )  =  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )
6463oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( A  x.  ( x ^ 2 ) )  =  ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) ) )
65 oveq2 5882 . . . . . . . . . . . . . . 15  |-  ( x  =  -u ( B  / 
( 2  x.  A
) )  ->  ( B  x.  x )  =  ( B  x.  -u ( B  /  (
2  x.  A ) ) ) )
6664, 65oveq12d 5892 . . . . . . . . . . . . . 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 ) ) ) ) )
6766oveq1d 5889 . . . . . . . . . . . . 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 ) )
6867breq2d 4051 . . . . . . . . . . . 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 ) ) )
6968rspcv 2893 . . . . . . . . . . 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 ) ) )
7059, 62, 69sylc 56 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( A  x.  ( -u ( B  / 
( 2  x.  A
) ) ^ 2 ) )  +  ( B  x.  -u ( B  /  ( 2  x.  A ) ) ) )  +  C ) )
7158recnd 8877 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( B  /  ( 2  x.  A ) )  e.  CC )
72 sqneg 11180 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  /  ( 2  x.  A ) )  e.  CC  ->  ( -u ( B  /  (
2  x.  A ) ) ^ 2 )  =  ( ( B  /  ( 2  x.  A ) ) ^
2 ) )
7371, 72syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B  /  (
2  x.  A ) ) ^ 2 ) )
742recnd 8877 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  B  e.  CC )
75 sqdiv 11185 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  e.  CC  /\  ( 2  x.  A
)  e.  CC  /\  ( 2  x.  A
)  =/=  0 )  ->  ( ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^ 2 ) ) )
7674, 51, 52, 75syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 2  x.  A ) ^ 2 ) ) )
77 sqval 11179 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 2  x.  A )  e.  CC  ->  (
( 2  x.  A
) ^ 2 )  =  ( ( 2  x.  A )  x.  ( 2  x.  A
) ) )
7851, 77syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
7951, 43, 25mulassd 8874 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 2  x.  A )  x.  (
2  x.  A ) ) )
8043, 25, 43mul32d 9038 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( ( 2  x.  2 )  x.  A
) )
8180, 41syl6eq 2344 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A )  x.  2 )  =  ( 4  x.  A
) )
8281oveq1d 5889 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  0  <  A )  ->  ( (
( 2  x.  A
)  x.  2 )  x.  A )  =  ( ( 4  x.  A )  x.  A
) )
8378, 79, 823eqtr2d 2334 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  0  <  A )  ->  ( (
2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A
) )
8483oveq2d 5890 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( ( 2  x.  A ) ^ 2 ) )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
8573, 76, 843eqtrd 2332 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( B ^ 2 )  /  ( ( 4  x.  A )  x.  A ) ) )
865, 21, 25, 22, 28divdiv1d 9583 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  /  A )  =  ( ( B ^
2 )  /  (
( 4  x.  A
)  x.  A ) ) )
8785, 86eqtr4d 2331 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( -u ( B  /  ( 2  x.  A ) ) ^
2 )  =  ( ( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) )
8887oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( A  x.  (
( ( B ^
2 )  /  (
4  x.  A ) )  /  A ) ) )
8938, 25, 28divcan2d 9554 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  /  A
) )  =  ( ( B ^ 2 )  /  ( 4  x.  A ) ) )
9088, 89eqtrd 2328 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( A  x.  ( -u ( B  /  ( 2  x.  A ) ) ^
2 ) )  =  ( ( B ^
2 )  /  (
4  x.  A ) ) )
9174, 71mulneg2d 9249 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
92 sqval 11179 . . . . . . . . . . . . . . . . . . 19  |-  ( B  e.  CC  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9374, 92syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  0  <  A )  ->  ( B ^ 2 )  =  ( B  x.  B
) )
9493oveq1d 5889 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( ( B  x.  B
)  /  ( 2  x.  A ) ) )
9574, 74, 51, 52divassd 9587 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  0  <  A )  ->  ( ( B  x.  B )  /  ( 2  x.  A ) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9694, 95eqtrd 2328 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  =  ( B  x.  ( B  /  ( 2  x.  A ) ) ) )
9796negeqd 9062 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  -u ( ( B ^ 2 )  /  ( 2  x.  A ) )  = 
-u ( B  x.  ( B  /  (
2  x.  A ) ) ) )
9891, 97eqtr4d 2331 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( B  x.  -u ( B  / 
( 2  x.  A
) ) )  = 
-u ( ( B ^ 2 )  / 
( 2  x.  A
) ) )
9990, 98oveq12d 5892 . . . . . . . . . . . . 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
) ) ) )
1004, 50rerpdivcld 10433 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  RR )
101100recnd 8877 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  e.  CC )
10238, 101negsubd 9179 . . . . . . . . . . . . 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 ) ) ) )
10399, 102eqtrd 2328 . . . . . . . . . . . 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
) ) ) )
104103oveq1d 5889 . . . . . . . . . . 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
) )
10538, 32, 101addsubd 9194 . . . . . . . . . . 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
) )
106104, 105eqtr4d 2331 . . . . . . . . . 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 ) ) ) )
10770, 106breqtrd 4063 . . . . . . . . 9  |-  ( (
ph  /\  0  <  A )  ->  0  <_  ( ( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
)  -  ( ( B ^ 2 )  /  ( 2  x.  A ) ) ) )
10837, 10readdcld 8878 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  C )  e.  RR )
109108, 100subge0d 9378 . . . . . . . . 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 ) ) )
110107, 109mpbid 201 . . . . . . . 8  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 2  x.  A
) )  <_  (
( ( B ^
2 )  /  (
4  x.  A ) )  +  C ) )
11157, 110eqbrtrd 4059 . . . . . . 7  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  +  ( ( B ^ 2 )  / 
( 4  x.  A
) ) )  <_ 
( ( ( B ^ 2 )  / 
( 4  x.  A
) )  +  C
) )
11237, 10, 37leadd2d 9383 . . . . . . 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
) ) )
113111, 112mpbird 223 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  <_  C
)
11437, 10suble0d 9379 . . . . . 6  |-  ( (
ph  /\  0  <  A )  ->  ( (
( ( B ^
2 )  /  (
4  x.  A ) )  -  C )  <_  0  <->  ( ( B ^ 2 )  / 
( 4  x.  A
) )  <_  C
) )
115113, 114mpbird 223 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  /  ( 4  x.  A ) )  -  C )  <_ 
0 )
11636, 115eqbrtrd 4059 . . . 4  |-  ( (
ph  /\  0  <  A )  ->  ( (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  /  ( 4  x.  A ) )  <_ 
0 )
1174, 13resubcld 9227 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  e.  RR )
118 0re 8854 . . . . . 6  |-  0  e.  RR
119118a1i 10 . . . . 5  |-  ( (
ph  /\  0  <  A )  ->  0  e.  RR )
120117, 119, 20ledivmuld 10455 . . . 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 ) ) )
121116, 120mpbid 201 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  (
( 4  x.  A
)  x.  0 ) )
12221mul01d 9027 . . 3  |-  ( (
ph  /\  0  <  A )  ->  ( (
4  x.  A )  x.  0 )  =  0 )
123121, 122breqtrd 4063 . 2  |-  ( (
ph  /\  0  <  A )  ->  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C )
) )  <_  0
)
1249adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  e.  RR )
125124ltp1d 9703 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  C  <  ( C  + 
1 ) )
126 peano2re 9001 . . . . . . . . . . . . . 14  |-  ( C  e.  RR  ->  ( C  +  1 )  e.  RR )
127124, 126syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  +  1 )  e.  RR )
128124, 127ltnegd 9366 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( C  <  ( C  +  1 )  <->  -u ( C  +  1 )  <  -u C
) )
129125, 128mpbid 201 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  -u C
)
130 df-neg 9056 . . . . . . . . . . 11  |-  -u C  =  ( 0  -  C )
131129, 130syl6breq 4078 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  <  ( 0  -  C ) )
132127renegcld 9226 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  RR )
133118a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  e.  RR )
134132, 124, 133ltaddsubd 9388 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( -u ( C  +  1 )  +  C )  <  0  <->  -u ( C  + 
1 )  <  (
0  -  C ) ) )
135131, 134mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  <  0 )
136135expr 598 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  ( B  =/=  0  ->  ( -u ( C  +  1 )  +  C )  <  0 ) )
1371adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  RR )
138 simprr 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  =/=  0 )
139132, 137, 138redivcld 9604 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  RR )
14061adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  A. x  e.  RR  0  <_  ( ( ( A  x.  ( x ^ 2 ) )  +  ( B  x.  x ) )  +  C ) )
141 oveq1 5881 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( x ^ 2 )  =  ( (
-u ( C  + 
1 )  /  B
) ^ 2 ) )
142141oveq2d 5890 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( A  x.  (
x ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
143 oveq2 5882 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( -u ( C  +  1 )  /  B )  -> 
( B  x.  x
)  =  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )
144142, 143oveq12d 5892 . . . . . . . . . . . . . . 15  |-  ( 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 ) ) ) )
145144oveq1d 5889 . . . . . . . . . . . . . 14  |-  ( 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 ) )
146145breq2d 4051 . . . . . . . . . . . . 13  |-  ( 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 ) ) )
147146rspcv 2893 . . . . . . . . . . . 12  |-  ( (
-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 ) ) )
148139, 140, 147sylc 56 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( (
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  +  C ) )
149 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  =  A )
150149oveq1d 5889 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) ) )
151139recnd 8877 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  /  B
)  e.  CC )
152 sqcl 11182 . . . . . . . . . . . . . . . . 17  |-  ( (
-u ( C  + 
1 )  /  B
)  e.  CC  ->  ( ( -u ( C  +  1 )  /  B ) ^ 2 )  e.  CC )
153151, 152syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( -u ( C  +  1 )  /  B ) ^
2 )  e.  CC )
154153mul02d 9026 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
155150, 154eqtr3d 2330 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( A  x.  (
( -u ( C  + 
1 )  /  B
) ^ 2 ) )  =  0 )
156132recnd 8877 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -u ( C  +  1 )  e.  CC )
157137recnd 8877 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  B  e.  CC )
158156, 157, 138divcan2d 9554 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( B  x.  ( -u ( C  +  1 )  /  B ) )  =  -u ( C  +  1 ) )
159155, 158oveq12d 5892 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  ( 0  +  -u ( C  +  1 ) ) )
160156addid2d 9029 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  +  -u ( C  +  1
) )  =  -u ( C  +  1
) )
161159, 160eqtrd 2328 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( A  x.  ( ( -u ( C  +  1 )  /  B ) ^
2 ) )  +  ( B  x.  ( -u ( C  +  1 )  /  B ) ) )  =  -u ( C  +  1
) )
162161oveq1d 5889 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( ( ( A  x.  ( ( -u ( C  +  1
)  /  B ) ^ 2 ) )  +  ( B  x.  ( -u ( C  + 
1 )  /  B
) ) )  +  C )  =  (
-u ( C  + 
1 )  +  C
) )
163148, 162breqtrd 4063 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
0  <_  ( -u ( C  +  1 )  +  C ) )
164132, 124readdcld 8878 . . . . . . . . . . 11  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( -u ( C  + 
1 )  +  C
)  e.  RR )
165 lenlt 8917 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  ( -u ( C  + 
1 )  +  C
)  e.  RR )  ->  ( 0  <_ 
( -u ( C  + 
1 )  +  C
)  <->  -.  ( -u ( C  +  1 )  +  C )  <  0 ) )
166118, 164, 165sylancr 644 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  -> 
( 0  <_  ( -u ( C  +  1 )  +  C )  <->  -.  ( -u ( C  +  1 )  +  C )  <  0
) )
167163, 166mpbid 201 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  =  A  /\  B  =/=  0 ) )  ->  -.  ( -u ( C  +  1 )  +  C )  <  0
)
168167expr 598 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  ( B  =/=  0  ->  -.  ( -u ( C  + 
1 )  +  C
)  <  0 ) )
169136, 168pm2.65d 166 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  -.  B  =/=  0 )
170 nne 2463 . . . . . . 7  |-  ( -.  B  =/=  0  <->  B  =  0 )
171169, 170sylib 188 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  B  =  0 )
172171oveq1d 5889 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  ( B ^ 2 )  =  ( 0 ^ 2 ) )
173 sq0 11211 . . . . 5  |-  ( 0 ^ 2 )  =  0
174172, 173syl6eq 2344 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  ( B ^ 2 )  =  0 )
175 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  0  =  A )
176175oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  ( A  x.  C ) )
1779recnd 8877 . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
178177adantr 451 . . . . . . . 8  |-  ( (
ph  /\  0  =  A )  ->  C  e.  CC )
179178mul02d 9026 . . . . . . 7  |-  ( (
ph  /\  0  =  A )  ->  (
0  x.  C )  =  0 )
180176, 179eqtr3d 2330 . . . . . 6  |-  ( (
ph  /\  0  =  A )  ->  ( A  x.  C )  =  0 )
181180oveq2d 5890 . . . . 5  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  ( 4  x.  0 ) )
18226mul01i 9018 . . . . 5  |-  ( 4  x.  0 )  =  0
183181, 182syl6eq 2344 . . . 4  |-  ( (
ph  /\  0  =  A )  ->  (
4  x.  ( A  x.  C ) )  =  0 )
184174, 183oveq12d 5892 . . 3  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  =  ( 0  -  0 ) )
185 0cn 8847 . . . . 5  |-  0  e.  CC
186185subidi 9133 . . . 4  |-  ( 0  -  0 )  =  0
187 0le0 9843 . . . 4  |-  0  <_  0
188186, 187eqbrtri 4058 . . 3  |-  ( 0  -  0 )  <_ 
0
189184, 188syl6eqbr 4076 . 2  |-  ( (
ph  /\  0  =  A )  ->  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) )  <_  0 )
190 eqid 2296 . . . 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 )
1917, 1, 9, 60, 190discr1 11253 . . 3  |-  ( ph  ->  0  <_  A )
192 leloe 8924 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
193118, 7, 192sylancr 644 . . 3  |-  ( ph  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
194191, 193mpbid 201 . 2  |-  ( ph  ->  ( 0  <  A  \/  0  =  A
) )
195123, 189, 194mpjaodan 761 1  |-  ( ph  ->  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) )  <_  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   ifcif 3578   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   4c4 9813   RR+crp 10370   ^cexp 11120
This theorem is referenced by:  normlem6  21710  csbrn  26565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121
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