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Theorem csbrn 26565
Description: Cauchy-Schwarz-Bunjakovsky inequality for R^n. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
Hypotheses
Ref Expression
csbrn.1  |-  ( ph  ->  A  e.  Fin )
csbrn.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
csbrn.3  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  RR )
Assertion
Ref Expression
csbrn  |-  ( ph  ->  ( sum_ k  e.  A  ( B  x.  C
) ^ 2 )  <_  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) )
Distinct variable groups:    A, k    ph, k
Allowed substitution hints:    B( k)    C( k)

Proof of Theorem csbrn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 2cn 9832 . . . . 5  |-  2  e.  CC
2 csbrn.1 . . . . . . 7  |-  ( ph  ->  A  e.  Fin )
3 csbrn.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
4 csbrn.3 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  RR )
53, 4remulcld 8879 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  ( B  x.  C )  e.  RR )
62, 5fsumrecl 12223 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  A  ( B  x.  C
)  e.  RR )
76recnd 8877 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  ( B  x.  C
)  e.  CC )
8 sqmul 11183 . . . . 5  |-  ( ( 2  e.  CC  /\  sum_ k  e.  A  ( B  x.  C )  e.  CC )  -> 
( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( sum_ k  e.  A  ( B  x.  C ) ^ 2 ) ) )
91, 7, 8sylancr 644 . . . 4  |-  ( ph  ->  ( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( sum_ k  e.  A  ( B  x.  C ) ^ 2 ) ) )
10 sq2 11215 . . . . 5  |-  ( 2 ^ 2 )  =  4
1110oveq1i 5884 . . . 4  |-  ( ( 2 ^ 2 )  x.  ( sum_ k  e.  A  ( B  x.  C ) ^ 2 ) )  =  ( 4  x.  ( sum_ k  e.  A  ( B  x.  C ) ^ 2 ) )
129, 11syl6eq 2344 . . 3  |-  ( ph  ->  ( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  =  ( 4  x.  ( sum_ k  e.  A  ( B  x.  C ) ^ 2 ) ) )
133resqcld 11287 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( B ^ 2 )  e.  RR )
142, 13fsumrecl 12223 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  ( B ^ 2 )  e.  RR )
15 2re 9831 . . . . . 6  |-  2  e.  RR
16 remulcl 8838 . . . . . 6  |-  ( ( 2  e.  RR  /\  sum_ k  e.  A  ( B  x.  C )  e.  RR )  -> 
( 2  x.  sum_ k  e.  A  ( B  x.  C )
)  e.  RR )
1715, 6, 16sylancr 644 . . . . 5  |-  ( ph  ->  ( 2  x.  sum_ k  e.  A  ( B  x.  C )
)  e.  RR )
184resqcld 11287 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  ( C ^ 2 )  e.  RR )
192, 18fsumrecl 12223 . . . . 5  |-  ( ph  -> 
sum_ k  e.  A  ( C ^ 2 )  e.  RR )
202adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  A  e. 
Fin )
2113adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( B ^ 2 )  e.  RR )
22 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  x  e.  RR )
2322resqcld 11287 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
x ^ 2 )  e.  RR )
2421, 23remulcld 8879 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( B ^ 2 )  x.  ( x ^ 2 ) )  e.  RR )
25 remulcl 8838 . . . . . . . . . . . 12  |-  ( ( 2  e.  RR  /\  ( B  x.  C
)  e.  RR )  ->  ( 2  x.  ( B  x.  C
) )  e.  RR )
2615, 5, 25sylancr 644 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  A )  ->  (
2  x.  ( B  x.  C ) )  e.  RR )
2726adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
2  x.  ( B  x.  C ) )  e.  RR )
2827, 22remulcld 8879 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( 2  x.  ( B  x.  C )
)  x.  x )  e.  RR )
2924, 28readdcld 8878 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( B ^
2 )  x.  (
x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  e.  RR )
3018adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( C ^ 2 )  e.  RR )
3129, 30readdcld 8878 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C )
)  x.  x ) )  +  ( C ^ 2 ) )  e.  RR )
323adantlr 695 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  B  e.  RR )
3332, 22remulcld 8879 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( B  x.  x )  e.  RR )
344adantlr 695 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  C  e.  RR )
3533, 34readdcld 8878 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( B  x.  x
)  +  C )  e.  RR )
3635sqge0d 11288 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  0  <_  ( ( ( B  x.  x )  +  C ) ^ 2 ) )
3733recnd 8877 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( B  x.  x )  e.  CC )
3834recnd 8877 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  C  e.  CC )
39 binom2 11234 . . . . . . . . . 10  |-  ( ( ( B  x.  x
)  e.  CC  /\  C  e.  CC )  ->  ( ( ( B  x.  x )  +  C ) ^ 2 )  =  ( ( ( ( B  x.  x ) ^ 2 )  +  ( 2  x.  ( ( B  x.  x )  x.  C ) ) )  +  ( C ^
2 ) ) )
4037, 38, 39syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( B  x.  x )  +  C
) ^ 2 )  =  ( ( ( ( B  x.  x
) ^ 2 )  +  ( 2  x.  ( ( B  x.  x )  x.  C
) ) )  +  ( C ^ 2 ) ) )
4132recnd 8877 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  B  e.  CC )
4222recnd 8877 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  x  e.  CC )
4341, 42sqmuld 11273 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( B  x.  x
) ^ 2 )  =  ( ( B ^ 2 )  x.  ( x ^ 2 ) ) )
4441, 42, 38mul32d 9038 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( B  x.  x
)  x.  C )  =  ( ( B  x.  C )  x.  x ) )
4544oveq2d 5890 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
2  x.  ( ( B  x.  x )  x.  C ) )  =  ( 2  x.  ( ( B  x.  C )  x.  x
) ) )
461a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  2  e.  CC )
475adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( B  x.  C )  e.  RR )
4847recnd 8877 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( B  x.  C )  e.  CC )
4946, 48, 42mulassd 8874 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( 2  x.  ( B  x.  C )
)  x.  x )  =  ( 2  x.  ( ( B  x.  C )  x.  x
) ) )
5045, 49eqtr4d 2331 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
2  x.  ( ( B  x.  x )  x.  C ) )  =  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )
5143, 50oveq12d 5892 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( B  x.  x ) ^ 2 )  +  ( 2  x.  ( ( B  x.  x )  x.  C ) ) )  =  ( ( ( B ^ 2 )  x.  ( x ^
2 ) )  +  ( ( 2  x.  ( B  x.  C
) )  x.  x
) ) )
5251oveq1d 5889 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( ( B  x.  x ) ^
2 )  +  ( 2  x.  ( ( B  x.  x )  x.  C ) ) )  +  ( C ^ 2 ) )  =  ( ( ( ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  +  ( C ^ 2 ) ) )
5340, 52eqtrd 2328 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( B  x.  x )  +  C
) ^ 2 )  =  ( ( ( ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  +  ( C ^ 2 ) ) )
5436, 53breqtrd 4063 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  0  <_  ( ( ( ( B ^ 2 )  x.  ( x ^
2 ) )  +  ( ( 2  x.  ( B  x.  C
) )  x.  x
) )  +  ( C ^ 2 ) ) )
5520, 31, 54fsumge0 12269 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  0  <_  sum_ k  e.  A  ( ( ( ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C )
)  x.  x ) )  +  ( C ^ 2 ) ) )
5624recnd 8877 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( B ^ 2 )  x.  ( x ^ 2 ) )  e.  CC )
5728recnd 8877 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( 2  x.  ( B  x.  C )
)  x.  x )  e.  CC )
5856, 57addcld 8870 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
( ( B ^
2 )  x.  (
x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  e.  CC )
5930recnd 8877 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( C ^ 2 )  e.  CC )
6020, 58, 59fsumadd 12227 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  sum_ k  e.  A  ( (
( ( B ^
2 )  x.  (
x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  +  ( C ^
2 ) )  =  ( sum_ k  e.  A  ( ( ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C )
)  x.  x ) )  +  sum_ k  e.  A  ( C ^ 2 ) ) )
6120, 56, 57fsumadd 12227 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  sum_ k  e.  A  ( (
( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  =  ( sum_ k  e.  A  ( ( B ^
2 )  x.  (
x ^ 2 ) )  +  sum_ k  e.  A  ( (
2  x.  ( B  x.  C ) )  x.  x ) ) )
62 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  x  e.  RR )
6362recnd 8877 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR )  ->  x  e.  CC )
6463sqcld 11259 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR )  ->  ( x ^ 2 )  e.  CC )
6521recnd 8877 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  ( B ^ 2 )  e.  CC )
6620, 64, 65fsummulc1 12263 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR )  ->  ( sum_ k  e.  A  ( B ^ 2 )  x.  ( x ^ 2 ) )  =  sum_ k  e.  A  (
( B ^ 2 )  x.  ( x ^ 2 ) ) )
671a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR )  ->  2  e.  CC )
6820, 67, 48fsummulc2 12262 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR )  ->  ( 2  x.  sum_ k  e.  A  ( B  x.  C
) )  =  sum_ k  e.  A  (
2  x.  ( B  x.  C ) ) )
6968oveq1d 5889 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( 2  x.  sum_ k  e.  A  ( B  x.  C ) )  x.  x )  =  (
sum_ k  e.  A  ( 2  x.  ( B  x.  C )
)  x.  x ) )
7026recnd 8877 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  (
2  x.  ( B  x.  C ) )  e.  CC )
7170adantlr 695 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR )  /\  k  e.  A )  ->  (
2  x.  ( B  x.  C ) )  e.  CC )
7220, 63, 71fsummulc1 12263 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  RR )  ->  ( sum_ k  e.  A  (
2  x.  ( B  x.  C ) )  x.  x )  = 
sum_ k  e.  A  ( ( 2  x.  ( B  x.  C
) )  x.  x
) )
7369, 72eqtrd 2328 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR )  ->  ( ( 2  x.  sum_ k  e.  A  ( B  x.  C ) )  x.  x )  =  sum_ k  e.  A  (
( 2  x.  ( B  x.  C )
)  x.  x ) )
7466, 73oveq12d 5892 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR )  ->  ( (
sum_ k  e.  A  ( B ^ 2 )  x.  ( x ^
2 ) )  +  ( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) )  x.  x
) )  =  (
sum_ k  e.  A  ( ( B ^
2 )  x.  (
x ^ 2 ) )  +  sum_ k  e.  A  ( (
2  x.  ( B  x.  C ) )  x.  x ) ) )
7561, 74eqtr4d 2331 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR )  ->  sum_ k  e.  A  ( (
( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  =  ( ( sum_ k  e.  A  ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  sum_ k  e.  A  ( B  x.  C )
)  x.  x ) ) )
7675oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( sum_ k  e.  A  (
( ( B ^
2 )  x.  (
x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  +  sum_ k  e.  A  ( C ^ 2 ) )  =  ( ( ( sum_ k  e.  A  ( B ^ 2 )  x.  ( x ^
2 ) )  +  ( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) )  x.  x
) )  +  sum_ k  e.  A  ( C ^ 2 ) ) )
7760, 76eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  sum_ k  e.  A  ( (
( ( B ^
2 )  x.  (
x ^ 2 ) )  +  ( ( 2  x.  ( B  x.  C ) )  x.  x ) )  +  ( C ^
2 ) )  =  ( ( ( sum_ k  e.  A  ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  sum_ k  e.  A  ( B  x.  C )
)  x.  x ) )  +  sum_ k  e.  A  ( C ^ 2 ) ) )
7855, 77breqtrd 4063 . . . . 5  |-  ( (
ph  /\  x  e.  RR )  ->  0  <_ 
( ( ( sum_ k  e.  A  ( B ^ 2 )  x.  ( x ^ 2 ) )  +  ( ( 2  x.  sum_ k  e.  A  ( B  x.  C )
)  x.  x ) )  +  sum_ k  e.  A  ( C ^ 2 ) ) )
7914, 17, 19, 78discr 11254 . . . 4  |-  ( ph  ->  ( ( ( 2  x.  sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  -  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) ) )  <_ 
0 )
8017resqcld 11287 . . . . 5  |-  ( ph  ->  ( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  e.  RR )
81 4re 9835 . . . . . 6  |-  4  e.  RR
8214, 19remulcld 8879 . . . . . 6  |-  ( ph  ->  ( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) )  e.  RR )
83 remulcl 8838 . . . . . 6  |-  ( ( 4  e.  RR  /\  ( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) )  e.  RR )  ->  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) ) )  e.  RR )
8481, 82, 83sylancr 644 . . . . 5  |-  ( ph  ->  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) ) )  e.  RR )
8580, 84suble0d 9379 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  sum_ k  e.  A  ( B  x.  C ) ) ^
2 )  -  (
4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) ) )  <_ 
0  <->  ( ( 2  x.  sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  <_  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) ) ) )
8679, 85mpbid 201 . . 3  |-  ( ph  ->  ( ( 2  x. 
sum_ k  e.  A  ( B  x.  C
) ) ^ 2 )  <_  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) ) )
8712, 86eqbrtrrd 4061 . 2  |-  ( ph  ->  ( 4  x.  ( sum_ k  e.  A  ( B  x.  C ) ^ 2 ) )  <_  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) ) ) )
886resqcld 11287 . . 3  |-  ( ph  ->  ( sum_ k  e.  A  ( B  x.  C
) ^ 2 )  e.  RR )
8981a1i 10 . . 3  |-  ( ph  ->  4  e.  RR )
90 4pos 9848 . . . 4  |-  0  <  4
9190a1i 10 . . 3  |-  ( ph  ->  0  <  4 )
92 lemul2 9625 . . 3  |-  ( ( ( sum_ k  e.  A  ( B  x.  C
) ^ 2 )  e.  RR  /\  ( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) )  e.  RR  /\  ( 4  e.  RR  /\  0  <  4 ) )  ->  ( ( sum_ k  e.  A  ( B  x.  C ) ^ 2 )  <_ 
( sum_ k  e.  A  ( B ^ 2 )  x.  sum_ k  e.  A  ( C ^ 2 ) )  <->  ( 4  x.  ( sum_ k  e.  A  ( B  x.  C
) ^ 2 ) )  <_  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) ) ) )
9388, 82, 89, 91, 92syl112anc 1186 . 2  |-  ( ph  ->  ( ( sum_ k  e.  A  ( B  x.  C ) ^ 2 )  <_  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) )  <->  ( 4  x.  ( sum_ k  e.  A  ( B  x.  C
) ^ 2 ) )  <_  ( 4  x.  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) ) ) )
9487, 93mpbird 223 1  |-  ( ph  ->  ( sum_ k  e.  A  ( B  x.  C
) ^ 2 )  <_  ( sum_ k  e.  A  ( B ^ 2 )  x. 
sum_ k  e.  A  ( C ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   2c2 9811   4c4 9813   ^cexp 11120   sum_csu 12174
This theorem is referenced by:  trirn  26566
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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  ax-pre-sup 8831
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-int 3879  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  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-ico 10678  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
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