Step |
Hyp |
Ref |
Expression |
1 |
|
divsqrtsum.2 |
|- F = ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( 1 / ( sqrt ` n ) ) - ( 2 x. ( sqrt ` x ) ) ) ) |
2 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
3 |
2
|
eqcomi |
|- RR+ = ( 0 (,) +oo ) |
4 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
5 |
|
1zzd |
|- ( T. -> 1 e. ZZ ) |
6 |
|
0red |
|- ( T. -> 0 e. RR ) |
7 |
|
1re |
|- 1 e. RR |
8 |
|
0nn0 |
|- 0 e. NN0 |
9 |
7 8
|
nn0addge2i |
|- 1 <_ ( 0 + 1 ) |
10 |
9
|
a1i |
|- ( T. -> 1 <_ ( 0 + 1 ) ) |
11 |
|
2re |
|- 2 e. RR |
12 |
|
rpsqrtcl |
|- ( x e. RR+ -> ( sqrt ` x ) e. RR+ ) |
13 |
12
|
adantl |
|- ( ( T. /\ x e. RR+ ) -> ( sqrt ` x ) e. RR+ ) |
14 |
13
|
rpred |
|- ( ( T. /\ x e. RR+ ) -> ( sqrt ` x ) e. RR ) |
15 |
|
remulcl |
|- ( ( 2 e. RR /\ ( sqrt ` x ) e. RR ) -> ( 2 x. ( sqrt ` x ) ) e. RR ) |
16 |
11 14 15
|
sylancr |
|- ( ( T. /\ x e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. RR ) |
17 |
13
|
rprecred |
|- ( ( T. /\ x e. RR+ ) -> ( 1 / ( sqrt ` x ) ) e. RR ) |
18 |
|
nnrp |
|- ( x e. NN -> x e. RR+ ) |
19 |
18 17
|
sylan2 |
|- ( ( T. /\ x e. NN ) -> ( 1 / ( sqrt ` x ) ) e. RR ) |
20 |
|
reelprrecn |
|- RR e. { RR , CC } |
21 |
20
|
a1i |
|- ( T. -> RR e. { RR , CC } ) |
22 |
13
|
rpcnd |
|- ( ( T. /\ x e. RR+ ) -> ( sqrt ` x ) e. CC ) |
23 |
|
2rp |
|- 2 e. RR+ |
24 |
|
rpmulcl |
|- ( ( 2 e. RR+ /\ ( sqrt ` x ) e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. RR+ ) |
25 |
23 13 24
|
sylancr |
|- ( ( T. /\ x e. RR+ ) -> ( 2 x. ( sqrt ` x ) ) e. RR+ ) |
26 |
25
|
rpreccld |
|- ( ( T. /\ x e. RR+ ) -> ( 1 / ( 2 x. ( sqrt ` x ) ) ) e. RR+ ) |
27 |
|
dvsqrt |
|- ( RR _D ( x e. RR+ |-> ( sqrt ` x ) ) ) = ( x e. RR+ |-> ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) |
28 |
27
|
a1i |
|- ( T. -> ( RR _D ( x e. RR+ |-> ( sqrt ` x ) ) ) = ( x e. RR+ |-> ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) |
29 |
|
2cnd |
|- ( T. -> 2 e. CC ) |
30 |
21 22 26 28 29
|
dvmptcmul |
|- ( T. -> ( RR _D ( x e. RR+ |-> ( 2 x. ( sqrt ` x ) ) ) ) = ( x e. RR+ |-> ( 2 x. ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) ) |
31 |
|
2cnd |
|- ( ( T. /\ x e. RR+ ) -> 2 e. CC ) |
32 |
|
1cnd |
|- ( ( T. /\ x e. RR+ ) -> 1 e. CC ) |
33 |
25
|
rpcnne0d |
|- ( ( T. /\ x e. RR+ ) -> ( ( 2 x. ( sqrt ` x ) ) e. CC /\ ( 2 x. ( sqrt ` x ) ) =/= 0 ) ) |
34 |
|
divass |
|- ( ( 2 e. CC /\ 1 e. CC /\ ( ( 2 x. ( sqrt ` x ) ) e. CC /\ ( 2 x. ( sqrt ` x ) ) =/= 0 ) ) -> ( ( 2 x. 1 ) / ( 2 x. ( sqrt ` x ) ) ) = ( 2 x. ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) |
35 |
31 32 33 34
|
syl3anc |
|- ( ( T. /\ x e. RR+ ) -> ( ( 2 x. 1 ) / ( 2 x. ( sqrt ` x ) ) ) = ( 2 x. ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) |
36 |
13
|
rpcnne0d |
|- ( ( T. /\ x e. RR+ ) -> ( ( sqrt ` x ) e. CC /\ ( sqrt ` x ) =/= 0 ) ) |
37 |
|
rpcnne0 |
|- ( 2 e. RR+ -> ( 2 e. CC /\ 2 =/= 0 ) ) |
38 |
23 37
|
mp1i |
|- ( ( T. /\ x e. RR+ ) -> ( 2 e. CC /\ 2 =/= 0 ) ) |
39 |
|
divcan5 |
|- ( ( 1 e. CC /\ ( ( sqrt ` x ) e. CC /\ ( sqrt ` x ) =/= 0 ) /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( 2 x. 1 ) / ( 2 x. ( sqrt ` x ) ) ) = ( 1 / ( sqrt ` x ) ) ) |
40 |
32 36 38 39
|
syl3anc |
|- ( ( T. /\ x e. RR+ ) -> ( ( 2 x. 1 ) / ( 2 x. ( sqrt ` x ) ) ) = ( 1 / ( sqrt ` x ) ) ) |
41 |
35 40
|
eqtr3d |
|- ( ( T. /\ x e. RR+ ) -> ( 2 x. ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) = ( 1 / ( sqrt ` x ) ) ) |
42 |
41
|
mpteq2dva |
|- ( T. -> ( x e. RR+ |-> ( 2 x. ( 1 / ( 2 x. ( sqrt ` x ) ) ) ) ) = ( x e. RR+ |-> ( 1 / ( sqrt ` x ) ) ) ) |
43 |
30 42
|
eqtrd |
|- ( T. -> ( RR _D ( x e. RR+ |-> ( 2 x. ( sqrt ` x ) ) ) ) = ( x e. RR+ |-> ( 1 / ( sqrt ` x ) ) ) ) |
44 |
|
fveq2 |
|- ( x = n -> ( sqrt ` x ) = ( sqrt ` n ) ) |
45 |
44
|
oveq2d |
|- ( x = n -> ( 1 / ( sqrt ` x ) ) = ( 1 / ( sqrt ` n ) ) ) |
46 |
|
simp3r |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> x <_ n ) |
47 |
|
simp2l |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> x e. RR+ ) |
48 |
47
|
rprege0d |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( x e. RR /\ 0 <_ x ) ) |
49 |
|
simp2r |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> n e. RR+ ) |
50 |
49
|
rprege0d |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( n e. RR /\ 0 <_ n ) ) |
51 |
|
sqrtle |
|- ( ( ( x e. RR /\ 0 <_ x ) /\ ( n e. RR /\ 0 <_ n ) ) -> ( x <_ n <-> ( sqrt ` x ) <_ ( sqrt ` n ) ) ) |
52 |
48 50 51
|
syl2anc |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( x <_ n <-> ( sqrt ` x ) <_ ( sqrt ` n ) ) ) |
53 |
46 52
|
mpbid |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( sqrt ` x ) <_ ( sqrt ` n ) ) |
54 |
47
|
rpsqrtcld |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( sqrt ` x ) e. RR+ ) |
55 |
49
|
rpsqrtcld |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( sqrt ` n ) e. RR+ ) |
56 |
54 55
|
lerecd |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( ( sqrt ` x ) <_ ( sqrt ` n ) <-> ( 1 / ( sqrt ` n ) ) <_ ( 1 / ( sqrt ` x ) ) ) ) |
57 |
53 56
|
mpbid |
|- ( ( T. /\ ( x e. RR+ /\ n e. RR+ ) /\ ( 0 <_ x /\ x <_ n ) ) -> ( 1 / ( sqrt ` n ) ) <_ ( 1 / ( sqrt ` x ) ) ) |
58 |
|
sqrtlim |
|- ( x e. RR+ |-> ( 1 / ( sqrt ` x ) ) ) ~~>r 0 |
59 |
58
|
a1i |
|- ( T. -> ( x e. RR+ |-> ( 1 / ( sqrt ` x ) ) ) ~~>r 0 ) |
60 |
|
fveq2 |
|- ( x = A -> ( sqrt ` x ) = ( sqrt ` A ) ) |
61 |
60
|
oveq2d |
|- ( x = A -> ( 1 / ( sqrt ` x ) ) = ( 1 / ( sqrt ` A ) ) ) |
62 |
3 4 5 6 10 6 16 17 19 43 45 57 1 59 61
|
dvfsumrlim3 |
|- ( T. -> ( F : RR+ --> RR /\ F e. dom ~~>r /\ ( ( F ~~>r L /\ A e. RR+ /\ 0 <_ A ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( 1 / ( sqrt ` A ) ) ) ) ) |
63 |
62
|
simp1d |
|- ( T. -> F : RR+ --> RR ) |
64 |
63
|
mptru |
|- F : RR+ --> RR |
65 |
62
|
simp2d |
|- ( T. -> F e. dom ~~>r ) |
66 |
65
|
mptru |
|- F e. dom ~~>r |
67 |
|
rpge0 |
|- ( A e. RR+ -> 0 <_ A ) |
68 |
67
|
adantl |
|- ( ( F ~~>r L /\ A e. RR+ ) -> 0 <_ A ) |
69 |
62
|
simp3d |
|- ( T. -> ( ( F ~~>r L /\ A e. RR+ /\ 0 <_ A ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( 1 / ( sqrt ` A ) ) ) ) |
70 |
69
|
mptru |
|- ( ( F ~~>r L /\ A e. RR+ /\ 0 <_ A ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( 1 / ( sqrt ` A ) ) ) |
71 |
68 70
|
mpd3an3 |
|- ( ( F ~~>r L /\ A e. RR+ ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( 1 / ( sqrt ` A ) ) ) |
72 |
64 66 71
|
3pm3.2i |
|- ( F : RR+ --> RR /\ F e. dom ~~>r /\ ( ( F ~~>r L /\ A e. RR+ ) -> ( abs ` ( ( F ` A ) - L ) ) <_ ( 1 / ( sqrt ` A ) ) ) ) |