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Theorem loglesqr 20098
Description: An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)
Assertion
Ref Expression
loglesqr  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  <_  ( sqr `  A ) )

Proof of Theorem loglesqr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 8838 . . . 4  |-  0  e.  RR
21a1i 10 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  RR )
3 simpl 443 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
4 elicc2 10715 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
51, 3, 4sylancr 644 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
65biimpa 470 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  A
) )
76simp1d 967 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  RR )
86simp2d 968 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  0  <_  x )
97, 8ge0p1rpd 10416 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  +  1 )  e.  RR+ )
10 fvres 5542 . . . . . 6  |-  ( ( x  +  1 )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( x  +  1
) )  =  ( log `  ( x  +  1 ) ) )
119, 10syl 15 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( ( log  |`  RR+ ) `  (
x  +  1 ) )  =  ( log `  ( x  +  1 ) ) )
1211mpteq2dva 4106 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) ) )
13 eqid 2283 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1413cnfldtopon 18292 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
157ex 423 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  RR ) )
1615ssrdv 3185 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  RR )
17 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
1816, 17syl6ss 3191 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  CC )
19 resttopon 16892 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  (
0 [,] A ) 
C_  CC )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
2014, 18, 19sylancr 644 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
21 eqid 2283 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )
229, 21fmptd 5684 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) ) : ( 0 [,] A ) -->
RR+ )
23 rpssre 10364 . . . . . . . . . 10  |-  RR+  C_  RR
2423, 17sstri 3188 . . . . . . . . 9  |-  RR+  C_  CC
2513addcn 18369 . . . . . . . . . . 11  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
2625a1i 10 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  +  e.  ( (
( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
27 ssid 3197 . . . . . . . . . . 11  |-  CC  C_  CC
28 cncfmptid 18416 . . . . . . . . . . 11  |-  ( ( ( 0 [,] A
)  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  ( 0 [,] A )  |->  x )  e.  ( ( 0 [,] A )
-cn-> CC ) )
2918, 27, 28sylancl 643 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  x )  e.  ( ( 0 [,] A
) -cn-> CC ) )
30 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
3130a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
1  e.  CC )
3227a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  CC  C_  CC )
33 cncfmptc 18415 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  ( 0 [,] A
)  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  ( 0 [,] A )  |->  1 )  e.  ( ( 0 [,] A )
-cn-> CC ) )
3431, 18, 32, 33syl3anc 1182 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  1 )  e.  ( ( 0 [,] A
) -cn-> CC ) )
3513, 26, 29, 34cncfmpt2f 18418 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> CC ) )
36 cncffvrn 18402 . . . . . . . . 9  |-  ( (
RR+  C_  CC  /\  (
x  e.  ( 0 [,] A )  |->  ( x  +  1 ) )  e.  ( ( 0 [,] A )
-cn-> CC ) )  -> 
( ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )  e.  ( ( 0 [,] A ) -cn-> RR+ )  <->  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) ) : ( 0 [,] A ) --> RR+ ) )
3724, 35, 36sylancr 644 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )  e.  ( ( 0 [,] A ) -cn-> RR+ )  <->  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) ) : ( 0 [,] A ) --> RR+ ) )
3822, 37mpbird 223 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> RR+ ) )
39 eqid 2283 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  ( 0 [,] A
) )  =  ( ( TopOpen ` fld )t  ( 0 [,] A ) )
40 eqid 2283 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
4113, 39, 40cncfcn 18413 . . . . . . . 8  |-  ( ( ( 0 [,] A
)  C_  CC  /\  RR+  C_  CC )  ->  ( ( 0 [,] A ) -cn-> RR+ )  =  ( (
( TopOpen ` fld )t  ( 0 [,] A ) )  Cn  ( ( TopOpen ` fld )t  RR+ ) ) )
4218, 24, 41sylancl 643 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR+ )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
4338, 42eleqtrd 2359 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( ( TopOpen ` fld )t  ( 0 [,] A ) )  Cn  ( ( TopOpen ` fld )t  RR+ ) ) )
44 relogcn 19985 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
45 eqid 2283 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
4613, 40, 45cncfcn 18413 . . . . . . . . 9  |-  ( (
RR+  C_  CC  /\  RR  C_  CC )  ->  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) ) )
4724, 17, 46mp2an 653 . . . . . . . 8  |-  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4844, 47eleqtri 2355 . . . . . . 7  |-  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4948a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5020, 43, 49cnmpt11f 17358 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  e.  ( ( ( TopOpen ` fld )t  (
0 [,] A ) )  Cn  ( (
TopOpen ` fld )t  RR ) ) )
5113, 39, 45cncfcn 18413 . . . . . 6  |-  ( ( ( 0 [,] A
)  C_  CC  /\  RR  C_  CC )  ->  (
( 0 [,] A
) -cn-> RR )  =  ( ( ( TopOpen ` fld )t  ( 0 [,] A ) )  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5218, 17, 51sylancl 643 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5350, 52eleqtrrd 2360 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
5412, 53eqeltrrd 2358 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
55 reex 8828 . . . . . 6  |-  RR  e.  _V
5655prid1 3734 . . . . 5  |-  RR  e.  { RR ,  CC }
5756a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  e.  { RR ,  CC } )
58 simpr 447 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
59 1rp 10358 . . . . . . 7  |-  1  e.  RR+
60 rpaddcl 10374 . . . . . . 7  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
6158, 59, 60sylancl 643 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR+ )
6261relogcld 19974 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  RR )
6362recnd 8861 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  CC )
6461rpreccld 10400 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  RR+ )
6530a1i 10 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  1  e.  CC )
66 relogcl 19932 . . . . . . . 8  |-  ( y  e.  RR+  ->  ( log `  y )  e.  RR )
6766adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  RR )
6867recnd 8861 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  CC )
69 rpreccl 10377 . . . . . . 7  |-  ( y  e.  RR+  ->  ( 1  /  y )  e.  RR+ )
7069adantl 452 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( 1  /  y
)  e.  RR+ )
71 peano2re 8985 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
7271adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  RR )
7372recnd 8861 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  CC )
7430a1i 10 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  1  e.  CC )
7517a1i 10 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  C_  CC )
7675sselda 3180 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  x  e.  CC )
7757dvmptid 19306 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  x ) )  =  ( x  e.  RR  |->  1 ) )
78 0cn 8831 . . . . . . . . . 10  |-  0  e.  CC
7978a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  0  e.  CC )
8057, 31dvmptc 19307 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  1 ) )  =  ( x  e.  RR  |->  0 ) )
8157, 76, 74, 77, 74, 79, 80dvmptadd 19309 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  ( 1  +  0 ) ) )
8230addid1i 8999 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
8382mpteq2i 4103 . . . . . . . 8  |-  ( x  e.  RR  |->  ( 1  +  0 ) )  =  ( x  e.  RR  |->  1 )
8481, 83syl6eq 2331 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  1 ) )
8523a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR+  C_  RR )
8613tgioo2 18309 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
87 ioorp 10727 . . . . . . . . 9  |-  ( 0 (,)  +oo )  =  RR+
88 iooretop 18275 . . . . . . . . 9  |-  ( 0 (,)  +oo )  e.  (
topGen `  ran  (,) )
8987, 88eqeltrri 2354 . . . . . . . 8  |-  RR+  e.  ( topGen `  ran  (,) )
9089a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR+ 
e.  ( topGen `  ran  (,) ) )
9157, 73, 74, 84, 85, 86, 13, 90dvmptres 19312 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( x  +  1 ) ) )  =  ( x  e.  RR+  |->  1 ) )
92 dvrelog 19984 . . . . . . 7  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( y  e.  RR+  |->  ( 1  /  y ) )
93 relogf1o 19924 . . . . . . . . . . 11  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
94 f1of 5472 . . . . . . . . . . 11  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
9593, 94mp1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ ) : RR+ --> RR )
9695feqmptd 5575 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y
) ) )
97 fvres 5542 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  ( ( log  |`  RR+ ) `  y )  =  ( log `  y ) )
9897mpteq2ia 4102 . . . . . . . . 9  |-  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y ) )  =  ( y  e.  RR+  |->  ( log `  y ) )
9996, 98syl6eq 2331 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( log `  y
) ) )
10099oveq2d 5874 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  ( log  |`  RR+ ) )  =  ( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) ) )
10192, 100syl5reqr 2330 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) )  =  ( y  e.  RR+  |->  ( 1  / 
y ) ) )
102 fveq2 5525 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  ( log `  y )  =  ( log `  (
x  +  1 ) ) )
103 oveq2 5866 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  (
1  /  y )  =  ( 1  / 
( x  +  1 ) ) )
10457, 57, 61, 65, 68, 70, 91, 101, 102, 103dvmptco 19321 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( ( 1  /  ( x  + 
1 ) )  x.  1 ) ) )
10564rpcnd 10392 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  CC )
106105mulid1d 8852 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 1  / 
( x  +  1 ) )  x.  1 )  =  ( 1  /  ( x  + 
1 ) ) )
107106mpteq2dva 4106 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  RR+  |->  ( ( 1  / 
( x  +  1 ) )  x.  1 ) )  =  ( x  e.  RR+  |->  ( 1  /  ( x  + 
1 ) ) ) )
108104, 107eqtrd 2315 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( x  +  1 ) ) ) )
109 ioossicc 10735 . . . . . . . . 9  |-  ( 0 (,) A )  C_  ( 0 [,] A
)
110109sseli 3176 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  x  e.  ( 0 [,] A
) )
111110, 7sylan2 460 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR )
112 eliooord 10710 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) A )  ->  (
0  <  x  /\  x  <  A ) )
113112simpld 445 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  0  <  x )
114113adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  0  <  x )
115111, 114elrpd 10388 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR+ )
116115ex 423 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 (,) A )  ->  x  e.  RR+ ) )
117116ssrdv 3185 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  C_  RR+ )
118 iooretop 18275 . . . . 5  |-  ( 0 (,) A )  e.  ( topGen `  ran  (,) )
119118a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  e.  ( topGen ` 
ran  (,) ) )
12057, 63, 64, 108, 117, 86, 13, 119dvmptres 19312 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( x  +  1 ) ) ) )
121 elrege0 10746 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 
+oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
1227, 8, 121sylanbrc 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  ( 0 [,)  +oo ) )
123122ex 423 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  ( 0 [,)  +oo )
) )
124123ssrdv 3185 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  ( 0 [,)  +oo ) )
125 resabs1 4984 . . . . . 6  |-  ( ( 0 [,] A ) 
C_  ( 0 [,) 
+oo )  ->  (
( sqr  |`  ( 0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( sqr  |`  ( 0 [,] A ) ) )
126124, 125syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( sqr  |`  ( 0 [,] A ) ) )
127 sqrf 11847 . . . . . . 7  |-  sqr : CC
--> CC
128127a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sqr : CC --> CC )
129128, 18feqresmpt 5576 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( sqr `  x
) ) )
130126, 129eqtrd 2315 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( x  e.  ( 0 [,] A )  |->  ( sqr `  x ) ) )
131 resqrcn 20089 . . . . 5  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
132 rescncf 18401 . . . . 5  |-  ( ( 0 [,] A ) 
C_  ( 0 [,) 
+oo )  ->  (
( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,) 
+oo ) -cn-> RR )  ->  ( ( sqr  |`  ( 0 [,)  +oo ) )  |`  (
0 [,] A ) )  e.  ( ( 0 [,] A )
-cn-> RR ) ) )
133124, 131, 132ee10 1366 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
134130, 133eqeltrrd 2358 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
135 rpcn 10362 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
136135adantl 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  CC )
137136sqrcld 11919 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  CC )
138 2rp 10359 . . . . . 6  |-  2  e.  RR+
139 rpsqrcl 11750 . . . . . . 7  |-  ( x  e.  RR+  ->  ( sqr `  x )  e.  RR+ )
140139adantl 452 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR+ )
141 rpmulcl 10375 . . . . . 6  |-  ( ( 2  e.  RR+  /\  ( sqr `  x )  e.  RR+ )  ->  ( 2  x.  ( sqr `  x
) )  e.  RR+ )
142138, 140, 141sylancr 644 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR+ )
143142rpreccld 10400 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
2  x.  ( sqr `  x ) ) )  e.  RR+ )
144 dvsqr 20084 . . . . 5  |-  ( RR 
_D  ( x  e.  RR+  |->  ( sqr `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  x ) ) ) )
145144a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( sqr `  x ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( 2  x.  ( sqr `  x ) ) ) ) )
14657, 137, 143, 145, 117, 86, 13, 119dvmptres 19312 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( sqr `  x ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
147140rpred 10390 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR )
148 1re 8837 . . . . . . . . 9  |-  1  e.  RR
149 resubcl 9111 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  RR  /\  1  e.  RR )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
150147, 148, 149sylancl 643 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
151150sqge0d 11272 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
( sqr `  x
)  -  1 ) ^ 2 ) )
152136sqsqrd 11921 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
) ^ 2 )  =  x )
153137mulid1d 8852 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  x.  1 )  =  ( sqr `  x
) )
154153oveq2d 5874 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  (
( sqr `  x
)  x.  1 ) )  =  ( 2  x.  ( sqr `  x
) ) )
155152, 154oveq12d 5876 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x ) ^ 2 )  -  ( 2  x.  ( ( sqr `  x )  x.  1 ) ) )  =  ( x  -  (
2  x.  ( sqr `  x ) ) ) )
156 sq1 11198 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
157156a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1 ^ 2 )  =  1 )
158155, 157oveq12d 5876 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( ( sqr `  x ) ^ 2 )  -  ( 2  x.  (
( sqr `  x
)  x.  1 ) ) )  +  ( 1 ^ 2 ) )  =  ( ( x  -  ( 2  x.  ( sqr `  x
) ) )  +  1 ) )
159 binom2sub 11220 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( ( ( sqr `  x
) ^ 2 )  -  ( 2  x.  ( ( sqr `  x
)  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
160137, 30, 159sylancl 643 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( ( ( sqr `  x
) ^ 2 )  -  ( 2  x.  ( ( sqr `  x
)  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
161142rpcnd 10392 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  CC )
162136, 65, 161addsubd 9178 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( x  + 
1 )  -  (
2  x.  ( sqr `  x ) ) )  =  ( ( x  -  ( 2  x.  ( sqr `  x
) ) )  +  1 ) )
163158, 160, 1623eqtr4d 2325 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( x  +  1 )  -  ( 2  x.  ( sqr `  x ) ) ) )
164151, 163breqtrd 4047 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) ) )
16561rpred 10390 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR )
166142rpred 10390 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR )
167165, 166subge0d 9362 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 0  <_  (
( x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) )  <->  ( 2  x.  ( sqr `  x
) )  <_  (
x  +  1 ) ) )
168164, 167mpbid 201 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  <_  ( x  + 
1 ) )
169142, 61lerecd 10409 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 2  x.  ( sqr `  x
) )  <_  (
x  +  1 )  <-> 
( 1  /  (
x  +  1 ) )  <_  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
170168, 169mpbid 201 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  <_  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) )
171115, 170syldan 456 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  ( 1  /  ( x  + 
1 ) )  <_ 
( 1  /  (
2  x.  ( sqr `  x ) ) ) )
172 rexr 8877 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
173 0xr 8878 . . . . 5  |-  0  e.  RR*
174 lbicc2 10752 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
175173, 174mp3an1 1264 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
176172, 175sylan 457 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  ( 0 [,] A ) )
177 ubicc2 10753 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
178173, 177mp3an1 1264 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
179172, 178sylan 457 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  ( 0 [,] A ) )
180 simpr 447 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  A )
181 oveq1 5865 . . . . . 6  |-  ( x  =  0  ->  (
x  +  1 )  =  ( 0  +  1 ) )
182 0p1e1 9839 . . . . . 6  |-  ( 0  +  1 )  =  1
183181, 182syl6eq 2331 . . . . 5  |-  ( x  =  0  ->  (
x  +  1 )  =  1 )
184183fveq2d 5529 . . . 4  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  ( log `  1
) )
185 log1 19939 . . . 4  |-  ( log `  1 )  =  0
186184, 185syl6eq 2331 . . 3  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  0 )
187 fveq2 5525 . . . 4  |-  ( x  =  0  ->  ( sqr `  x )  =  ( sqr `  0
) )
188 sqr0 11727 . . . 4  |-  ( sqr `  0 )  =  0
189187, 188syl6eq 2331 . . 3  |-  ( x  =  0  ->  ( sqr `  x )  =  0 )
190 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
191190fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( log `  ( x  + 
1 ) )  =  ( log `  ( A  +  1 ) ) )
192 fveq2 5525 . . 3  |-  ( x  =  A  ->  ( sqr `  x )  =  ( sqr `  A
) )
1932, 3, 54, 120, 134, 146, 171, 176, 179, 180, 186, 189, 191, 192dvle 19354 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  -  0 )  <_  ( ( sqr `  A )  -  0 ) )
194 ge0p1rp 10382 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  +  1 )  e.  RR+ )
195194relogcld 19974 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  e.  RR )
196 resqrcl 11739 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
197195, 196, 2lesub1d 9379 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  <_  ( sqr `  A )  <->  ( ( log `  ( A  + 
1 ) )  - 
0 )  <_  (
( sqr `  A
)  -  0 ) ) )
198193, 197mpbird 223 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  <_  ( sqr `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   {cpr 3641   class class class wbr 4023    e. cmpt 4077   ran crn 4690    |` cres 4691   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   RR+crp 10354   (,)cioo 10656   [,)cico 10658   [,]cicc 10659   ^cexp 11104   sqrcsqr 11718   ↾t crest 13325   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   -cn->ccncf 18380    _D cdv 19213   logclog 19912
This theorem is referenced by:  rplogsumlem1  20633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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