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Theorem loglesqr 20595
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 9047 . . . 4  |-  0  e.  RR
21a1i 11 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  RR )
3 simpl 444 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
4 elicc2 10931 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
51, 3, 4sylancr 645 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
65biimpa 471 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  A
) )
76simp1d 969 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  RR )
86simp2d 970 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  0  <_  x )
97, 8ge0p1rpd 10630 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  +  1 )  e.  RR+ )
10 fvres 5704 . . . . . 6  |-  ( ( x  +  1 )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( x  +  1
) )  =  ( log `  ( x  +  1 ) ) )
119, 10syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( ( log  |`  RR+ ) `  (
x  +  1 ) )  =  ( log `  ( x  +  1 ) ) )
1211mpteq2dva 4255 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) ) )
13 eqid 2404 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1413cnfldtopon 18770 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
157ex 424 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  RR ) )
1615ssrdv 3314 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  RR )
17 ax-resscn 9003 . . . . . . . 8  |-  RR  C_  CC
1816, 17syl6ss 3320 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  CC )
19 resttopon 17179 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  (
0 [,] A ) 
C_  CC )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
2014, 18, 19sylancr 645 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
21 eqid 2404 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )
229, 21fmptd 5852 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) ) : ( 0 [,] A ) -->
RR+ )
23 rpssre 10578 . . . . . . . . . 10  |-  RR+  C_  RR
2423, 17sstri 3317 . . . . . . . . 9  |-  RR+  C_  CC
2513addcn 18848 . . . . . . . . . . 11  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
2625a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  +  e.  ( (
( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
27 ssid 3327 . . . . . . . . . . 11  |-  CC  C_  CC
28 cncfmptid 18895 . . . . . . . . . . 11  |-  ( ( ( 0 [,] A
)  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  ( 0 [,] A )  |->  x )  e.  ( ( 0 [,] A )
-cn-> CC ) )
2918, 27, 28sylancl 644 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  x )  e.  ( ( 0 [,] A
) -cn-> CC ) )
30 ax-1cn 9004 . . . . . . . . . . . 12  |-  1  e.  CC
3130a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
1  e.  CC )
3227a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  CC  C_  CC )
33 cncfmptc 18894 . . . . . . . . . . 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 1184 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  1 )  e.  ( ( 0 [,] A
) -cn-> CC ) )
3513, 26, 29, 34cncfmpt2f 18897 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> CC ) )
36 cncffvrn 18881 . . . . . . . . 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 645 . . . . . . . 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 224 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> RR+ ) )
39 eqid 2404 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  ( 0 [,] A
) )  =  ( ( TopOpen ` fld )t  ( 0 [,] A ) )
40 eqid 2404 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
4113, 39, 40cncfcn 18892 . . . . . . . 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 644 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR+ )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
4338, 42eleqtrd 2480 . . . . . 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 20482 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
45 eqid 2404 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
4613, 40, 45cncfcn 18892 . . . . . . . . 9  |-  ( (
RR+  C_  CC  /\  RR  C_  CC )  ->  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) ) )
4724, 17, 46mp2an 654 . . . . . . . 8  |-  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4844, 47eleqtri 2476 . . . . . . 7  |-  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4948a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5020, 43, 49cnmpt11f 17649 . . . . 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 18892 . . . . . 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 644 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5350, 52eleqtrrd 2481 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
5412, 53eqeltrrd 2479 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
55 reex 9037 . . . . . 6  |-  RR  e.  _V
5655prid1 3872 . . . . 5  |-  RR  e.  { RR ,  CC }
5756a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  e.  { RR ,  CC } )
58 simpr 448 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
59 1rp 10572 . . . . . . 7  |-  1  e.  RR+
60 rpaddcl 10588 . . . . . . 7  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
6158, 59, 60sylancl 644 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR+ )
6261relogcld 20471 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  RR )
6362recnd 9070 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  CC )
6461rpreccld 10614 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  RR+ )
6530a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  1  e.  CC )
66 relogcl 20426 . . . . . . . 8  |-  ( y  e.  RR+  ->  ( log `  y )  e.  RR )
6766adantl 453 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  RR )
6867recnd 9070 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  CC )
69 rpreccl 10591 . . . . . . 7  |-  ( y  e.  RR+  ->  ( 1  /  y )  e.  RR+ )
7069adantl 453 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( 1  /  y
)  e.  RR+ )
71 peano2re 9195 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
7271adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  RR )
7372recnd 9070 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  CC )
7430a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  1  e.  CC )
7517a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  C_  CC )
7675sselda 3308 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  x  e.  CC )
7757dvmptid 19796 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  x ) )  =  ( x  e.  RR  |->  1 ) )
78 0cn 9040 . . . . . . . . . 10  |-  0  e.  CC
7978a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  0  e.  CC )
8057, 31dvmptc 19797 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  1 ) )  =  ( x  e.  RR  |->  0 ) )
8157, 76, 74, 77, 74, 79, 80dvmptadd 19799 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  ( 1  +  0 ) ) )
8230addid1i 9209 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
8382mpteq2i 4252 . . . . . . . 8  |-  ( x  e.  RR  |->  ( 1  +  0 ) )  =  ( x  e.  RR  |->  1 )
8481, 83syl6eq 2452 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  1 ) )
8523a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR+  C_  RR )
8613tgioo2 18787 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
87 ioorp 10944 . . . . . . . . 9  |-  ( 0 (,)  +oo )  =  RR+
88 iooretop 18753 . . . . . . . . 9  |-  ( 0 (,)  +oo )  e.  (
topGen `  ran  (,) )
8987, 88eqeltrri 2475 . . . . . . . 8  |-  RR+  e.  ( topGen `  ran  (,) )
9089a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR+ 
e.  ( topGen `  ran  (,) ) )
9157, 73, 74, 84, 85, 86, 13, 90dvmptres 19802 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( x  +  1 ) ) )  =  ( x  e.  RR+  |->  1 ) )
92 dvrelog 20481 . . . . . . 7  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( y  e.  RR+  |->  ( 1  /  y ) )
93 relogf1o 20417 . . . . . . . . . . 11  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
94 f1of 5633 . . . . . . . . . . 11  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
9593, 94mp1i 12 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ ) : RR+ --> RR )
9695feqmptd 5738 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y
) ) )
97 fvres 5704 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  ( ( log  |`  RR+ ) `  y )  =  ( log `  y ) )
9897mpteq2ia 4251 . . . . . . . . 9  |-  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y ) )  =  ( y  e.  RR+  |->  ( log `  y ) )
9996, 98syl6eq 2452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( log `  y
) ) )
10099oveq2d 6056 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  ( log  |`  RR+ ) )  =  ( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) ) )
10192, 100syl5reqr 2451 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) )  =  ( y  e.  RR+  |->  ( 1  / 
y ) ) )
102 fveq2 5687 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  ( log `  y )  =  ( log `  (
x  +  1 ) ) )
103 oveq2 6048 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  (
1  /  y )  =  ( 1  / 
( x  +  1 ) ) )
10457, 57, 61, 65, 68, 70, 91, 101, 102, 103dvmptco 19811 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( ( 1  /  ( x  + 
1 ) )  x.  1 ) ) )
10564rpcnd 10606 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  CC )
106105mulid1d 9061 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 1  / 
( x  +  1 ) )  x.  1 )  =  ( 1  /  ( x  + 
1 ) ) )
107106mpteq2dva 4255 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  RR+  |->  ( ( 1  / 
( x  +  1 ) )  x.  1 ) )  =  ( x  e.  RR+  |->  ( 1  /  ( x  + 
1 ) ) ) )
108104, 107eqtrd 2436 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( x  +  1 ) ) ) )
109 ioossicc 10952 . . . . . . . . 9  |-  ( 0 (,) A )  C_  ( 0 [,] A
)
110109sseli 3304 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  x  e.  ( 0 [,] A
) )
111110, 7sylan2 461 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR )
112 eliooord 10926 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) A )  ->  (
0  <  x  /\  x  <  A ) )
113112simpld 446 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  0  <  x )
114113adantl 453 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  0  <  x )
115111, 114elrpd 10602 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR+ )
116115ex 424 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 (,) A )  ->  x  e.  RR+ ) )
117116ssrdv 3314 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  C_  RR+ )
118 iooretop 18753 . . . . 5  |-  ( 0 (,) A )  e.  ( topGen `  ran  (,) )
119118a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  e.  ( topGen ` 
ran  (,) ) )
12057, 63, 64, 108, 117, 86, 13, 119dvmptres 19802 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( x  +  1 ) ) ) )
121 elrege0 10963 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) 
+oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
1227, 8, 121sylanbrc 646 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  ( 0 [,)  +oo ) )
123122ex 424 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  ( 0 [,)  +oo )
) )
124123ssrdv 3314 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  ( 0 [,)  +oo ) )
125 resabs1 5134 . . . . . 6  |-  ( ( 0 [,] A ) 
C_  ( 0 [,) 
+oo )  ->  (
( sqr  |`  ( 0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( sqr  |`  ( 0 [,] A ) ) )
126124, 125syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( sqr  |`  ( 0 [,] A ) ) )
127 sqrf 12122 . . . . . . 7  |-  sqr : CC
--> CC
128127a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sqr : CC --> CC )
129128, 18feqresmpt 5739 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( sqr `  x
) ) )
130126, 129eqtrd 2436 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( x  e.  ( 0 [,] A )  |->  ( sqr `  x ) ) )
131 resqrcn 20586 . . . . 5  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
132 rescncf 18880 . . . . 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 1382 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
134130, 133eqeltrrd 2479 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
135 rpcn 10576 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
136135adantl 453 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  CC )
137136sqrcld 12194 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  CC )
138 2rp 10573 . . . . . 6  |-  2  e.  RR+
139 rpsqrcl 12025 . . . . . . 7  |-  ( x  e.  RR+  ->  ( sqr `  x )  e.  RR+ )
140139adantl 453 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR+ )
141 rpmulcl 10589 . . . . . 6  |-  ( ( 2  e.  RR+  /\  ( sqr `  x )  e.  RR+ )  ->  ( 2  x.  ( sqr `  x
) )  e.  RR+ )
142138, 140, 141sylancr 645 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR+ )
143142rpreccld 10614 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
2  x.  ( sqr `  x ) ) )  e.  RR+ )
144 dvsqr 20581 . . . . 5  |-  ( RR 
_D  ( x  e.  RR+  |->  ( sqr `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  x ) ) ) )
145144a1i 11 . . . 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 19802 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( sqr `  x ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
147140rpred 10604 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR )
148 1re 9046 . . . . . . . . 9  |-  1  e.  RR
149 resubcl 9321 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  RR  /\  1  e.  RR )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
150147, 148, 149sylancl 644 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
151150sqge0d 11505 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
( sqr `  x
)  -  1 ) ^ 2 ) )
152136sqsqrd 12196 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
) ^ 2 )  =  x )
153137mulid1d 9061 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  x.  1 )  =  ( sqr `  x
) )
154153oveq2d 6056 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  (
( sqr `  x
)  x.  1 ) )  =  ( 2  x.  ( sqr `  x
) ) )
155152, 154oveq12d 6058 . . . . . . . . 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 11431 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
157156a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1 ^ 2 )  =  1 )
158155, 157oveq12d 6058 . . . . . . . 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 11453 . . . . . . . . 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 644 . . . . . . . 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 10606 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  CC )
162136, 65, 161addsubd 9388 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( x  + 
1 )  -  (
2  x.  ( sqr `  x ) ) )  =  ( ( x  -  ( 2  x.  ( sqr `  x
) ) )  +  1 ) )
163158, 160, 1623eqtr4d 2446 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( x  +  1 )  -  ( 2  x.  ( sqr `  x ) ) ) )
164151, 163breqtrd 4196 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) ) )
16561rpred 10604 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR )
166142rpred 10604 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR )
167165, 166subge0d 9572 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 0  <_  (
( x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) )  <->  ( 2  x.  ( sqr `  x
) )  <_  (
x  +  1 ) ) )
168164, 167mpbid 202 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  <_  ( x  + 
1 ) )
169142, 61lerecd 10623 . . . . 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 202 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  <_  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) )
171115, 170syldan 457 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  ( 1  /  ( x  + 
1 ) )  <_ 
( 1  /  (
2  x.  ( sqr `  x ) ) ) )
172 rexr 9086 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
173 0xr 9087 . . . . 5  |-  0  e.  RR*
174 lbicc2 10969 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
175173, 174mp3an1 1266 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
176172, 175sylan 458 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  ( 0 [,] A ) )
177 ubicc2 10970 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
178173, 177mp3an1 1266 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
179172, 178sylan 458 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  ( 0 [,] A ) )
180 simpr 448 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  A )
181 oveq1 6047 . . . . . 6  |-  ( x  =  0  ->  (
x  +  1 )  =  ( 0  +  1 ) )
182 0p1e1 10049 . . . . . 6  |-  ( 0  +  1 )  =  1
183181, 182syl6eq 2452 . . . . 5  |-  ( x  =  0  ->  (
x  +  1 )  =  1 )
184183fveq2d 5691 . . . 4  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  ( log `  1
) )
185 log1 20433 . . . 4  |-  ( log `  1 )  =  0
186184, 185syl6eq 2452 . . 3  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  0 )
187 fveq2 5687 . . . 4  |-  ( x  =  0  ->  ( sqr `  x )  =  ( sqr `  0
) )
188 sqr0 12002 . . . 4  |-  ( sqr `  0 )  =  0
189187, 188syl6eq 2452 . . 3  |-  ( x  =  0  ->  ( sqr `  x )  =  0 )
190 oveq1 6047 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
191190fveq2d 5691 . . 3  |-  ( x  =  A  ->  ( log `  ( x  + 
1 ) )  =  ( log `  ( A  +  1 ) ) )
192 fveq2 5687 . . 3  |-  ( x  =  A  ->  ( sqr `  x )  =  ( sqr `  A
) )
1932, 3, 54, 120, 134, 146, 171, 176, 179, 180, 186, 189, 191, 192dvle 19844 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  -  0 )  <_  ( ( sqr `  A )  -  0 ) )
194 ge0p1rp 10596 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  +  1 )  e.  RR+ )
195194relogcld 20471 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  e.  RR )
196 resqrcl 12014 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
197195, 196, 2lesub1d 9589 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  <_  ( sqr `  A )  <->  ( ( log `  ( A  + 
1 ) )  - 
0 )  <_  (
( sqr `  A
)  -  0 ) ) )
198193, 197mpbird 224 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  <_  ( sqr `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226   ran crn 4838    |` cres 4839   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   RR+crp 10568   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   ^cexp 11337   sqrcsqr 11993   ↾t crest 13603   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   -cn->ccncf 18859    _D cdv 19703   logclog 20405
This theorem is referenced by:  rplogsumlem1  21131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-tan 12629  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408
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