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Theorem loglesqrt 23777
Description: An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)
Assertion
Ref Expression
loglesqrt  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  <_  ( sqr `  A ) )

Proof of Theorem loglesqrt
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 9661 . . . 4  |-  0  e.  RR
21a1i 11 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  RR )
3 simpl 464 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
4 elicc2 11724 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
51, 3, 4sylancr 676 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
65biimpa 492 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  A
) )
76simp1d 1042 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  RR )
86simp2d 1043 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  0  <_  x )
97, 8ge0p1rpd 11391 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  +  1 )  e.  RR+ )
10 fvres 5893 . . . . . 6  |-  ( ( x  +  1 )  e.  RR+  ->  ( ( log  |`  RR+ ) `  ( x  +  1
) )  =  ( log `  ( x  +  1 ) ) )
119, 10syl 17 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( ( log  |`  RR+ ) `  (
x  +  1 ) )  =  ( log `  ( x  +  1 ) ) )
1211mpteq2dva 4482 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) ) )
13 eqid 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1413cnfldtopon 21881 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
157ex 441 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  RR ) )
1615ssrdv 3424 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  RR )
17 ax-resscn 9614 . . . . . . . 8  |-  RR  C_  CC
1816, 17syl6ss 3430 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  CC )
19 resttopon 20254 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  (
0 [,] A ) 
C_  CC )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
2014, 18, 19sylancr 676 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
21 eqid 2471 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )
229, 21fmptd 6061 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) ) : ( 0 [,] A ) -->
RR+ )
23 rpssre 11335 . . . . . . . . . 10  |-  RR+  C_  RR
2423, 17sstri 3427 . . . . . . . . 9  |-  RR+  C_  CC
2513addcn 21975 . . . . . . . . . . 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 3437 . . . . . . . . . . 11  |-  CC  C_  CC
28 cncfmptid 22022 . . . . . . . . . . 11  |-  ( ( ( 0 [,] A
)  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  ( 0 [,] A )  |->  x )  e.  ( ( 0 [,] A )
-cn-> CC ) )
2918, 27, 28sylancl 675 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  x )  e.  ( ( 0 [,] A
) -cn-> CC ) )
30 1cnd 9677 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
1  e.  CC )
3127a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  CC  C_  CC )
32 cncfmptc 22021 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  ( 0 [,] A
)  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  ( 0 [,] A )  |->  1 )  e.  ( ( 0 [,] A )
-cn-> CC ) )
3330, 18, 31, 32syl3anc 1292 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  1 )  e.  ( ( 0 [,] A
) -cn-> CC ) )
3413, 26, 29, 33cncfmpt2f 22024 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> CC ) )
35 cncffvrn 22008 . . . . . . . . 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+ ) )
3624, 34, 35sylancr 676 . . . . . . . 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+ ) )
3722, 36mpbird 240 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> RR+ ) )
38 eqid 2471 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  ( 0 [,] A
) )  =  ( ( TopOpen ` fld )t  ( 0 [,] A ) )
39 eqid 2471 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
4013, 38, 39cncfcn 22019 . . . . . . . 8  |-  ( ( ( 0 [,] A
)  C_  CC  /\  RR+  C_  CC )  ->  ( ( 0 [,] A ) -cn-> RR+ )  =  ( (
( TopOpen ` fld )t  ( 0 [,] A ) )  Cn  ( ( TopOpen ` fld )t  RR+ ) ) )
4118, 24, 40sylancl 675 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR+ )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
4237, 41eleqtrd 2551 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( ( TopOpen ` fld )t  ( 0 [,] A ) )  Cn  ( ( TopOpen ` fld )t  RR+ ) ) )
43 relogcn 23662 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
44 eqid 2471 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
4513, 39, 44cncfcn 22019 . . . . . . . . 9  |-  ( (
RR+  C_  CC  /\  RR  C_  CC )  ->  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) ) )
4624, 17, 45mp2an 686 . . . . . . . 8  |-  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4743, 46eleqtri 2547 . . . . . . 7  |-  ( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4847a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  e.  ( ( ( TopOpen ` fld )t  RR+ )  Cn  ( ( TopOpen ` fld )t  RR ) ) )
4920, 42, 48cnmpt11f 20756 . . . . 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 ) ) )
5013, 38, 44cncfcn 22019 . . . . . 6  |-  ( ( ( 0 [,] A
)  C_  CC  /\  RR  C_  CC )  ->  (
( 0 [,] A
) -cn-> RR )  =  ( ( ( TopOpen ` fld )t  ( 0 [,] A ) )  Cn  ( ( TopOpen ` fld )t  RR ) ) )
5118, 17, 50sylancl 675 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5249, 51eleqtrrd 2552 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
5312, 52eqeltrrd 2550 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
54 reelprrecn 9649 . . . . 5  |-  RR  e.  { RR ,  CC }
5554a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  e.  { RR ,  CC } )
56 simpr 468 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
57 1rp 11329 . . . . . . 7  |-  1  e.  RR+
58 rpaddcl 11346 . . . . . . 7  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
5956, 57, 58sylancl 675 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR+ )
6059relogcld 23651 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  RR )
6160recnd 9687 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  CC )
6259rpreccld 11374 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  RR+ )
63 1cnd 9677 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  1  e.  CC )
64 relogcl 23604 . . . . . . . 8  |-  ( y  e.  RR+  ->  ( log `  y )  e.  RR )
6564adantl 473 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  RR )
6665recnd 9687 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  CC )
67 rpreccl 11349 . . . . . . 7  |-  ( y  e.  RR+  ->  ( 1  /  y )  e.  RR+ )
6867adantl 473 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( 1  /  y
)  e.  RR+ )
69 peano2re 9824 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
7069adantl 473 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  RR )
7170recnd 9687 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  CC )
72 1cnd 9677 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  1  e.  CC )
7317a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  C_  CC )
7473sselda 3418 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  x  e.  CC )
7555dvmptid 22990 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  x ) )  =  ( x  e.  RR  |->  1 ) )
76 0cnd 9654 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  0  e.  CC )
7755, 30dvmptc 22991 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  1 ) )  =  ( x  e.  RR  |->  0 ) )
7855, 74, 72, 75, 72, 76, 77dvmptadd 22993 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  ( 1  +  0 ) ) )
79 1p0e1 10744 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
8079mpteq2i 4479 . . . . . . . 8  |-  ( x  e.  RR  |->  ( 1  +  0 ) )  =  ( x  e.  RR  |->  1 )
8178, 80syl6eq 2521 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  1 ) )
8223a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR+  C_  RR )
8313tgioo2 21899 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
84 ioorp 11737 . . . . . . . . 9  |-  ( 0 (,) +oo )  = 
RR+
85 iooretop 21864 . . . . . . . . 9  |-  ( 0 (,) +oo )  e.  ( topGen `  ran  (,) )
8684, 85eqeltrri 2546 . . . . . . . 8  |-  RR+  e.  ( topGen `  ran  (,) )
8786a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR+ 
e.  ( topGen `  ran  (,) ) )
8855, 71, 72, 81, 82, 83, 13, 87dvmptres 22996 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( x  +  1 ) ) )  =  ( x  e.  RR+  |->  1 ) )
89 dvrelog 23661 . . . . . . 7  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( y  e.  RR+  |->  ( 1  /  y ) )
90 relogf1o 23595 . . . . . . . . . . 11  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
91 f1of 5828 . . . . . . . . . . 11  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
9290, 91mp1i 13 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ ) : RR+ --> RR )
9392feqmptd 5932 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y
) ) )
94 fvres 5893 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  ( ( log  |`  RR+ ) `  y )  =  ( log `  y ) )
9594mpteq2ia 4478 . . . . . . . . 9  |-  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y ) )  =  ( y  e.  RR+  |->  ( log `  y ) )
9693, 95syl6eq 2521 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( log `  y
) ) )
9796oveq2d 6324 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  ( log  |`  RR+ ) )  =  ( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) ) )
9889, 97syl5reqr 2520 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) )  =  ( y  e.  RR+  |->  ( 1  / 
y ) ) )
99 fveq2 5879 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  ( log `  y )  =  ( log `  (
x  +  1 ) ) )
100 oveq2 6316 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  (
1  /  y )  =  ( 1  / 
( x  +  1 ) ) )
10155, 55, 59, 63, 66, 68, 88, 98, 99, 100dvmptco 23005 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( ( 1  /  ( x  + 
1 ) )  x.  1 ) ) )
10262rpcnd 11366 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  CC )
103102mulid1d 9678 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 1  / 
( x  +  1 ) )  x.  1 )  =  ( 1  /  ( x  + 
1 ) ) )
104103mpteq2dva 4482 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  RR+  |->  ( ( 1  / 
( x  +  1 ) )  x.  1 ) )  =  ( x  e.  RR+  |->  ( 1  /  ( x  + 
1 ) ) ) )
105101, 104eqtrd 2505 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( x  +  1 ) ) ) )
106 ioossicc 11745 . . . . . . . . 9  |-  ( 0 (,) A )  C_  ( 0 [,] A
)
107106sseli 3414 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  x  e.  ( 0 [,] A
) )
108107, 7sylan2 482 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR )
109 eliooord 11719 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) A )  ->  (
0  <  x  /\  x  <  A ) )
110109simpld 466 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  0  <  x )
111110adantl 473 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  0  <  x )
112108, 111elrpd 11361 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR+ )
113112ex 441 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 (,) A )  ->  x  e.  RR+ ) )
114113ssrdv 3424 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  C_  RR+ )
115 iooretop 21864 . . . . 5  |-  ( 0 (,) A )  e.  ( topGen `  ran  (,) )
116115a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  e.  ( topGen ` 
ran  (,) ) )
11755, 61, 62, 105, 114, 83, 13, 116dvmptres 22996 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( x  +  1 ) ) ) )
118 elrege0 11764 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
1197, 8, 118sylanbrc 677 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  ( 0 [,) +oo ) )
120119ex 441 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  ( 0 [,) +oo )
) )
121120ssrdv 3424 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  ( 0 [,) +oo ) )
122121resabs1d 5140 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,) +oo )
)  |`  ( 0 [,] A ) )  =  ( sqr  |`  (
0 [,] A ) ) )
123 sqrtf 13503 . . . . . . 7  |-  sqr : CC
--> CC
124123a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sqr : CC --> CC )
125124, 18feqresmpt 5933 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( sqr `  x
) ) )
126122, 125eqtrd 2505 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,) +oo )
)  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) ) )
127 resqrtcn 23768 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )
128 rescncf 22007 . . . . 5  |-  ( ( 0 [,] A ) 
C_  ( 0 [,) +oo )  ->  ( ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )  ->  ( ( sqr  |`  ( 0 [,) +oo ) )  |`  (
0 [,] A ) )  e.  ( ( 0 [,] A )
-cn-> RR ) ) )
129121, 127, 128mpisyl 21 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,) +oo )
)  |`  ( 0 [,] A ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
130126, 129eqeltrrd 2550 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
131 rpcn 11333 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
132131adantl 473 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  CC )
133132sqrtcld 13576 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  CC )
134 2rp 11330 . . . . . 6  |-  2  e.  RR+
135 rpsqrtcl 13405 . . . . . . 7  |-  ( x  e.  RR+  ->  ( sqr `  x )  e.  RR+ )
136135adantl 473 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR+ )
137 rpmulcl 11347 . . . . . 6  |-  ( ( 2  e.  RR+  /\  ( sqr `  x )  e.  RR+ )  ->  ( 2  x.  ( sqr `  x
) )  e.  RR+ )
138134, 136, 137sylancr 676 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR+ )
139138rpreccld 11374 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
2  x.  ( sqr `  x ) ) )  e.  RR+ )
140 dvsqrt 23761 . . . . 5  |-  ( RR 
_D  ( x  e.  RR+  |->  ( sqr `  x
) ) )  =  ( x  e.  RR+  |->  ( 1  /  (
2  x.  ( sqr `  x ) ) ) )
141140a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( sqr `  x ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( 2  x.  ( sqr `  x ) ) ) ) )
14255, 133, 139, 141, 114, 83, 13, 116dvmptres 22996 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( sqr `  x ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
143136rpred 11364 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR )
144 1re 9660 . . . . . . . . 9  |-  1  e.  RR
145 resubcl 9958 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  RR  /\  1  e.  RR )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
146143, 144, 145sylancl 675 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
147146sqge0d 12481 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
( sqr `  x
)  -  1 ) ^ 2 ) )
148132sqsqrtd 13578 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
) ^ 2 )  =  x )
149133mulid1d 9678 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  x.  1 )  =  ( sqr `  x
) )
150149oveq2d 6324 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  (
( sqr `  x
)  x.  1 ) )  =  ( 2  x.  ( sqr `  x
) ) )
151148, 150oveq12d 6326 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x ) ^ 2 )  -  ( 2  x.  ( ( sqr `  x )  x.  1 ) ) )  =  ( x  -  (
2  x.  ( sqr `  x ) ) ) )
152 sq1 12407 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
153152a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1 ^ 2 )  =  1 )
154151, 153oveq12d 6326 . . . . . . . 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 ) )
155 ax-1cn 9615 . . . . . . . . 9  |-  1  e.  CC
156 binom2sub 12429 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( ( ( sqr `  x
) ^ 2 )  -  ( 2  x.  ( ( sqr `  x
)  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
157133, 155, 156sylancl 675 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( ( ( sqr `  x
) ^ 2 )  -  ( 2  x.  ( ( sqr `  x
)  x.  1 ) ) )  +  ( 1 ^ 2 ) ) )
158138rpcnd 11366 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  CC )
159132, 63, 158addsubd 10026 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( x  + 
1 )  -  (
2  x.  ( sqr `  x ) ) )  =  ( ( x  -  ( 2  x.  ( sqr `  x
) ) )  +  1 ) )
160154, 157, 1593eqtr4d 2515 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( x  +  1 )  -  ( 2  x.  ( sqr `  x ) ) ) )
161147, 160breqtrd 4420 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) ) )
16259rpred 11364 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR )
163138rpred 11364 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR )
164162, 163subge0d 10224 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 0  <_  (
( x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) )  <->  ( 2  x.  ( sqr `  x
) )  <_  (
x  +  1 ) ) )
165161, 164mpbid 215 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  <_  ( x  + 
1 ) )
166138, 59lerecd 11383 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 2  x.  ( sqr `  x
) )  <_  (
x  +  1 )  <-> 
( 1  /  (
x  +  1 ) )  <_  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
167165, 166mpbid 215 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  <_  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) )
168112, 167syldan 478 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  ( 1  /  ( x  + 
1 ) )  <_ 
( 1  /  (
2  x.  ( sqr `  x ) ) ) )
169 rexr 9704 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
170 0xr 9705 . . . . 5  |-  0  e.  RR*
171 lbicc2 11774 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
172170, 171mp3an1 1377 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
173169, 172sylan 479 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  ( 0 [,] A ) )
174 ubicc2 11775 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
175170, 174mp3an1 1377 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
176169, 175sylan 479 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  ( 0 [,] A ) )
177 simpr 468 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  A )
178 oveq1 6315 . . . . . 6  |-  ( x  =  0  ->  (
x  +  1 )  =  ( 0  +  1 ) )
179 0p1e1 10743 . . . . . 6  |-  ( 0  +  1 )  =  1
180178, 179syl6eq 2521 . . . . 5  |-  ( x  =  0  ->  (
x  +  1 )  =  1 )
181180fveq2d 5883 . . . 4  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  ( log `  1
) )
182 log1 23614 . . . 4  |-  ( log `  1 )  =  0
183181, 182syl6eq 2521 . . 3  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  0 )
184 fveq2 5879 . . . 4  |-  ( x  =  0  ->  ( sqr `  x )  =  ( sqr `  0
) )
185 sqrt0 13382 . . . 4  |-  ( sqr `  0 )  =  0
186184, 185syl6eq 2521 . . 3  |-  ( x  =  0  ->  ( sqr `  x )  =  0 )
187 oveq1 6315 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
188187fveq2d 5883 . . 3  |-  ( x  =  A  ->  ( log `  ( x  + 
1 ) )  =  ( log `  ( A  +  1 ) ) )
189 fveq2 5879 . . 3  |-  ( x  =  A  ->  ( sqr `  x )  =  ( sqr `  A
) )
1902, 3, 53, 117, 130, 142, 168, 173, 176, 177, 183, 186, 188, 189dvle 23038 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  -  0 )  <_  ( ( sqr `  A )  -  0 ) )
191 ge0p1rp 11354 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  +  1 )  e.  RR+ )
192191relogcld 23651 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  e.  RR )
193 resqrtcl 13394 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
194192, 193, 2lesub1d 10241 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  <_  ( sqr `  A )  <->  ( ( log `  ( A  + 
1 ) )  - 
0 )  <_  (
( sqr `  A
)  -  0 ) ) )
195190, 194mpbird 240 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  <_  ( sqr `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    C_ wss 3390   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   ran crn 4840    |` cres 4841   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   2c2 10681   RR+crp 11325   (,)cioo 11660   [,)cico 11662   [,]cicc 11663   ^cexp 12310   sqrcsqrt 13373   ↾t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047  TopOnctopon 19995    Cn ccn 20317    tX ctx 20652   -cn->ccncf 21986    _D cdv 22897   logclog 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586
This theorem is referenced by:  rplogsumlem1  24401
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