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Theorem loglesqr 20114
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 8854 . . . 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 10731 . . . . . . . . . 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 10432 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  +  1 )  e.  RR+ )
10 fvres 5558 . . . . . 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 4122 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) ) )
13 eqid 2296 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1413cnfldtopon 18308 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
157ex 423 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  RR ) )
1615ssrdv 3198 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  RR )
17 ax-resscn 8810 . . . . . . . 8  |-  RR  C_  CC
1816, 17syl6ss 3204 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  CC )
19 resttopon 16908 . . . . . . 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 2296 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )
229, 21fmptd 5700 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) ) : ( 0 [,] A ) -->
RR+ )
23 rpssre 10380 . . . . . . . . . 10  |-  RR+  C_  RR
2423, 17sstri 3201 . . . . . . . . 9  |-  RR+  C_  CC
2513addcn 18385 . . . . . . . . . . 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 3210 . . . . . . . . . . 11  |-  CC  C_  CC
28 cncfmptid 18432 . . . . . . . . . . 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 8811 . . . . . . . . . . . 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 18431 . . . . . . . . . . 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 18434 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> CC ) )
36 cncffvrn 18418 . . . . . . . . 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 2296 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  ( 0 [,] A
) )  =  ( ( TopOpen ` fld )t  ( 0 [,] A ) )
40 eqid 2296 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
4113, 39, 40cncfcn 18429 . . . . . . . 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 2372 . . . . . 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 20001 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
45 eqid 2296 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
4613, 40, 45cncfcn 18429 . . . . . . . . 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 2368 . . . . . . 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 17374 . . . . 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 18429 . . . . . 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 2373 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
5412, 53eqeltrrd 2371 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
55 reex 8844 . . . . . 6  |-  RR  e.  _V
5655prid1 3747 . . . . 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 10374 . . . . . . 7  |-  1  e.  RR+
60 rpaddcl 10390 . . . . . . 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 19990 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  RR )
6362recnd 8877 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  CC )
6461rpreccld 10416 . . . 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 19948 . . . . . . . 8  |-  ( y  e.  RR+  ->  ( log `  y )  e.  RR )
6766adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  RR )
6867recnd 8877 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  CC )
69 rpreccl 10393 . . . . . . 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 9001 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
7271adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  RR )
7372recnd 8877 . . . . . . 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 3193 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  x  e.  CC )
7757dvmptid 19322 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  x ) )  =  ( x  e.  RR  |->  1 ) )
78 0cn 8847 . . . . . . . . . 10  |-  0  e.  CC
7978a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  0  e.  CC )
8057, 31dvmptc 19323 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  1 ) )  =  ( x  e.  RR  |->  0 ) )
8157, 76, 74, 77, 74, 79, 80dvmptadd 19325 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  ( 1  +  0 ) ) )
8230addid1i 9015 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
8382mpteq2i 4119 . . . . . . . 8  |-  ( x  e.  RR  |->  ( 1  +  0 ) )  =  ( x  e.  RR  |->  1 )
8481, 83syl6eq 2344 . . . . . . 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 18325 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
87 ioorp 10743 . . . . . . . . 9  |-  ( 0 (,)  +oo )  =  RR+
88 iooretop 18291 . . . . . . . . 9  |-  ( 0 (,)  +oo )  e.  (
topGen `  ran  (,) )
8987, 88eqeltrri 2367 . . . . . . . 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 19328 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( x  +  1 ) ) )  =  ( x  e.  RR+  |->  1 ) )
92 dvrelog 20000 . . . . . . 7  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( y  e.  RR+  |->  ( 1  /  y ) )
93 relogf1o 19940 . . . . . . . . . . 11  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
94 f1of 5488 . . . . . . . . . . 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 5591 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y
) ) )
97 fvres 5558 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  ( ( log  |`  RR+ ) `  y )  =  ( log `  y ) )
9897mpteq2ia 4118 . . . . . . . . 9  |-  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y ) )  =  ( y  e.  RR+  |->  ( log `  y ) )
9996, 98syl6eq 2344 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( log `  y
) ) )
10099oveq2d 5890 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  ( log  |`  RR+ ) )  =  ( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) ) )
10192, 100syl5reqr 2343 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) )  =  ( y  e.  RR+  |->  ( 1  / 
y ) ) )
102 fveq2 5541 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  ( log `  y )  =  ( log `  (
x  +  1 ) ) )
103 oveq2 5882 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  (
1  /  y )  =  ( 1  / 
( x  +  1 ) ) )
10457, 57, 61, 65, 68, 70, 91, 101, 102, 103dvmptco 19337 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( ( 1  /  ( x  + 
1 ) )  x.  1 ) ) )
10564rpcnd 10408 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  CC )
106105mulid1d 8868 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 1  / 
( x  +  1 ) )  x.  1 )  =  ( 1  /  ( x  + 
1 ) ) )
107106mpteq2dva 4122 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  RR+  |->  ( ( 1  / 
( x  +  1 ) )  x.  1 ) )  =  ( x  e.  RR+  |->  ( 1  /  ( x  + 
1 ) ) ) )
108104, 107eqtrd 2328 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( x  +  1 ) ) ) )
109 ioossicc 10751 . . . . . . . . 9  |-  ( 0 (,) A )  C_  ( 0 [,] A
)
110109sseli 3189 . . . . . . . 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 10726 . . . . . . . . 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 10404 . . . . . 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 3198 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  C_  RR+ )
118 iooretop 18291 . . . . 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 19328 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( x  +  1 ) ) ) )
121 elrege0 10762 . . . . . . . . 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 3198 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  ( 0 [,)  +oo ) )
125 resabs1 5000 . . . . . 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 11863 . . . . . . 7  |-  sqr : CC
--> CC
128127a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sqr : CC --> CC )
129128, 18feqresmpt 5592 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( sqr `  x
) ) )
130126, 129eqtrd 2328 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,)  +oo ) )  |`  ( 0 [,] A
) )  =  ( x  e.  ( 0 [,] A )  |->  ( sqr `  x ) ) )
131 resqrcn 20105 . . . . 5  |-  ( sqr  |`  ( 0 [,)  +oo ) )  e.  ( ( 0 [,)  +oo ) -cn-> RR )
132 rescncf 18417 . . . . 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 2371 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
135 rpcn 10378 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
136135adantl 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  CC )
137136sqrcld 11935 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  CC )
138 2rp 10375 . . . . . 6  |-  2  e.  RR+
139 rpsqrcl 11766 . . . . . . 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 10391 . . . . . 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 10416 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
2  x.  ( sqr `  x ) ) )  e.  RR+ )
144 dvsqr 20100 . . . . 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 19328 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( sqr `  x ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
147140rpred 10406 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR )
148 1re 8853 . . . . . . . . 9  |-  1  e.  RR
149 resubcl 9127 . . . . . . . . 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 11288 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
( sqr `  x
)  -  1 ) ^ 2 ) )
152136sqsqrd 11937 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
) ^ 2 )  =  x )
153137mulid1d 8868 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  x.  1 )  =  ( sqr `  x
) )
154153oveq2d 5890 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  (
( sqr `  x
)  x.  1 ) )  =  ( 2  x.  ( sqr `  x
) ) )
155152, 154oveq12d 5892 . . . . . . . . 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 11214 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
157156a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1 ^ 2 )  =  1 )
158155, 157oveq12d 5892 . . . . . . . 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 11236 . . . . . . . . 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 10408 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  CC )
162136, 65, 161addsubd 9194 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( x  + 
1 )  -  (
2  x.  ( sqr `  x ) ) )  =  ( ( x  -  ( 2  x.  ( sqr `  x
) ) )  +  1 ) )
163158, 160, 1623eqtr4d 2338 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( x  +  1 )  -  ( 2  x.  ( sqr `  x ) ) ) )
164151, 163breqtrd 4063 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) ) )
16561rpred 10406 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR )
166142rpred 10406 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR )
167165, 166subge0d 9378 . . . . . 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 10425 . . . . 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 8893 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
173 0xr 8894 . . . . 5  |-  0  e.  RR*
174 lbicc2 10768 . . . . 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 10769 . . . . 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 5881 . . . . . 6  |-  ( x  =  0  ->  (
x  +  1 )  =  ( 0  +  1 ) )
182 0p1e1 9855 . . . . . 6  |-  ( 0  +  1 )  =  1
183181, 182syl6eq 2344 . . . . 5  |-  ( x  =  0  ->  (
x  +  1 )  =  1 )
184183fveq2d 5545 . . . 4  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  ( log `  1
) )
185 log1 19955 . . . 4  |-  ( log `  1 )  =  0
186184, 185syl6eq 2344 . . 3  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  0 )
187 fveq2 5541 . . . 4  |-  ( x  =  0  ->  ( sqr `  x )  =  ( sqr `  0
) )
188 sqr0 11743 . . . 4  |-  ( sqr `  0 )  =  0
189187, 188syl6eq 2344 . . 3  |-  ( x  =  0  ->  ( sqr `  x )  =  0 )
190 oveq1 5881 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
191190fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( log `  ( x  + 
1 ) )  =  ( log `  ( A  +  1 ) ) )
192 fveq2 5541 . . 3  |-  ( x  =  A  ->  ( sqr `  x )  =  ( sqr `  A
) )
1932, 3, 54, 120, 134, 146, 171, 176, 179, 180, 186, 189, 191, 192dvle 19370 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  -  0 )  <_  ( ( sqr `  A )  -  0 ) )
194 ge0p1rp 10398 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  +  1 )  e.  RR+ )
195194relogcld 19990 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  e.  RR )
196 resqrcl 11755 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
197195, 196, 2lesub1d 9395 . 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 1632    e. wcel 1696    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093   ran crn 4706    |` cres 4707   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370   (,)cioo 10672   [,)cico 10674   [,]cicc 10675   ^cexp 11120   sqrcsqr 11734   ↾t crest 13341   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271   -cn->ccncf 18396    _D cdv 19229   logclog 19928
This theorem is referenced by:  rplogsumlem1  20649
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  ax-addf 8832  ax-mulf 8833
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-iin 3924  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-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  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-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-tan 12369  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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