MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  loglesqrt Structured version   Unicode version

Theorem loglesqrt 23257
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 9613 . . . 4  |-  0  e.  RR
21a1i 11 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  RR )
3 simpl 457 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
4 elicc2 11614 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
51, 3, 4sylancr 663 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
65biimpa 484 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  A
) )
76simp1d 1008 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  RR )
86simp2d 1009 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  0  <_  x )
97, 8ge0p1rpd 11307 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  ( x  +  1 )  e.  RR+ )
10 fvres 5886 . . . . . 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 4543 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) ) )
13 eqid 2457 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
1413cnfldtopon 21415 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
157ex 434 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  RR ) )
1615ssrdv 3505 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  RR )
17 ax-resscn 9566 . . . . . . . 8  |-  RR  C_  CC
1816, 17syl6ss 3511 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  CC )
19 resttopon 19788 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  (
0 [,] A ) 
C_  CC )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
2014, 18, 19sylancr 663 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( TopOpen ` fld )t  ( 0 [,] A ) )  e.  (TopOn `  ( 0 [,] A ) ) )
21 eqid 2457 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] A )  |->  ( x  +  1 ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( x  + 
1 ) )
229, 21fmptd 6056 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) ) : ( 0 [,] A ) -->
RR+ )
23 rpssre 11255 . . . . . . . . . 10  |-  RR+  C_  RR
2423, 17sstri 3508 . . . . . . . . 9  |-  RR+  C_  CC
2513addcn 21494 . . . . . . . . . . 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 3518 . . . . . . . . . . 11  |-  CC  C_  CC
28 cncfmptid 21541 . . . . . . . . . . 11  |-  ( ( ( 0 [,] A
)  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  ( 0 [,] A )  |->  x )  e.  ( ( 0 [,] A )
-cn-> CC ) )
2918, 27, 28sylancl 662 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  x )  e.  ( ( 0 [,] A
) -cn-> CC ) )
30 1cnd 9629 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
1  e.  CC )
3127a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  CC  C_  CC )
32 cncfmptc 21540 . . . . . . . . . . 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 1228 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  1 )  e.  ( ( 0 [,] A
) -cn-> CC ) )
3413, 26, 29, 33cncfmpt2f 21543 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> CC ) )
35 cncffvrn 21527 . . . . . . . . 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 663 . . . . . . . 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 232 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( x  +  1 ) )  e.  ( ( 0 [,] A
) -cn-> RR+ ) )
38 eqid 2457 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t  ( 0 [,] A
) )  =  ( ( TopOpen ` fld )t  ( 0 [,] A ) )
39 eqid 2457 . . . . . . . . 9  |-  ( (
TopOpen ` fld )t 
RR+ )  =  ( ( TopOpen ` fld )t  RR+ )
4013, 38, 39cncfcn 21538 . . . . . . . 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 662 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR+ )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR+ ) ) )
4237, 41eleqtrd 2547 . . . . . 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 23144 . . . . . . . 8  |-  ( log  |`  RR+ )  e.  (
RR+ -cn-> RR )
44 eqid 2457 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )t  RR )  =  ( ( TopOpen ` fld )t  RR )
4513, 39, 44cncfcn 21538 . . . . . . . . 9  |-  ( (
RR+  C_  CC  /\  RR  C_  CC )  ->  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) ) )
4624, 17, 45mp2an 672 . . . . . . . 8  |-  ( RR+ -cn-> RR )  =  ( ( ( TopOpen ` fld )t  RR+ )  Cn  (
( TopOpen ` fld )t  RR ) )
4743, 46eleqtri 2543 . . . . . . 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 20290 . . . . 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 21538 . . . . . 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 662 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( 0 [,] A ) -cn-> RR )  =  ( ( (
TopOpen ` fld )t  ( 0 [,] A
) )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5249, 51eleqtrrd 2548 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( ( log  |`  RR+ ) `  ( x  +  1 ) ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
5312, 52eqeltrrd 2546 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( log `  (
x  +  1 ) ) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
54 reelprrecn 9601 . . . . 5  |-  RR  e.  { RR ,  CC }
5554a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  e.  { RR ,  CC } )
56 simpr 461 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
57 1rp 11249 . . . . . . 7  |-  1  e.  RR+
58 rpaddcl 11265 . . . . . . 7  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
5956, 57, 58sylancl 662 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR+ )
6059relogcld 23133 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  RR )
6160recnd 9639 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( log `  (
x  +  1 ) )  e.  CC )
6259rpreccld 11291 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  RR+ )
63 1cnd 9629 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  1  e.  CC )
64 relogcl 23088 . . . . . . . 8  |-  ( y  e.  RR+  ->  ( log `  y )  e.  RR )
6564adantl 466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  RR )
6665recnd 9639 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( log `  y
)  e.  CC )
67 rpreccl 11268 . . . . . . 7  |-  ( y  e.  RR+  ->  ( 1  /  y )  e.  RR+ )
6867adantl 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  y  e.  RR+ )  ->  ( 1  /  y
)  e.  RR+ )
69 peano2re 9770 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
7069adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  RR )
7170recnd 9639 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  ( x  + 
1 )  e.  CC )
72 1cnd 9629 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  1  e.  CC )
7317a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  RR  C_  CC )
7473sselda 3499 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  x  e.  CC )
7555dvmptid 22485 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  x ) )  =  ( x  e.  RR  |->  1 ) )
76 0cnd 9606 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR )  ->  0  e.  CC )
7755, 30dvmptc 22486 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  1 ) )  =  ( x  e.  RR  |->  0 ) )
7855, 74, 72, 75, 72, 76, 77dvmptadd 22488 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR  |->  ( x  +  1 ) ) )  =  ( x  e.  RR  |->  ( 1  +  0 ) ) )
79 1p0e1 10669 . . . . . . . . 9  |-  ( 1  +  0 )  =  1
8079mpteq2i 4540 . . . . . . . 8  |-  ( x  e.  RR  |->  ( 1  +  0 ) )  =  ( x  e.  RR  |->  1 )
8178, 80syl6eq 2514 . . . . . . 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 21433 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
84 ioorp 11627 . . . . . . . . 9  |-  ( 0 (,) +oo )  = 
RR+
85 iooretop 21398 . . . . . . . . 9  |-  ( 0 (,) +oo )  e.  ( topGen `  ran  (,) )
8684, 85eqeltrri 2542 . . . . . . . 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 22491 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( x  +  1 ) ) )  =  ( x  e.  RR+  |->  1 ) )
89 dvrelog 23143 . . . . . . 7  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( y  e.  RR+  |->  ( 1  /  y ) )
90 relogf1o 23079 . . . . . . . . . . 11  |-  ( log  |`  RR+ ) : RR+ -1-1-onto-> RR
91 f1of 5822 . . . . . . . . . . 11  |-  ( ( log  |`  RR+ ) :
RR+
-1-1-onto-> RR  ->  ( log  |`  RR+ ) : RR+ --> RR )
9290, 91mp1i 12 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ ) : RR+ --> RR )
9392feqmptd 5926 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y
) ) )
94 fvres 5886 . . . . . . . . . 10  |-  ( y  e.  RR+  ->  ( ( log  |`  RR+ ) `  y )  =  ( log `  y ) )
9594mpteq2ia 4539 . . . . . . . . 9  |-  ( y  e.  RR+  |->  ( ( log  |`  RR+ ) `  y ) )  =  ( y  e.  RR+  |->  ( log `  y ) )
9693, 95syl6eq 2514 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log  |`  RR+ )  =  ( y  e.  RR+  |->  ( log `  y
) ) )
9796oveq2d 6312 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  ( log  |`  RR+ ) )  =  ( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) ) )
9889, 97syl5reqr 2513 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
y  e.  RR+  |->  ( log `  y ) ) )  =  ( y  e.  RR+  |->  ( 1  / 
y ) ) )
99 fveq2 5872 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  ( log `  y )  =  ( log `  (
x  +  1 ) ) )
100 oveq2 6304 . . . . . 6  |-  ( y  =  ( x  + 
1 )  ->  (
1  /  y )  =  ( 1  / 
( x  +  1 ) ) )
10155, 55, 59, 63, 66, 68, 88, 98, 99, 100dvmptco 22500 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( ( 1  /  ( x  + 
1 ) )  x.  1 ) ) )
10262rpcnd 11283 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  e.  CC )
103102mulid1d 9630 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( 1  / 
( x  +  1 ) )  x.  1 )  =  ( 1  /  ( x  + 
1 ) ) )
104103mpteq2dva 4543 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  RR+  |->  ( ( 1  / 
( x  +  1 ) )  x.  1 ) )  =  ( x  e.  RR+  |->  ( 1  /  ( x  + 
1 ) ) ) )
105101, 104eqtrd 2498 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  RR+  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  RR+  |->  ( 1  / 
( x  +  1 ) ) ) )
106 ioossicc 11635 . . . . . . . . 9  |-  ( 0 (,) A )  C_  ( 0 [,] A
)
107106sseli 3495 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  x  e.  ( 0 [,] A
) )
108107, 7sylan2 474 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR )
109 eliooord 11609 . . . . . . . . 9  |-  ( x  e.  ( 0 (,) A )  ->  (
0  <  x  /\  x  <  A ) )
110109simpld 459 . . . . . . . 8  |-  ( x  e.  ( 0 (,) A )  ->  0  <  x )
111110adantl 466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  0  <  x )
112108, 111elrpd 11279 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  x  e.  RR+ )
113112ex 434 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 (,) A )  ->  x  e.  RR+ ) )
114113ssrdv 3505 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 (,) A
)  C_  RR+ )
115 iooretop 21398 . . . . 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 22491 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( log `  ( x  +  1 ) ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( x  +  1 ) ) ) )
118 elrege0 11652 . . . . . . . . 9  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR  /\  0  <_  x ) )
1197, 8, 118sylanbrc 664 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 [,] A ) )  ->  x  e.  ( 0 [,) +oo ) )
120119ex 434 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A )  ->  x  e.  ( 0 [,) +oo )
) )
121120ssrdv 3505 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 0 [,] A
)  C_  ( 0 [,) +oo ) )
122121resabs1d 5313 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,) +oo )
)  |`  ( 0 [,] A ) )  =  ( sqr  |`  (
0 [,] A ) ) )
123 sqrtf 13207 . . . . . . 7  |-  sqr : CC
--> CC
124123a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sqr : CC --> CC )
125124, 18feqresmpt 5927 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A
)  |->  ( sqr `  x
) ) )
126122, 125eqtrd 2498 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,) +oo )
)  |`  ( 0 [,] A ) )  =  ( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) ) )
127 resqrtcn 23248 . . . . 5  |-  ( sqr  |`  ( 0 [,) +oo ) )  e.  ( ( 0 [,) +oo ) -cn-> RR )
128 rescncf 21526 . . . . 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 18 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( sqr  |`  (
0 [,) +oo )
)  |`  ( 0 [,] A ) )  e.  ( ( 0 [,] A ) -cn-> RR ) )
130126, 129eqeltrrd 2546 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( x  e.  ( 0 [,] A ) 
|->  ( sqr `  x
) )  e.  ( ( 0 [,] A
) -cn-> RR ) )
131 rpcn 11253 . . . . . 6  |-  ( x  e.  RR+  ->  x  e.  CC )
132131adantl 466 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  x  e.  CC )
133132sqrtcld 13279 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  CC )
134 2rp 11250 . . . . . 6  |-  2  e.  RR+
135 rpsqrtcl 13109 . . . . . . 7  |-  ( x  e.  RR+  ->  ( sqr `  x )  e.  RR+ )
136135adantl 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR+ )
137 rpmulcl 11266 . . . . . 6  |-  ( ( 2  e.  RR+  /\  ( sqr `  x )  e.  RR+ )  ->  ( 2  x.  ( sqr `  x
) )  e.  RR+ )
138134, 136, 137sylancr 663 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR+ )
139138rpreccld 11291 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
2  x.  ( sqr `  x ) ) )  e.  RR+ )
140 dvsqrt 23243 . . . . 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 22491 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( RR  _D  (
x  e.  ( 0 (,) A )  |->  ( sqr `  x ) ) )  =  ( x  e.  ( 0 (,) A )  |->  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) ) )
143136rpred 11281 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( sqr `  x
)  e.  RR )
144 1re 9612 . . . . . . . . 9  |-  1  e.  RR
145 resubcl 9902 . . . . . . . . 9  |-  ( ( ( sqr `  x
)  e.  RR  /\  1  e.  RR )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
146143, 144, 145sylancl 662 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  -  1 )  e.  RR )
147146sqge0d 12339 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
( sqr `  x
)  -  1 ) ^ 2 ) )
148132sqsqrtd 13281 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
) ^ 2 )  =  x )
149133mulid1d 9630 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( sqr `  x
)  x.  1 )  =  ( sqr `  x
) )
150149oveq2d 6312 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  (
( sqr `  x
)  x.  1 ) )  =  ( 2  x.  ( sqr `  x
) ) )
151148, 150oveq12d 6314 . . . . . . . . 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 12264 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
153152a1i 11 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1 ^ 2 )  =  1 )
154151, 153oveq12d 6314 . . . . . . . 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 9567 . . . . . . . . 9  |-  1  e.  CC
156 binom2sub 12287 . . . . . . . . 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 662 . . . . . . . 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 11283 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  CC )
159132, 63, 158addsubd 9971 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( x  + 
1 )  -  (
2  x.  ( sqr `  x ) ) )  =  ( ( x  -  ( 2  x.  ( sqr `  x
) ) )  +  1 ) )
160154, 157, 1593eqtr4d 2508 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( ( ( sqr `  x )  -  1 ) ^ 2 )  =  ( ( x  +  1 )  -  ( 2  x.  ( sqr `  x ) ) ) )
161147, 160breqtrd 4480 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  0  <_  ( (
x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) ) )
16259rpred 11281 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( x  +  1 )  e.  RR )
163138rpred 11281 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  e.  RR )
164162, 163subge0d 10163 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 0  <_  (
( x  +  1 )  -  ( 2  x.  ( sqr `  x
) ) )  <->  ( 2  x.  ( sqr `  x
) )  <_  (
x  +  1 ) ) )
165161, 164mpbid 210 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 2  x.  ( sqr `  x ) )  <_  ( x  + 
1 ) )
166138, 59lerecd 11300 . . . . 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 210 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  RR+ )  ->  ( 1  /  (
x  +  1 ) )  <_  ( 1  /  ( 2  x.  ( sqr `  x
) ) ) )
168112, 167syldan 470 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  x  e.  (
0 (,) A ) )  ->  ( 1  /  ( x  + 
1 ) )  <_ 
( 1  /  (
2  x.  ( sqr `  x ) ) ) )
169 rexr 9656 . . . 4  |-  ( A  e.  RR  ->  A  e.  RR* )
170 0xr 9657 . . . . 5  |-  0  e.  RR*
171 lbicc2 11661 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
172170, 171mp3an1 1311 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  0  e.  ( 0 [,] A
) )
173169, 172sylan 471 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  e.  ( 0 [,] A ) )
174 ubicc2 11662 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
175170, 174mp3an1 1311 . . . 4  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  A  e.  ( 0 [,] A
) )
176169, 175sylan 471 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  ( 0 [,] A ) )
177 simpr 461 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
0  <_  A )
178 oveq1 6303 . . . . . 6  |-  ( x  =  0  ->  (
x  +  1 )  =  ( 0  +  1 ) )
179 0p1e1 10668 . . . . . 6  |-  ( 0  +  1 )  =  1
180178, 179syl6eq 2514 . . . . 5  |-  ( x  =  0  ->  (
x  +  1 )  =  1 )
181180fveq2d 5876 . . . 4  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  ( log `  1
) )
182 log1 23095 . . . 4  |-  ( log `  1 )  =  0
183181, 182syl6eq 2514 . . 3  |-  ( x  =  0  ->  ( log `  ( x  + 
1 ) )  =  0 )
184 fveq2 5872 . . . 4  |-  ( x  =  0  ->  ( sqr `  x )  =  ( sqr `  0
) )
185 sqrt0 13086 . . . 4  |-  ( sqr `  0 )  =  0
186184, 185syl6eq 2514 . . 3  |-  ( x  =  0  ->  ( sqr `  x )  =  0 )
187 oveq1 6303 . . . 4  |-  ( x  =  A  ->  (
x  +  1 )  =  ( A  + 
1 ) )
188187fveq2d 5876 . . 3  |-  ( x  =  A  ->  ( log `  ( x  + 
1 ) )  =  ( log `  ( A  +  1 ) ) )
189 fveq2 5872 . . 3  |-  ( x  =  A  ->  ( sqr `  x )  =  ( sqr `  A
) )
1902, 3, 53, 117, 130, 142, 168, 173, 176, 177, 183, 186, 188, 189dvle 22533 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  -  0 )  <_  ( ( sqr `  A )  -  0 ) )
191 ge0p1rp 11273 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  +  1 )  e.  RR+ )
192191relogcld 23133 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  e.  RR )
193 resqrtcl 13098 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( sqr `  A
)  e.  RR )
194192, 193, 2lesub1d 10180 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( log `  ( A  +  1 ) )  <_  ( sqr `  A )  <->  ( ( log `  ( A  + 
1 ) )  - 
0 )  <_  (
( sqr `  A
)  -  0 ) ) )
195190, 194mpbird 232 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( A  +  1 ) )  <_  ( sqr `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   {cpr 4034   class class class wbr 4456    |-> cmpt 4515   ran crn 5009    |` cres 5010   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   2c2 10606   RR+crp 11245   (,)cioo 11554   [,)cico 11556   [,]cicc 11557   ^cexp 12168   sqrcsqrt 13077   ↾t crest 14837   TopOpenctopn 14838   topGenctg 14854  ℂfldccnfld 18546  TopOnctopon 19521    Cn ccn 19851    tX ctx 20186   -cn->ccncf 21505    _D cdv 22392   logclog 23067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-sin 13816  df-cos 13817  df-tan 13818  df-pi 13819  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-limc 22395  df-dv 22396  df-log 23069  df-cxp 23070
This theorem is referenced by:  rplogsumlem1  23794
  Copyright terms: Public domain W3C validator