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Theorem ovolicc1 18875
Description: The measure of a closed interval is lower bounded by its length. (Contributed by Mario Carneiro, 13-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
Hypotheses
Ref Expression
ovolicc.1  |-  ( ph  ->  A  e.  RR )
ovolicc.2  |-  ( ph  ->  B  e.  RR )
ovolicc.3  |-  ( ph  ->  A  <_  B )
ovolicc1.4  |-  G  =  ( n  e.  NN  |->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
Assertion
Ref Expression
ovolicc1  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  <_  ( B  -  A ) )
Distinct variable groups:    A, n    B, n    n, G    ph, n

Proof of Theorem ovolicc1
Dummy variables  k  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolicc.1 . . . 4  |-  ( ph  ->  A  e.  RR )
2 ovolicc.2 . . . 4  |-  ( ph  ->  B  e.  RR )
3 iccssre 10731 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 642 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ovolcl 18837 . . 3  |-  ( ( A [,] B ) 
C_  RR  ->  ( vol
* `  ( A [,] B ) )  e. 
RR* )
64, 5syl 15 . 2  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  e.  RR* )
7 ovolicc.3 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
8 df-br 4024 . . . . . . . . . . 11  |-  ( A  <_  B  <->  <. A ,  B >.  e.  <_  )
97, 8sylib 188 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e. 
<_  )
10 opelxpi 4721 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
111, 2, 10syl2anc 642 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
12 elin 3358 . . . . . . . . . 10  |-  ( <. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <-> 
( <. A ,  B >.  e.  <_  /\  <. A ,  B >.  e.  ( RR 
X.  RR ) ) )
139, 11, 12sylanbrc 645 . . . . . . . . 9  |-  ( ph  -> 
<. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1413adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  <. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
15 0le0 9827 . . . . . . . . . 10  |-  0  <_  0
16 df-br 4024 . . . . . . . . . 10  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
1715, 16mpbi 199 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  <_
18 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
19 opelxpi 4721 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
2018, 18, 19mp2an 653 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
21 elin 3358 . . . . . . . . 9  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <-> 
( <. 0 ,  0
>.  e.  <_  /\  <. 0 ,  0 >.  e.  ( RR  X.  RR ) ) )
2217, 20, 21mpbir2an 886 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )
23 ifcl 3601 . . . . . . . 8  |-  ( (
<. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) )  /\  <. 0 ,  0
>.  e.  (  <_  i^i  ( RR  X.  RR ) ) )  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
2414, 22, 23sylancl 643 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  if ( n  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
25 ovolicc1.4 . . . . . . 7  |-  G  =  ( n  e.  NN  |->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
2624, 25fmptd 5684 . . . . . 6  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
27 eqid 2283 . . . . . . 7  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
28 eqid 2283 . . . . . . 7  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
2927, 28ovolsf 18832 . . . . . 6  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,)  +oo ) )
3026, 29syl 15 . . . . 5  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) 
+oo ) )
31 frn 5395 . . . . 5  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,)  +oo )  ->  ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  C_  ( 0 [,)  +oo ) )
3230, 31syl 15 . . . 4  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  ( 0 [,)  +oo ) )
33 icossxr 10734 . . . 4  |-  ( 0 [,)  +oo )  C_  RR*
3432, 33syl6ss 3191 . . 3  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  RR* )
35 supxrcl 10633 . . 3  |-  ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  C_  RR* 
->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
3634, 35syl 15 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
372, 1resubcld 9211 . . 3  |-  ( ph  ->  ( B  -  A
)  e.  RR )
3837rexrd 8881 . 2  |-  ( ph  ->  ( B  -  A
)  e.  RR* )
39 1nn 9757 . . . . . . 7  |-  1  e.  NN
4039a1i 10 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  1  e.  NN )
41 op1stg 6132 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
421, 2, 41syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
4342adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  =  A )
44 elicc2 10715 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
451, 2, 44syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4645biimpa 470 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4746simp2d 968 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
4843, 47eqbrtrd 4043 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  <_  x )
4946simp3d 969 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
50 op2ndg 6133 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
511, 2, 50syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
5251adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 2nd ` 
<. A ,  B >. )  =  B )
5349, 52breqtrrd 4049 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  ( 2nd `  <. A ,  B >. ) )
54 fveq2 5525 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
55 iftrue 3571 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. A ,  B >. )
56 opex 4237 . . . . . . . . . . . . 13  |-  <. A ,  B >.  e.  _V
5755, 25, 56fvmpt 5602 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  ( G `  1 )  =  <. A ,  B >. )
5839, 57ax-mp 8 . . . . . . . . . . 11  |-  ( G `
 1 )  = 
<. A ,  B >.
5954, 58syl6eq 2331 . . . . . . . . . 10  |-  ( n  =  1  ->  ( G `  n )  =  <. A ,  B >. )
6059fveq2d 5529 . . . . . . . . 9  |-  ( n  =  1  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. A ,  B >. )
)
6160breq1d 4033 . . . . . . . 8  |-  ( n  =  1  ->  (
( 1st `  ( G `  n )
)  <_  x  <->  ( 1st ` 
<. A ,  B >. )  <_  x ) )
6259fveq2d 5529 . . . . . . . . 9  |-  ( n  =  1  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. A ,  B >. )
)
6362breq2d 4035 . . . . . . . 8  |-  ( n  =  1  ->  (
x  <_  ( 2nd `  ( G `  n
) )  <->  x  <_  ( 2nd `  <. A ,  B >. ) ) )
6461, 63anbi12d 691 . . . . . . 7  |-  ( n  =  1  ->  (
( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) )  <->  ( ( 1st `  <. A ,  B >. )  <_  x  /\  x  <_  ( 2nd `  <. A ,  B >. )
) ) )
6564rspcev 2884 . . . . . 6  |-  ( ( 1  e.  NN  /\  ( ( 1st `  <. A ,  B >. )  <_  x  /\  x  <_ 
( 2nd `  <. A ,  B >. )
) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <_  x  /\  x  <_  ( 2nd `  ( G `  n
) ) ) )
6640, 48, 53, 65syl12anc 1180 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <_  x  /\  x  <_  ( 2nd `  ( G `  n
) ) ) )
6766ralrimiva 2626 . . . 4  |-  ( ph  ->  A. x  e.  ( A [,] B ) E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) )
68 ovolficc 18828 . . . . 5  |-  ( ( ( A [,] B
)  C_  RR  /\  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )  -> 
( ( A [,] B )  C_  U. ran  ( [,]  o.  G )  <->  A. x  e.  ( A [,] B ) E. n  e.  NN  (
( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) ) )
694, 26, 68syl2anc 642 . . . 4  |-  ( ph  ->  ( ( A [,] B )  C_  U. ran  ( [,]  o.  G )  <->  A. x  e.  ( A [,] B ) E. n  e.  NN  (
( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) ) )
7067, 69mpbird 223 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  U. ran  ( [,]  o.  G ) )
7128ovollb2 18848 . . 3  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  ( A [,] B )  C_  U.
ran  ( [,]  o.  G ) )  -> 
( vol * `  ( A [,] B ) )  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  ) )
7226, 70, 71syl2anc 642 . 2  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  ) )
73 addid1 8992 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
k  +  0 )  =  k )
7473adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  CC )  ->  (
k  +  0 )  =  k )
75 nnuz 10263 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
7639, 75eleqtri 2355 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  1 )
7776a1i 10 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  1  e.  ( ZZ>= `  1 )
)
78 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  NN )
7978, 75syl6eleq 2373 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  ( ZZ>= `  1 )
)
80 pnfxr 10455 . . . . . . . . . . 11  |-  +oo  e.  RR*
81 icossre 10730 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
8218, 80, 81mp2an 653 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  RR
8330adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) : NN --> ( 0 [,)  +oo ) )
84 ffvelrn 5663 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) 
+oo )  /\  1  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,)  +oo ) )
8583, 39, 84sylancl 643 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,)  +oo ) )
8682, 85sseldi 3178 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  RR )
8786recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  CC )
8826ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
89 elfzuz 10794 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 1  +  1 ) ... x )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
9089adantl 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
91 df-2 9804 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
9291fveq2i 5528 . . . . . . . . . . . 12  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
9390, 92syl6eleqr 2374 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  2 )
)
94 eluz2b3 10291 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( k  e.  NN  /\  k  =/=  1 ) )
9594simplbi 446 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  e.  NN )
9693, 95syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  NN )
9727ovolfsval 18830 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  k  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
9888, 96, 97syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
99 eqeq1 2289 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
n  =  1  <->  k  =  1 ) )
10099ifbid 3583 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
101 opex 4237 . . . . . . . . . . . . . . . . 17  |-  <. 0 ,  0 >.  e.  _V
10256, 101ifex 3623 . . . . . . . . . . . . . . . 16  |-  if ( k  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  _V
103100, 25, 102fvmpt 5602 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10496, 103syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10594simprbi 450 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  =/=  1 )
10693, 105syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  =/=  1 )
107106neneqd 2462 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  -.  k  =  1 )
108 iffalse 3572 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  1  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
109107, 108syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
110104, 109eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  <. 0 ,  0
>. )
111110fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  <. 0 ,  0 >. ) )
112 c0ex 8832 . . . . . . . . . . . . 13  |-  0  e.  _V
113112, 112op2nd 6129 . . . . . . . . . . . 12  |-  ( 2nd `  <. 0 ,  0
>. )  =  0
114111, 113syl6eq 2331 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  0 )
115110fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  <. 0 ,  0 >. ) )
116112, 112op1st 6128 . . . . . . . . . . . 12  |-  ( 1st `  <. 0 ,  0
>. )  =  0
117115, 116syl6eq 2331 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  0 )
118114, 117oveq12d 5876 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  ( 0  -  0 ) )
119 0cn 8831 . . . . . . . . . . 11  |-  0  e.  CC
120119subidi 9117 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
121118, 120syl6eq 2331 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  0 )
12298, 121eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  0 )
12374, 77, 79, 87, 122seqid2 11092 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
) )
124 1z 10053 . . . . . . . 8  |-  1  e.  ZZ
12526adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  G : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
12627ovolfsval 18830 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  1  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  1 )  =  ( ( 2nd `  ( G `  1
) )  -  ( 1st `  ( G ` 
1 ) ) ) )
127125, 39, 126sylancl 643 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( ( 2nd `  ( G `  1 )
)  -  ( 1st `  ( G `  1
) ) ) )
12858fveq2i 5528 . . . . . . . . . . 11  |-  ( 2nd `  ( G `  1
) )  =  ( 2nd `  <. A ,  B >. )
12951adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  <. A ,  B >. )  =  B )
130128, 129syl5eq 2327 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  ( G `  1
) )  =  B )
13158fveq2i 5528 . . . . . . . . . . 11  |-  ( 1st `  ( G `  1
) )  =  ( 1st `  <. A ,  B >. )
13242adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  <. A ,  B >. )  =  A )
133131, 132syl5eq 2327 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  ( G `  1
) )  =  A )
134130, 133oveq12d 5876 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd `  ( G `
 1 ) )  -  ( 1st `  ( G `  1 )
) )  =  ( B  -  A ) )
135127, 134eqtrd 2315 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( B  -  A
) )
136124, 135seq1i 11060 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  ( B  -  A ) )
137123, 136eqtr3d 2317 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  =  ( B  -  A ) )
13837leidd 9339 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  <_  ( B  -  A ) )
139138adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( B  -  A )  <_ 
( B  -  A
) )
140137, 139eqbrtrd 4043 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  <_  ( B  -  A ) )
141140ralrimiva 2626 . . . 4  |-  ( ph  ->  A. x  e.  NN  (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
)
142 ffn 5389 . . . . . 6  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,)  +oo )  ->  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  Fn  NN )
14330, 142syl 15 . . . . 5  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  Fn  NN )
144 breq1 4026 . . . . . 6  |-  ( z  =  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  ->  ( z  <_  ( B  -  A )  <->  (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
) )
145144ralrn 5668 . . . . 5  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) )  Fn  NN  ->  ( A. z  e. 
ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) z  <_  ( B  -  A )  <->  A. x  e.  NN  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  <_  ( B  -  A
) ) )
146143, 145syl 15 . . . 4  |-  ( ph  ->  ( A. z  e. 
ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) z  <_  ( B  -  A )  <->  A. x  e.  NN  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  <_  ( B  -  A
) ) )
147141, 146mpbird 223 . . 3  |-  ( ph  ->  A. z  e.  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) z  <_ 
( B  -  A
) )
148 supxrleub 10645 . . . 4  |-  ( ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  RR* 
/\  ( B  -  A )  e.  RR* )  ->  ( sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  )  <_ 
( B  -  A
)  <->  A. z  e.  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) z  <_ 
( B  -  A
) ) )
14934, 38, 148syl2anc 642 . . 3  |-  ( ph  ->  ( sup ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  )  <_ 
( B  -  A
)  <->  A. z  e.  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) z  <_ 
( B  -  A
) ) )
150147, 149mpbird 223 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  <_  ( B  -  A )
)
1516, 36, 38, 72, 150xrletrd 10493 1  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  <_  ( B  -  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ifcif 3565   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   ZZ>=cuz 10230   [,)cico 10658   [,]cicc 10659   ...cfz 10782    seq cseq 11046   abscabs 11719   vol
*covol 18822
This theorem is referenced by:  ovolicc  18882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ovol 18824
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