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Theorem ovolicc1 19404
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 10984 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 643 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ovolcl 19366 . . 3  |-  ( ( A [,] B ) 
C_  RR  ->  ( vol
* `  ( A [,] B ) )  e. 
RR* )
64, 5syl 16 . 2  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  e.  RR* )
7 ovolicc.3 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
8 df-br 4205 . . . . . . . . . . 11  |-  ( A  <_  B  <->  <. A ,  B >.  e.  <_  )
97, 8sylib 189 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e. 
<_  )
10 opelxpi 4902 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
111, 2, 10syl2anc 643 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
12 elin 3522 . . . . . . . . . 10  |-  ( <. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <-> 
( <. A ,  B >.  e.  <_  /\  <. A ,  B >.  e.  ( RR 
X.  RR ) ) )
139, 11, 12sylanbrc 646 . . . . . . . . 9  |-  ( ph  -> 
<. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1413adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  <. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
15 0le0 10073 . . . . . . . . . 10  |-  0  <_  0
16 df-br 4205 . . . . . . . . . 10  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
1715, 16mpbi 200 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  <_
18 0re 9083 . . . . . . . . . 10  |-  0  e.  RR
19 opelxpi 4902 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
2018, 18, 19mp2an 654 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
21 elin 3522 . . . . . . . . 9  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <-> 
( <. 0 ,  0
>.  e.  <_  /\  <. 0 ,  0 >.  e.  ( RR  X.  RR ) ) )
2217, 20, 21mpbir2an 887 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )
23 ifcl 3767 . . . . . . . 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 644 . . . . . . 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 5885 . . . . . 6  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
27 eqid 2435 . . . . . . 7  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
28 eqid 2435 . . . . . . 7  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
2927, 28ovolsf 19361 . . . . . 6  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,)  +oo ) )
3026, 29syl 16 . . . . 5  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) 
+oo ) )
31 frn 5589 . . . . 5  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,)  +oo )  ->  ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  C_  ( 0 [,)  +oo ) )
3230, 31syl 16 . . . 4  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  ( 0 [,)  +oo ) )
33 icossxr 10987 . . . 4  |-  ( 0 [,)  +oo )  C_  RR*
3432, 33syl6ss 3352 . . 3  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  RR* )
35 supxrcl 10885 . . 3  |-  ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  C_  RR* 
->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
3634, 35syl 16 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
372, 1resubcld 9457 . . 3  |-  ( ph  ->  ( B  -  A
)  e.  RR )
3837rexrd 9126 . 2  |-  ( ph  ->  ( B  -  A
)  e.  RR* )
39 1nn 10003 . . . . . . 7  |-  1  e.  NN
4039a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  1  e.  NN )
41 op1stg 6351 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
421, 2, 41syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
4342adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  =  A )
44 elicc2 10967 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
451, 2, 44syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4645biimpa 471 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4746simp2d 970 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
4843, 47eqbrtrd 4224 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  <_  x )
4946simp3d 971 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
50 op2ndg 6352 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
511, 2, 50syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
5251adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 2nd ` 
<. A ,  B >. )  =  B )
5349, 52breqtrrd 4230 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  ( 2nd `  <. A ,  B >. ) )
54 fveq2 5720 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
55 iftrue 3737 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. A ,  B >. )
56 opex 4419 . . . . . . . . . . . . 13  |-  <. A ,  B >.  e.  _V
5755, 25, 56fvmpt 5798 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  ( G `  1 )  =  <. A ,  B >. )
5839, 57ax-mp 8 . . . . . . . . . . 11  |-  ( G `
 1 )  = 
<. A ,  B >.
5954, 58syl6eq 2483 . . . . . . . . . 10  |-  ( n  =  1  ->  ( G `  n )  =  <. A ,  B >. )
6059fveq2d 5724 . . . . . . . . 9  |-  ( n  =  1  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. A ,  B >. )
)
6160breq1d 4214 . . . . . . . 8  |-  ( n  =  1  ->  (
( 1st `  ( G `  n )
)  <_  x  <->  ( 1st ` 
<. A ,  B >. )  <_  x ) )
6259fveq2d 5724 . . . . . . . . 9  |-  ( n  =  1  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. A ,  B >. )
)
6362breq2d 4216 . . . . . . . 8  |-  ( n  =  1  ->  (
x  <_  ( 2nd `  ( G `  n
) )  <->  x  <_  ( 2nd `  <. A ,  B >. ) ) )
6461, 63anbi12d 692 . . . . . . 7  |-  ( n  =  1  ->  (
( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) )  <->  ( ( 1st `  <. A ,  B >. )  <_  x  /\  x  <_  ( 2nd `  <. A ,  B >. )
) ) )
6564rspcev 3044 . . . . . 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 1182 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <_  x  /\  x  <_  ( 2nd `  ( G `  n
) ) ) )
6766ralrimiva 2781 . . . 4  |-  ( ph  ->  A. x  e.  ( A [,] B ) E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) )
68 ovolficc 19357 . . . . 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 643 . . . 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 224 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  U. ran  ( [,]  o.  G ) )
7128ovollb2 19377 . . 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 643 . 2  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  <_  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  ) )
73 addid1 9238 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
k  +  0 )  =  k )
7473adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  CC )  ->  (
k  +  0 )  =  k )
75 nnuz 10513 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
7639, 75eleqtri 2507 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  1 )
7776a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  1  e.  ( ZZ>= `  1 )
)
78 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  NN )
7978, 75syl6eleq 2525 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  ( ZZ>= `  1 )
)
80 pnfxr 10705 . . . . . . . . . . 11  |-  +oo  e.  RR*
81 icossre 10983 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
8218, 80, 81mp2an 654 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  RR
8330adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) : NN --> ( 0 [,)  +oo ) )
84 ffvelrn 5860 . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,)  +oo ) )
8682, 85sseldi 3338 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  RR )
8786recnd 9106 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  CC )
8826ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
89 elfzuz 11047 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 1  +  1 ) ... x )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
9089adantl 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
91 df-2 10050 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
9291fveq2i 5723 . . . . . . . . . . . 12  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
9390, 92syl6eleqr 2526 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  2 )
)
94 eluz2b3 10541 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( k  e.  NN  /\  k  =/=  1 ) )
9594simplbi 447 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  e.  NN )
9693, 95syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  NN )
9727ovolfsval 19359 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
99 eqeq1 2441 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
n  =  1  <->  k  =  1 ) )
10099ifbid 3749 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
101 opex 4419 . . . . . . . . . . . . . . . . 17  |-  <. 0 ,  0 >.  e.  _V
10256, 101ifex 3789 . . . . . . . . . . . . . . . 16  |-  if ( k  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  _V
103100, 25, 102fvmpt 5798 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10496, 103syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10594simprbi 451 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  =/=  1 )
10693, 105syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  =/=  1 )
107106neneqd 2614 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  -.  k  =  1 )
108 iffalse 3738 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  1  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
109107, 108syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
110104, 109eqtrd 2467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  <. 0 ,  0
>. )
111110fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  <. 0 ,  0 >. ) )
112 c0ex 9077 . . . . . . . . . . . . 13  |-  0  e.  _V
113112, 112op2nd 6348 . . . . . . . . . . . 12  |-  ( 2nd `  <. 0 ,  0
>. )  =  0
114111, 113syl6eq 2483 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  0 )
115110fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  <. 0 ,  0 >. ) )
116112, 112op1st 6347 . . . . . . . . . . . 12  |-  ( 1st `  <. 0 ,  0
>. )  =  0
117115, 116syl6eq 2483 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  0 )
118114, 117oveq12d 6091 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  ( 0  -  0 ) )
119 0cn 9076 . . . . . . . . . . 11  |-  0  e.  CC
120119subidi 9363 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
121118, 120syl6eq 2483 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  0 )
12298, 121eqtrd 2467 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  0 )
12374, 77, 79, 87, 122seqid2 11361 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
) )
124 1z 10303 . . . . . . . 8  |-  1  e.  ZZ
12526adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  G : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
12627ovolfsval 19359 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( ( 2nd `  ( G `  1 )
)  -  ( 1st `  ( G `  1
) ) ) )
12858fveq2i 5723 . . . . . . . . . . 11  |-  ( 2nd `  ( G `  1
) )  =  ( 2nd `  <. A ,  B >. )
12951adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  <. A ,  B >. )  =  B )
130128, 129syl5eq 2479 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  ( G `  1
) )  =  B )
13158fveq2i 5723 . . . . . . . . . . 11  |-  ( 1st `  ( G `  1
) )  =  ( 1st `  <. A ,  B >. )
13242adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  <. A ,  B >. )  =  A )
133131, 132syl5eq 2479 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  ( G `  1
) )  =  A )
134130, 133oveq12d 6091 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd `  ( G `
 1 ) )  -  ( 1st `  ( G `  1 )
) )  =  ( B  -  A ) )
135127, 134eqtrd 2467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( B  -  A
) )
136124, 135seq1i 11329 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  ( B  -  A ) )
137123, 136eqtr3d 2469 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  =  ( B  -  A ) )
13837leidd 9585 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  <_  ( B  -  A ) )
139138adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( B  -  A )  <_ 
( B  -  A
) )
140137, 139eqbrtrd 4224 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  <_  ( B  -  A ) )
141140ralrimiva 2781 . . . 4  |-  ( ph  ->  A. x  e.  NN  (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
)
142 ffn 5583 . . . . . 6  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,)  +oo )  ->  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  Fn  NN )
14330, 142syl 16 . . . . 5  |-  ( ph  ->  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  Fn  NN )
144 breq1 4207 . . . . . 6  |-  ( z  =  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  ->  ( z  <_  ( B  -  A )  <->  (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
) )
145144ralrn 5865 . . . . 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 16 . . . 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 224 . . 3  |-  ( ph  ->  A. z  e.  ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) z  <_ 
( B  -  A
) )
148 supxrleub 10897 . . . 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 643 . . 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 224 . 2  |-  ( ph  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  <_  ( B  -  A )
)
1516, 36, 38, 72, 150xrletrd 10744 1  |-  ( ph  ->  ( vol * `  ( A [,] B ) )  <_  ( B  -  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   ifcif 3731   <.cop 3809   U.cuni 4007   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   ran crn 4871    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   supcsup 7437   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113    - cmin 9283   NNcn 9992   2c2 10041   ZZ>=cuz 10480   [,)cico 10910   [,]cicc 10911   ...cfz 11035    seq cseq 11315   abscabs 12031   vol
*covol 19351
This theorem is referenced by:  ovolicc  19411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ovol 19353
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