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Theorem ovolicc1 19281
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 10926 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 643 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ovolcl 19243 . . 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 4156 . . . . . . . . . . 11  |-  ( A  <_  B  <->  <. A ,  B >.  e.  <_  )
97, 8sylib 189 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e. 
<_  )
10 opelxpi 4852 . . . . . . . . . . 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 3475 . . . . . . . . . 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 10015 . . . . . . . . . 10  |-  0  <_  0
16 df-br 4156 . . . . . . . . . 10  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
1715, 16mpbi 200 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  <_
18 0re 9026 . . . . . . . . . 10  |-  0  e.  RR
19 opelxpi 4852 . . . . . . . . . 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 3475 . . . . . . . . 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 3720 . . . . . . . 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 5834 . . . . . 6  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
27 eqid 2389 . . . . . . 7  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
28 eqid 2389 . . . . . . 7  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
2927, 28ovolsf 19238 . . . . . 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 5539 . . . . 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 10929 . . . 4  |-  ( 0 [,)  +oo )  C_  RR*
3432, 33syl6ss 3305 . . 3  |-  ( ph  ->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  RR* )
35 supxrcl 10827 . . 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 9399 . . 3  |-  ( ph  ->  ( B  -  A
)  e.  RR )
3837rexrd 9069 . 2  |-  ( ph  ->  ( B  -  A
)  e.  RR* )
39 1nn 9945 . . . . . . 7  |-  1  e.  NN
4039a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  1  e.  NN )
41 op1stg 6300 . . . . . . . . 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 10909 . . . . . . . . . 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 4175 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  <_  x )
4946simp3d 971 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
50 op2ndg 6301 . . . . . . . . 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 4181 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  ( 2nd `  <. A ,  B >. ) )
54 fveq2 5670 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
55 iftrue 3690 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. A ,  B >. )
56 opex 4370 . . . . . . . . . . . . 13  |-  <. A ,  B >.  e.  _V
5755, 25, 56fvmpt 5747 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  ( G `  1 )  =  <. A ,  B >. )
5839, 57ax-mp 8 . . . . . . . . . . 11  |-  ( G `
 1 )  = 
<. A ,  B >.
5954, 58syl6eq 2437 . . . . . . . . . 10  |-  ( n  =  1  ->  ( G `  n )  =  <. A ,  B >. )
6059fveq2d 5674 . . . . . . . . 9  |-  ( n  =  1  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. A ,  B >. )
)
6160breq1d 4165 . . . . . . . 8  |-  ( n  =  1  ->  (
( 1st `  ( G `  n )
)  <_  x  <->  ( 1st ` 
<. A ,  B >. )  <_  x ) )
6259fveq2d 5674 . . . . . . . . 9  |-  ( n  =  1  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. A ,  B >. )
)
6362breq2d 4167 . . . . . . . 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 2997 . . . . . 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 2734 . . . 4  |-  ( ph  ->  A. x  e.  ( A [,] B ) E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) )
68 ovolficc 19234 . . . . 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 19254 . . 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 9180 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
k  +  0 )  =  k )
7473adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  CC )  ->  (
k  +  0 )  =  k )
75 nnuz 10455 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
7639, 75eleqtri 2461 . . . . . . . . 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 2479 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  ( ZZ>= `  1 )
)
80 pnfxr 10647 . . . . . . . . . . 11  |-  +oo  e.  RR*
81 icossre 10925 . . . . . . . . . . 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 5809 . . . . . . . . . . 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 3291 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  RR )
8786recnd 9049 . . . . . . . 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 10989 . . . . . . . . . . . . 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 9992 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
9291fveq2i 5673 . . . . . . . . . . . 12  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
9390, 92syl6eleqr 2480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  2 )
)
94 eluz2b3 10483 . . . . . . . . . . . 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 19236 . . . . . . . . . 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 2395 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
n  =  1  <->  k  =  1 ) )
10099ifbid 3702 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
101 opex 4370 . . . . . . . . . . . . . . . . 17  |-  <. 0 ,  0 >.  e.  _V
10256, 101ifex 3742 . . . . . . . . . . . . . . . 16  |-  if ( k  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  _V
103100, 25, 102fvmpt 5747 . . . . . . . . . . . . . . 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 2568 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  -.  k  =  1 )
108 iffalse 3691 . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  <. 0 ,  0
>. )
111110fveq2d 5674 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  <. 0 ,  0 >. ) )
112 c0ex 9020 . . . . . . . . . . . . 13  |-  0  e.  _V
113112, 112op2nd 6297 . . . . . . . . . . . 12  |-  ( 2nd `  <. 0 ,  0
>. )  =  0
114111, 113syl6eq 2437 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  0 )
115110fveq2d 5674 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  <. 0 ,  0 >. ) )
116112, 112op1st 6296 . . . . . . . . . . . 12  |-  ( 1st `  <. 0 ,  0
>. )  =  0
117115, 116syl6eq 2437 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  0 )
118114, 117oveq12d 6040 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  ( 0  -  0 ) )
119 0cn 9019 . . . . . . . . . . 11  |-  0  e.  CC
120119subidi 9305 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
121118, 120syl6eq 2437 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  0 )
12298, 121eqtrd 2421 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  0 )
12374, 77, 79, 87, 122seqid2 11298 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
) )
124 1z 10245 . . . . . . . 8  |-  1  e.  ZZ
12526adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  G : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
12627ovolfsval 19236 . . . . . . . . . 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 5673 . . . . . . . . . . 11  |-  ( 2nd `  ( G `  1
) )  =  ( 2nd `  <. A ,  B >. )
12951adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  <. A ,  B >. )  =  B )
130128, 129syl5eq 2433 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  ( G `  1
) )  =  B )
13158fveq2i 5673 . . . . . . . . . . 11  |-  ( 1st `  ( G `  1
) )  =  ( 1st `  <. A ,  B >. )
13242adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  <. A ,  B >. )  =  A )
133131, 132syl5eq 2433 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  ( G `  1
) )  =  A )
134130, 133oveq12d 6040 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd `  ( G `
 1 ) )  -  ( 1st `  ( G `  1 )
) )  =  ( B  -  A ) )
135127, 134eqtrd 2421 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( B  -  A
) )
136124, 135seq1i 11266 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  ( B  -  A ) )
137123, 136eqtr3d 2423 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  =  ( B  -  A ) )
13837leidd 9527 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  <_  ( B  -  A ) )
139138adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( B  -  A )  <_ 
( B  -  A
) )
140137, 139eqbrtrd 4175 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  <_  ( B  -  A ) )
141140ralrimiva 2734 . . . 4  |-  ( ph  ->  A. x  e.  NN  (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
)
142 ffn 5533 . . . . . 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 4158 . . . . . 6  |-  ( z  =  (  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  ->  ( z  <_  ( B  -  A )  <->  (  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
) )
145144ralrn 5814 . . . . 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 10839 . . . 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 10686 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 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652    i^i cin 3264    C_ wss 3265   ifcif 3684   <.cop 3762   U.cuni 3959   class class class wbr 4155    e. cmpt 4209    X. cxp 4818   ran crn 4821    o. ccom 4824    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   supcsup 7382   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    +oocpnf 9052   RR*cxr 9054    < clt 9055    <_ cle 9056    - cmin 9225   NNcn 9934   2c2 9983   ZZ>=cuz 10422   [,)cico 10852   [,]cicc 10853   ...cfz 10977    seq cseq 11252   abscabs 11968   vol
*covol 19228
This theorem is referenced by:  ovolicc  19288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-ioo 10854  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409  df-ovol 19230
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