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Theorem ovolicc1 20997
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 11375 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ovolcl 20959 . . 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 4291 . . . . . . . . . . 11  |-  ( A  <_  B  <->  <. A ,  B >.  e.  <_  )
97, 8sylib 196 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e. 
<_  )
10 opelxpi 4869 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
111, 2, 10syl2anc 661 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
129, 11elind 3538 . . . . . . . . 9  |-  ( ph  -> 
<. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1312adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  <. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
14 0le0 10409 . . . . . . . . . 10  |-  0  <_  0
15 df-br 4291 . . . . . . . . . 10  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
1614, 15mpbi 208 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  <_
17 0re 9384 . . . . . . . . . 10  |-  0  e.  RR
18 opelxpi 4869 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
1917, 17, 18mp2an 672 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
20 elin 3537 . . . . . . . . 9  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <-> 
( <. 0 ,  0
>.  e.  <_  /\  <. 0 ,  0 >.  e.  ( RR  X.  RR ) ) )
2116, 19, 20mpbir2an 911 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )
22 ifcl 3829 . . . . . . . 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 ) ) )
2313, 21, 22sylancl 662 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  if ( n  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
24 ovolicc1.4 . . . . . . 7  |-  G  =  ( n  e.  NN  |->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
2523, 24fmptd 5865 . . . . . 6  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
26 eqid 2441 . . . . . . 7  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
27 eqid 2441 . . . . . . 7  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
2826, 27ovolsf 20954 . . . . . 6  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,) +oo ) )
2925, 28syl 16 . . . . 5  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) +oo ) )
30 frn 5563 . . . . 5  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,) +oo )  ->  ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  C_  ( 0 [,) +oo ) )
3129, 30syl 16 . . . 4  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  ( 0 [,) +oo ) )
32 icossxr 11378 . . . 4  |-  ( 0 [,) +oo )  C_  RR*
3331, 32syl6ss 3366 . . 3  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  RR* )
34 supxrcl 11275 . . 3  |-  ( ran 
seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  C_  RR* 
->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
3533, 34syl 16 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
362, 1resubcld 9774 . . 3  |-  ( ph  ->  ( B  -  A
)  e.  RR )
3736rexrd 9431 . 2  |-  ( ph  ->  ( B  -  A
)  e.  RR* )
38 1nn 10331 . . . . . . 7  |-  1  e.  NN
3938a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  1  e.  NN )
40 op1stg 6587 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
411, 2, 40syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
4241adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  =  A )
43 elicc2 11358 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
441, 2, 43syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4544biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4645simp2d 1001 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
4742, 46eqbrtrd 4310 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  <_  x )
4845simp3d 1002 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
49 op2ndg 6588 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
501, 2, 49syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
5150adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 2nd ` 
<. A ,  B >. )  =  B )
5248, 51breqtrrd 4316 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  ( 2nd `  <. A ,  B >. ) )
53 fveq2 5689 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
54 iftrue 3795 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. A ,  B >. )
55 opex 4554 . . . . . . . . . . . . 13  |-  <. A ,  B >.  e.  _V
5654, 24, 55fvmpt 5772 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  ( G `  1 )  =  <. A ,  B >. )
5738, 56ax-mp 5 . . . . . . . . . . 11  |-  ( G `
 1 )  = 
<. A ,  B >.
5853, 57syl6eq 2489 . . . . . . . . . 10  |-  ( n  =  1  ->  ( G `  n )  =  <. A ,  B >. )
5958fveq2d 5693 . . . . . . . . 9  |-  ( n  =  1  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. A ,  B >. )
)
6059breq1d 4300 . . . . . . . 8  |-  ( n  =  1  ->  (
( 1st `  ( G `  n )
)  <_  x  <->  ( 1st ` 
<. A ,  B >. )  <_  x ) )
6158fveq2d 5693 . . . . . . . . 9  |-  ( n  =  1  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. A ,  B >. )
)
6261breq2d 4302 . . . . . . . 8  |-  ( n  =  1  ->  (
x  <_  ( 2nd `  ( G `  n
) )  <->  x  <_  ( 2nd `  <. A ,  B >. ) ) )
6360, 62anbi12d 710 . . . . . . 7  |-  ( n  =  1  ->  (
( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) )  <->  ( ( 1st `  <. A ,  B >. )  <_  x  /\  x  <_  ( 2nd `  <. A ,  B >. )
) ) )
6463rspcev 3071 . . . . . 6  |-  ( ( 1  e.  NN  /\  ( ( 1st `  <. A ,  B >. )  <_  x  /\  x  <_ 
( 2nd `  <. A ,  B >. )
) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <_  x  /\  x  <_  ( 2nd `  ( G `  n
) ) ) )
6539, 47, 52, 64syl12anc 1216 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <_  x  /\  x  <_  ( 2nd `  ( G `  n
) ) ) )
6665ralrimiva 2797 . . . 4  |-  ( ph  ->  A. x  e.  ( A [,] B ) E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) )
67 ovolficc 20950 . . . . 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 )
) ) ) )
684, 25, 67syl2anc 661 . . . 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 )
) ) ) )
6966, 68mpbird 232 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  U. ran  ( [,]  o.  G ) )
7027ovollb2 20970 . . 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* ,  <  ) )
7125, 69, 70syl2anc 661 . 2  |-  ( ph  ->  ( vol* `  ( A [,] B ) )  <_  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  ) )
72 addid1 9547 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
k  +  0 )  =  k )
7372adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  CC )  ->  (
k  +  0 )  =  k )
74 nnuz 10894 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
7538, 74eleqtri 2513 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  1 )
7675a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  1  e.  ( ZZ>= `  1 )
)
77 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  NN )
7877, 74syl6eleq 2531 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  ( ZZ>= `  1 )
)
79 pnfxr 11090 . . . . . . . . . . 11  |- +oo  e.  RR*
80 icossre 11374 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
8117, 79, 80mp2an 672 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
8229adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) : NN --> ( 0 [,) +oo ) )
83 ffvelrn 5839 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) +oo )  /\  1  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,) +oo ) )
8482, 38, 83sylancl 662 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,) +oo ) )
8581, 84sseldi 3352 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  RR )
8685recnd 9410 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  CC )
8725ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
88 elfzuz 11447 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 1  +  1 ) ... x )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
8988adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
90 df-2 10378 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
9190fveq2i 5692 . . . . . . . . . . . 12  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
9289, 91syl6eleqr 2532 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  2 )
)
93 eluz2b3 10926 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( k  e.  NN  /\  k  =/=  1 ) )
9493simplbi 460 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  e.  NN )
9592, 94syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  NN )
9626ovolfsval 20952 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  k  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
9787, 95, 96syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
98 eqeq1 2447 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
n  =  1  <->  k  =  1 ) )
9998ifbid 3809 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
100 opex 4554 . . . . . . . . . . . . . . . . 17  |-  <. 0 ,  0 >.  e.  _V
10155, 100ifex 3856 . . . . . . . . . . . . . . . 16  |-  if ( k  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  _V
10299, 24, 101fvmpt 5772 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10395, 102syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10493simprbi 464 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  =/=  1 )
10592, 104syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  =/=  1 )
106105neneqd 2622 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  -.  k  =  1 )
107 iffalse 3797 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  1  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
108106, 107syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
109103, 108eqtrd 2473 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  <. 0 ,  0
>. )
110109fveq2d 5693 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  <. 0 ,  0 >. ) )
111 c0ex 9378 . . . . . . . . . . . . 13  |-  0  e.  _V
112111, 111op2nd 6584 . . . . . . . . . . . 12  |-  ( 2nd `  <. 0 ,  0
>. )  =  0
113110, 112syl6eq 2489 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  0 )
114109fveq2d 5693 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  <. 0 ,  0 >. ) )
115111, 111op1st 6583 . . . . . . . . . . . 12  |-  ( 1st `  <. 0 ,  0
>. )  =  0
116114, 115syl6eq 2489 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  0 )
117113, 116oveq12d 6107 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  ( 0  -  0 ) )
118 0m0e0 10429 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
119117, 118syl6eq 2489 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  0 )
12097, 119eqtrd 2473 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  0 )
12173, 76, 78, 86, 120seqid2 11850 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
) )
122 1z 10674 . . . . . . . 8  |-  1  e.  ZZ
12325adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  G : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
12426ovolfsval 20952 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  1  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  1 )  =  ( ( 2nd `  ( G `  1
) )  -  ( 1st `  ( G ` 
1 ) ) ) )
125123, 38, 124sylancl 662 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( ( 2nd `  ( G `  1 )
)  -  ( 1st `  ( G `  1
) ) ) )
12657fveq2i 5692 . . . . . . . . . . 11  |-  ( 2nd `  ( G `  1
) )  =  ( 2nd `  <. A ,  B >. )
12750adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  <. A ,  B >. )  =  B )
128126, 127syl5eq 2485 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  ( G `  1
) )  =  B )
12957fveq2i 5692 . . . . . . . . . . 11  |-  ( 1st `  ( G `  1
) )  =  ( 1st `  <. A ,  B >. )
13041adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  <. A ,  B >. )  =  A )
131129, 130syl5eq 2485 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  ( G `  1
) )  =  A )
132128, 131oveq12d 6107 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd `  ( G `
 1 ) )  -  ( 1st `  ( G `  1 )
) )  =  ( B  -  A ) )
133125, 132eqtrd 2473 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( B  -  A
) )
134122, 133seq1i 11818 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  ( B  -  A ) )
135121, 134eqtr3d 2475 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  =  ( B  -  A ) )
13636leidd 9904 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  <_  ( B  -  A ) )
137136adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( B  -  A )  <_ 
( B  -  A
) )
138135, 137eqbrtrd 4310 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  <_  ( B  -  A ) )
139138ralrimiva 2797 . . . 4  |-  ( ph  ->  A. x  e.  NN  (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
)
140 ffn 5557 . . . . . 6  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  Fn  NN )
14129, 140syl 16 . . . . 5  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  Fn  NN )
142 breq1 4293 . . . . . 6  |-  ( z  =  (  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  ->  ( z  <_  ( B  -  A )  <->  (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
) )
143142ralrn 5844 . . . . 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
) ) )
144141, 143syl 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
) ) )
145139, 144mpbird 232 . . 3  |-  ( ph  ->  A. z  e.  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) z  <_ 
( B  -  A
) )
146 supxrleub 11287 . . . 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
) ) )
14733, 37, 146syl2anc 661 . . 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
) ) )
148145, 147mpbird 232 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  <_  ( B  -  A )
)
1496, 35, 37, 71, 148xrletrd 11134 1  |-  ( ph  ->  ( vol* `  ( A [,] B ) )  <_  ( B  -  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714    i^i cin 3325    C_ wss 3326   ifcif 3789   <.cop 3881   U.cuni 4089   class class class wbr 4290    e. cmpt 4348    X. cxp 4836   ran crn 4839    o. ccom 4842    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574   supcsup 7688   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    + caddc 9283   +oocpnf 9413   RR*cxr 9415    < clt 9416    <_ cle 9417    - cmin 9593   NNcn 10320   2c2 10369   ZZ>=cuz 10859   [,)cico 11300   [,]cicc 11301   ...cfz 11435    seqcseq 11804   abscabs 12721   vol*covol 20944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-q 10952  df-rp 10990  df-ioo 11302  df-ico 11304  df-icc 11305  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162  df-ovol 20946
This theorem is referenced by:  ovolicc  21004
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