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Theorem ovolicc1 22469
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 11716 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
41, 2, 3syl2anc 667 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  RR )
5 ovolcl 22431 . . 3  |-  ( ( A [,] B ) 
C_  RR  ->  ( vol* `  ( A [,] B ) )  e. 
RR* )
64, 5syl 17 . 2  |-  ( ph  ->  ( vol* `  ( A [,] B ) )  e.  RR* )
7 ovolicc.3 . . . . . . . . . . 11  |-  ( ph  ->  A  <_  B )
8 df-br 4403 . . . . . . . . . . 11  |-  ( A  <_  B  <->  <. A ,  B >.  e.  <_  )
97, 8sylib 200 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e. 
<_  )
10 opelxpi 4866 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
111, 2, 10syl2anc 667 . . . . . . . . . 10  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
129, 11elind 3618 . . . . . . . . 9  |-  ( ph  -> 
<. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
1312adantr 467 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  <. A ,  B >.  e.  (  <_  i^i  ( RR  X.  RR ) ) )
14 0le0 10699 . . . . . . . . . 10  |-  0  <_  0
15 df-br 4403 . . . . . . . . . 10  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
1614, 15mpbi 212 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  <_
17 0re 9643 . . . . . . . . . 10  |-  0  e.  RR
18 opelxpi 4866 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
1917, 17, 18mp2an 678 . . . . . . . . 9  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
20 elin 3617 . . . . . . . . 9  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )  <-> 
( <. 0 ,  0
>.  e.  <_  /\  <. 0 ,  0 >.  e.  ( RR  X.  RR ) ) )
2116, 19, 20mpbir2an 931 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR  X.  RR ) )
22 ifcl 3923 . . . . . . . 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 668 . . . . . . 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 6046 . . . . . 6  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
26 eqid 2451 . . . . . . 7  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
27 eqid 2451 . . . . . . 7  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)
2826, 27ovolsf 22425 . . . . . 6  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,) +oo ) )
2925, 28syl 17 . . . . 5  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) +oo ) )
30 frn 5735 . . . . 5  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,) +oo )  ->  ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
)  C_  ( 0 [,) +oo ) )
3129, 30syl 17 . . . 4  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  ( 0 [,) +oo ) )
32 icossxr 11719 . . . 4  |-  ( 0 [,) +oo )  C_  RR*
3331, 32syl6ss 3444 . . 3  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  G ) )  C_  RR* )
34 supxrcl 11600 . . 3  |-  ( ran 
seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  C_  RR* 
->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
3533, 34syl 17 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  e.  RR* )
362, 1resubcld 10047 . . 3  |-  ( ph  ->  ( B  -  A
)  e.  RR )
3736rexrd 9690 . 2  |-  ( ph  ->  ( B  -  A
)  e.  RR* )
38 1nn 10620 . . . . . . 7  |-  1  e.  NN
3938a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  1  e.  NN )
40 op1stg 6805 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
411, 2, 40syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
4241adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  =  A )
43 elicc2 11699 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
441, 2, 43syl2anc 667 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( A [,] B )  <-> 
( x  e.  RR  /\  A  <_  x  /\  x  <_  B ) ) )
4544biimpa 487 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  B
) )
4645simp2d 1021 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A  <_  x )
4742, 46eqbrtrd 4423 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 1st ` 
<. A ,  B >. )  <_  x )
4845simp3d 1022 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  B )
49 op2ndg 6806 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
501, 2, 49syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
5150adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( 2nd ` 
<. A ,  B >. )  =  B )
5248, 51breqtrrd 4429 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  x  <_  ( 2nd `  <. A ,  B >. ) )
53 fveq2 5865 . . . . . . . . . . 11  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
54 iftrue 3887 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. A ,  B >. )
55 opex 4664 . . . . . . . . . . . . 13  |-  <. A ,  B >.  e.  _V
5654, 24, 55fvmpt 5948 . . . . . . . . . . . 12  |-  ( 1  e.  NN  ->  ( G `  1 )  =  <. A ,  B >. )
5738, 56ax-mp 5 . . . . . . . . . . 11  |-  ( G `
 1 )  = 
<. A ,  B >.
5853, 57syl6eq 2501 . . . . . . . . . 10  |-  ( n  =  1  ->  ( G `  n )  =  <. A ,  B >. )
5958fveq2d 5869 . . . . . . . . 9  |-  ( n  =  1  ->  ( 1st `  ( G `  n ) )  =  ( 1st `  <. A ,  B >. )
)
6059breq1d 4412 . . . . . . . 8  |-  ( n  =  1  ->  (
( 1st `  ( G `  n )
)  <_  x  <->  ( 1st ` 
<. A ,  B >. )  <_  x ) )
6158fveq2d 5869 . . . . . . . . 9  |-  ( n  =  1  ->  ( 2nd `  ( G `  n ) )  =  ( 2nd `  <. A ,  B >. )
)
6261breq2d 4414 . . . . . . . 8  |-  ( n  =  1  ->  (
x  <_  ( 2nd `  ( G `  n
) )  <->  x  <_  ( 2nd `  <. A ,  B >. ) ) )
6360, 62anbi12d 717 . . . . . . 7  |-  ( n  =  1  ->  (
( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) )  <->  ( ( 1st `  <. A ,  B >. )  <_  x  /\  x  <_  ( 2nd `  <. A ,  B >. )
) ) )
6463rspcev 3150 . . . . . 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 1266 . . . . 5  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  E. n  e.  NN  ( ( 1st `  ( G `  n
) )  <_  x  /\  x  <_  ( 2nd `  ( G `  n
) ) ) )
6665ralrimiva 2802 . . . 4  |-  ( ph  ->  A. x  e.  ( A [,] B ) E. n  e.  NN  ( ( 1st `  ( G `  n )
)  <_  x  /\  x  <_  ( 2nd `  ( G `  n )
) ) )
67 ovolficc 22421 . . . . 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 667 . . . 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 236 . . 3  |-  ( ph  ->  ( A [,] B
)  C_  U. ran  ( [,]  o.  G ) )
7027ovollb2 22442 . . 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 667 . 2  |-  ( ph  ->  ( vol* `  ( A [,] B ) )  <_  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) , 
RR* ,  <  ) )
72 addid1 9813 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
k  +  0 )  =  k )
7372adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  CC )  ->  (
k  +  0 )  =  k )
74 nnuz 11194 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
7538, 74eleqtri 2527 . . . . . . . . 9  |-  1  e.  ( ZZ>= `  1 )
7675a1i 11 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  1  e.  ( ZZ>= `  1 )
)
77 simpr 463 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  NN )
7877, 74syl6eleq 2539 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  x  e.  ( ZZ>= `  1 )
)
79 rge0ssre 11740 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
8029adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) : NN --> ( 0 [,) +oo ) )
81 ffvelrn 6020 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) : NN --> ( 0 [,) +oo )  /\  1  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,) +oo ) )
8280, 38, 81sylancl 668 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  ( 0 [,) +oo ) )
8379, 82sseldi 3430 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  RR )
8483recnd 9669 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  e.  CC )
8525ad2antrr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
86 elfzuz 11796 . . . . . . . . . . . . 13  |-  ( k  e.  ( ( 1  +  1 ) ... x )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
8786adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  ( 1  +  1 ) ) )
88 df-2 10668 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
8988fveq2i 5868 . . . . . . . . . . . 12  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
9087, 89syl6eleqr 2540 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  ( ZZ>= `  2 )
)
91 eluz2nn 11197 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  e.  NN )
9290, 91syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  e.  NN )
9326ovolfsval 22423 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  k  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
9485, 92, 93syl2anc 667 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  ( ( 2nd `  ( G `  k
) )  -  ( 1st `  ( G `  k ) ) ) )
95 eqeq1 2455 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  (
n  =  1  <->  k  =  1 ) )
9695ifbid 3903 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  if ( n  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. ) )
97 opex 4664 . . . . . . . . . . . . . . . . 17  |-  <. 0 ,  0 >.  e.  _V
9855, 97ifex 3949 . . . . . . . . . . . . . . . 16  |-  if ( k  =  1 , 
<. A ,  B >. , 
<. 0 ,  0
>. )  e.  _V
9996, 24, 98fvmpt 5948 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
10092, 99syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0 >. )
)
101 eluz2b3 11232 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  ( ZZ>= `  2
)  <->  ( k  e.  NN  /\  k  =/=  1 ) )
102101simprbi 466 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( ZZ>= `  2
)  ->  k  =/=  1 )
10390, 102syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  k  =/=  1 )
104103neneqd 2629 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  -.  k  =  1 )
105104iffalsed 3892 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  if ( k  =  1 ,  <. A ,  B >. ,  <. 0 ,  0
>. )  =  <. 0 ,  0 >. )
106100, 105eqtrd 2485 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( G `  k )  =  <. 0 ,  0
>. )
107106fveq2d 5869 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  <. 0 ,  0 >. ) )
108 c0ex 9637 . . . . . . . . . . . . 13  |-  0  e.  _V
109108, 108op2nd 6802 . . . . . . . . . . . 12  |-  ( 2nd `  <. 0 ,  0
>. )  =  0
110107, 109syl6eq 2501 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 2nd `  ( G `  k ) )  =  0 )
111106fveq2d 5869 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  <. 0 ,  0 >. ) )
112108, 108op1st 6801 . . . . . . . . . . . 12  |-  ( 1st `  <. 0 ,  0
>. )  =  0
113111, 112syl6eq 2501 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  ( 1st `  ( G `  k ) )  =  0 )
114110, 113oveq12d 6308 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  ( 0  -  0 ) )
115 0m0e0 10719 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
116114, 115syl6eq 2501 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( 2nd `  ( G `  k )
)  -  ( 1st `  ( G `  k
) ) )  =  0 )
11794, 116eqtrd 2485 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN )  /\  k  e.  ( ( 1  +  1 ) ... x
) )  ->  (
( ( abs  o.  -  )  o.  G
) `  k )  =  0 )
11873, 76, 78, 84, 117seqid2 12259 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
) )
119 1z 10967 . . . . . . . 8  |-  1  e.  ZZ
12025adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  G : NN
--> (  <_  i^i  ( RR  X.  RR ) ) )
12126ovolfsval 22423 . . . . . . . . . 10  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  1  e.  NN )  ->  (
( ( abs  o.  -  )  o.  G
) `  1 )  =  ( ( 2nd `  ( G `  1
) )  -  ( 1st `  ( G ` 
1 ) ) ) )
122120, 38, 121sylancl 668 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( ( 2nd `  ( G `  1 )
)  -  ( 1st `  ( G `  1
) ) ) )
12357fveq2i 5868 . . . . . . . . . . 11  |-  ( 2nd `  ( G `  1
) )  =  ( 2nd `  <. A ,  B >. )
12450adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  <. A ,  B >. )  =  B )
125123, 124syl5eq 2497 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 2nd `  ( G `  1
) )  =  B )
12657fveq2i 5868 . . . . . . . . . . 11  |-  ( 1st `  ( G `  1
) )  =  ( 1st `  <. A ,  B >. )
12741adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  <. A ,  B >. )  =  A )
128126, 127syl5eq 2497 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN )  ->  ( 1st `  ( G `  1
) )  =  A )
129125, 128oveq12d 6308 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( 2nd `  ( G `
 1 ) )  -  ( 1st `  ( G `  1 )
) )  =  ( B  -  A ) )
130122, 129eqtrd 2485 . . . . . . . 8  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( ( abs  o.  -  )  o.  G ) `  1 )  =  ( B  -  A
) )
131119, 130seq1i 12227 . . . . . . 7  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  1
)  =  ( B  -  A ) )
132118, 131eqtr3d 2487 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  =  ( B  -  A ) )
13336leidd 10180 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  <_  ( B  -  A ) )
134133adantr 467 . . . . . 6  |-  ( (
ph  /\  x  e.  NN )  ->  ( B  -  A )  <_ 
( B  -  A
) )
135132, 134eqbrtrd 4423 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) `  x
)  <_  ( B  -  A ) )
136135ralrimiva 2802 . . . 4  |-  ( ph  ->  A. x  e.  NN  (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
)
137 ffn 5728 . . . . . 6  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) : NN --> ( 0 [,) +oo )  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  Fn  NN )
13829, 137syl 17 . . . . 5  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )  Fn  NN )
139 breq1 4405 . . . . . 6  |-  ( z  =  (  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  G )
) `  x )  ->  ( z  <_  ( B  -  A )  <->  (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) ) `  x )  <_  ( B  -  A )
) )
140139ralrn 6025 . . . . 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
) ) )
141138, 140syl 17 . . . 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
) ) )
142136, 141mpbird 236 . . 3  |-  ( ph  ->  A. z  e.  ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) z  <_ 
( B  -  A
) )
143 supxrleub 11612 . . . 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
) ) )
14433, 37, 143syl2anc 667 . . 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
) ) )
145142, 144mpbird 236 . 2  |-  ( ph  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  G
) ) ,  RR* ,  <  )  <_  ( B  -  A )
)
1466, 35, 37, 71, 145xrletrd 11459 1  |-  ( ph  ->  ( vol* `  ( A [,] B ) )  <_  ( B  -  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    i^i cin 3403    C_ wss 3404   ifcif 3881   <.cop 3974   U.cuni 4198   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   ran crn 4835    o. ccom 4838    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   2c2 10659   ZZ>=cuz 11159   [,)cico 11637   [,]cicc 11638   ...cfz 11784    seqcseq 12213   abscabs 13297   vol*covol 22413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-ovol 22416
This theorem is referenced by:  ovolicc  22477
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