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Theorem ovolval2lem 38583
Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypothesis
Ref Expression
ovolval2lem.1  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
Assertion
Ref Expression
ovolval2lem  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  F ) )  =  ran  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n ) ( vol `  (
( [,)  o.  F
) `  k )
) ) )
Distinct variable groups:    k, F, n    ph, k
Allowed substitution hint:    ph( n)

Proof of Theorem ovolval2lem
StepHypRef Expression
1 reex 9648 . . . . . . 7  |-  RR  e.  _V
21, 1xpex 6614 . . . . . 6  |-  ( RR 
X.  RR )  e. 
_V
3 inss2 3644 . . . . . 6  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
4 mapss 7532 . . . . . 6  |-  ( ( ( RR  X.  RR )  e.  _V  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR ) )  ->  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  C_  ( ( RR 
X.  RR )  ^m  NN ) )
52, 3, 4mp2an 686 . . . . 5  |-  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  C_  ( ( RR  X.  RR )  ^m  NN )
6 ovolval2lem.1 . . . . . 6  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
72inex2 4538 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
87a1i 11 . . . . . . 7  |-  ( ph  ->  (  <_  i^i  ( RR  X.  RR ) )  e.  _V )
9 nnex 10637 . . . . . . . 8  |-  NN  e.  _V
109a1i 11 . . . . . . 7  |-  ( ph  ->  NN  e.  _V )
118, 10elmapd 7504 . . . . . 6  |-  ( ph  ->  ( F  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  <->  F : NN
--> (  <_  i^i  ( RR  X.  RR ) ) ) )
126, 11mpbird 240 . . . . 5  |-  ( ph  ->  F  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
135, 12sseldi 3416 . . . 4  |-  ( ph  ->  F  e.  ( ( RR  X.  RR )  ^m  NN ) )
14 1zzd 10992 . . . . 5  |-  ( F  e.  ( ( RR 
X.  RR )  ^m  NN )  ->  1  e.  ZZ )
15 nnuz 11218 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
16 elmapi 7511 . . . . . . . . . 10  |-  ( F  e.  ( ( RR 
X.  RR )  ^m  NN )  ->  F : NN
--> ( RR  X.  RR ) )
1716adantr 472 . . . . . . . . 9  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  F : NN --> ( RR 
X.  RR ) )
18 simpr 468 . . . . . . . . 9  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  k  e.  NN )
1917, 18fvovco 37540 . . . . . . . 8  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( ( [,)  o.  F ) `  k
)  =  ( ( 1st `  ( F `
 k ) ) [,) ( 2nd `  ( F `  k )
) ) )
2019fveq2d 5883 . . . . . . 7  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( vol `  (
( [,)  o.  F
) `  k )
)  =  ( vol `  ( ( 1st `  ( F `  k )
) [,) ( 2nd `  ( F `  k
) ) ) ) )
2116ffvelrnda 6037 . . . . . . . . 9  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( F `  k
)  e.  ( RR 
X.  RR ) )
22 xp1st 6842 . . . . . . . . 9  |-  ( ( F `  k )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  k
) )  e.  RR )
2321, 22syl 17 . . . . . . . 8  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( 1st `  ( F `  k )
)  e.  RR )
24 xp2nd 6843 . . . . . . . . 9  |-  ( ( F `  k )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  k
) )  e.  RR )
2521, 24syl 17 . . . . . . . 8  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( 2nd `  ( F `  k )
)  e.  RR )
26 volicore 38521 . . . . . . . 8  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR )  -> 
( vol `  (
( 1st `  ( F `  k )
) [,) ( 2nd `  ( F `  k
) ) ) )  e.  RR )
2723, 25, 26syl2anc 673 . . . . . . 7  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( vol `  (
( 1st `  ( F `  k )
) [,) ( 2nd `  ( F `  k
) ) ) )  e.  RR )
2820, 27eqeltrd 2549 . . . . . 6  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( vol `  (
( [,)  o.  F
) `  k )
)  e.  RR )
2928recnd 9687 . . . . 5  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( vol `  (
( [,)  o.  F
) `  k )
)  e.  CC )
30 eqid 2471 . . . . 5  |-  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n
) ( vol `  (
( [,)  o.  F
) `  k )
) )  =  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n ) ( vol `  ( ( [,)  o.  F ) `
 k ) ) )
31 eqid 2471 . . . . 5  |-  seq 1
(  +  ,  ( k  e.  NN  |->  ( vol `  ( ( [,)  o.  F ) `
 k ) ) ) )  =  seq 1 (  +  , 
( k  e.  NN  |->  ( vol `  ( ( [,)  o.  F ) `
 k ) ) ) )
3214, 15, 29, 30, 31fsumsermpt 37754 . . . 4  |-  ( F  e.  ( ( RR 
X.  RR )  ^m  NN )  ->  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n
) ( vol `  (
( [,)  o.  F
) `  k )
) )  =  seq 1 (  +  , 
( k  e.  NN  |->  ( vol `  ( ( [,)  o.  F ) `
 k ) ) ) ) )
3313, 32syl 17 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n ) ( vol `  ( ( [,)  o.  F ) `
 k ) ) )  =  seq 1
(  +  ,  ( k  e.  NN  |->  ( vol `  ( ( [,)  o.  F ) `
 k ) ) ) ) )
34 simpr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( 1st `  ( F `  k ) )  < 
( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  < 
( 2nd `  ( F `  k )
) )
3534iftrued 3880 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  ( 1st `  ( F `  k ) )  < 
( 2nd `  ( F `  k )
) )  ->  if ( ( 1st `  ( F `  k )
)  <  ( 2nd `  ( F `  k
) ) ,  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) ,  0 )  =  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) )
3613, 23sylan 479 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1st `  ( F `  k
) )  e.  RR )
3736adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  e.  RR )
3813, 25sylan 479 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( 2nd `  ( F `  k
) )  e.  RR )
3938adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 2nd `  ( F `  k ) )  e.  RR )
40 ressxr 9702 . . . . . . . . . . . 12  |-  RR  C_  RR*
4140, 37sseldi 3416 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  e. 
RR* )
4240, 39sseldi 3416 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 2nd `  ( F `  k ) )  e. 
RR* )
43 xpss 4946 . . . . . . . . . . . . . . . . . 18  |-  ( RR 
X.  RR )  C_  ( _V  X.  _V )
4443, 21sseldi 3416 . . . . . . . . . . . . . . . . 17  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( F `  k
)  e.  ( _V 
X.  _V ) )
45 1st2ndb 6850 . . . . . . . . . . . . . . . . 17  |-  ( ( F `  k )  e.  ( _V  X.  _V )  <->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
4644, 45sylib 201 . . . . . . . . . . . . . . . 16  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( F `  k
)  =  <. ( 1st `  ( F `  k ) ) ,  ( 2nd `  ( F `  k )
) >. )
4713, 46sylan 479 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  = 
<. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
4847eqcomd 2477 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  NN )  ->  <. ( 1st `  ( F `  k ) ) ,  ( 2nd `  ( F `  k )
) >.  =  ( F `
 k ) )
49 inss1 3643 . . . . . . . . . . . . . . . . 17  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  <_
5049a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (  <_  i^i  ( RR  X.  RR ) ) 
C_  <_  )
516, 50fssd 5750 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : NN -->  <_  )
5251ffvelrnda 6037 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e. 
<_  )
5348, 52eqeltrd 2549 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  <. ( 1st `  ( F `  k ) ) ,  ( 2nd `  ( F `  k )
) >.  e.  <_  )
54 df-br 4396 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( F `
 k ) )  <_  ( 2nd `  ( F `  k )
)  <->  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >.  e.  <_  )
5553, 54sylibr 217 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1st `  ( F `  k
) )  <_  ( 2nd `  ( F `  k ) ) )
5655adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  <_ 
( 2nd `  ( F `  k )
) )
57 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )
5839, 37lenltd 9798 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  (
( 2nd `  ( F `  k )
)  <_  ( 1st `  ( F `  k
) )  <->  -.  ( 1st `  ( F `  k ) )  < 
( 2nd `  ( F `  k )
) ) )
5957, 58mpbird 240 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 2nd `  ( F `  k ) )  <_ 
( 1st `  ( F `  k )
) )
6041, 42, 56, 59xrletrid 11475 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )
61 simp3 1032 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )
62 simp1 1030 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  ( 1st `  ( F `  k ) )  e.  RR )
63 simp2 1031 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  ( 2nd `  ( F `  k ) )  e.  RR )
6462, 63eqleltd 9796 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  (
( 1st `  ( F `  k )
)  =  ( 2nd `  ( F `  k
) )  <->  ( ( 1st `  ( F `  k ) )  <_ 
( 2nd `  ( F `  k )
)  /\  -.  ( 1st `  ( F `  k ) )  < 
( 2nd `  ( F `  k )
) ) ) )
6561, 64mpbid 215 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  (
( 1st `  ( F `  k )
)  <_  ( 2nd `  ( F `  k
) )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) ) )
6665simprd 470 . . . . . . . . . . . 12  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )
6766iffalsed 3883 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  if ( ( 1st `  ( F `  k )
)  <  ( 2nd `  ( F `  k
) ) ,  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) ,  0 )  =  0 )
6863recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  ( 2nd `  ( F `  k ) )  e.  CC )
6961eqcomd 2477 . . . . . . . . . . . 12  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  ( 2nd `  ( F `  k ) )  =  ( 1st `  ( F `  k )
) )
7068, 69subeq0bd 10066 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  (
( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) )  =  0 )
7167, 70eqtr4d 2508 . . . . . . . . . 10  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR  /\  ( 1st `  ( F `  k ) )  =  ( 2nd `  ( F `  k )
) )  ->  if ( ( 1st `  ( F `  k )
)  <  ( 2nd `  ( F `  k
) ) ,  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) ,  0 )  =  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) )
7237, 39, 60, 71syl3anc 1292 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  -.  ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) )  ->  if ( ( 1st `  ( F `  k )
)  <  ( 2nd `  ( F `  k
) ) ,  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) ,  0 )  =  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) )
7335, 72pm2.61dan 808 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  if ( ( 1st `  ( F `  k )
)  <  ( 2nd `  ( F `  k
) ) ,  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) ,  0 )  =  ( ( 2nd `  ( F `  k )
)  -  ( 1st `  ( F `  k
) ) ) )
74 volico 37958 . . . . . . . . 9  |-  ( ( ( 1st `  ( F `  k )
)  e.  RR  /\  ( 2nd `  ( F `
 k ) )  e.  RR )  -> 
( vol `  (
( 1st `  ( F `  k )
) [,) ( 2nd `  ( F `  k
) ) ) )  =  if ( ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) ,  ( ( 2nd `  ( F `
 k ) )  -  ( 1st `  ( F `  k )
) ) ,  0 ) )
7536, 38, 74syl2anc 673 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( vol `  ( ( 1st `  ( F `  k )
) [,) ( 2nd `  ( F `  k
) ) ) )  =  if ( ( 1st `  ( F `
 k ) )  <  ( 2nd `  ( F `  k )
) ,  ( ( 2nd `  ( F `
 k ) )  -  ( 1st `  ( F `  k )
) ) ,  0 ) )
7636, 38, 55abssuble0d 13571 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( ( 1st `  ( F `  k )
)  -  ( 2nd `  ( F `  k
) ) ) )  =  ( ( 2nd `  ( F `  k
) )  -  ( 1st `  ( F `  k ) ) ) )
7773, 75, 763eqtr4d 2515 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( vol `  ( ( 1st `  ( F `  k )
) [,) ( 2nd `  ( F `  k
) ) ) )  =  ( abs `  (
( 1st `  ( F `  k )
)  -  ( 2nd `  ( F `  k
) ) ) ) )
7813adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  F  e.  ( ( RR  X.  RR )  ^m  NN ) )
79 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
8078, 79, 20syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( vol `  ( ( [,)  o.  F ) `  k
) )  =  ( vol `  ( ( 1st `  ( F `
 k ) ) [,) ( 2nd `  ( F `  k )
) ) ) )
8146fveq2d 5883 . . . . . . . . 9  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( ( abs  o.  -  ) `  ( F `  k )
)  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
)
82 df-ov 6311 . . . . . . . . . . 11  |-  ( ( 1st `  ( F `
 k ) ) ( abs  o.  -  ) ( 2nd `  ( F `  k )
) )  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
8382eqcomi 2480 . . . . . . . . . 10  |-  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )  =  ( ( 1st `  ( F `  k
) ) ( abs 
o.  -  ) ( 2nd `  ( F `  k ) ) )
8483a1i 11 . . . . . . . . 9  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )  =  ( ( 1st `  ( F `  k )
) ( abs  o.  -  ) ( 2nd `  ( F `  k
) ) ) )
8523recnd 9687 . . . . . . . . . 10  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( 1st `  ( F `  k )
)  e.  CC )
8625recnd 9687 . . . . . . . . . 10  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( 2nd `  ( F `  k )
)  e.  CC )
87 eqid 2471 . . . . . . . . . . 11  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
8887cnmetdval 21869 . . . . . . . . . 10  |-  ( ( ( 1st `  ( F `  k )
)  e.  CC  /\  ( 2nd `  ( F `
 k ) )  e.  CC )  -> 
( ( 1st `  ( F `  k )
) ( abs  o.  -  ) ( 2nd `  ( F `  k
) ) )  =  ( abs `  (
( 1st `  ( F `  k )
)  -  ( 2nd `  ( F `  k
) ) ) ) )
8985, 86, 88syl2anc 673 . . . . . . . . 9  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( ( 1st `  ( F `  k )
) ( abs  o.  -  ) ( 2nd `  ( F `  k
) ) )  =  ( abs `  (
( 1st `  ( F `  k )
)  -  ( 2nd `  ( F `  k
) ) ) ) )
9081, 84, 893eqtrd 2509 . . . . . . . 8  |-  ( ( F  e.  ( ( RR  X.  RR )  ^m  NN )  /\  k  e.  NN )  ->  ( ( abs  o.  -  ) `  ( F `  k )
)  =  ( abs `  ( ( 1st `  ( F `  k )
)  -  ( 2nd `  ( F `  k
) ) ) ) )
9178, 79, 90syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( abs  o.  -  ) `  ( F `  k
) )  =  ( abs `  ( ( 1st `  ( F `
 k ) )  -  ( 2nd `  ( F `  k )
) ) ) )
9277, 80, 913eqtr4d 2515 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( vol `  ( ( [,)  o.  F ) `  k
) )  =  ( ( abs  o.  -  ) `  ( F `  k ) ) )
9392mpteq2dva 4482 . . . . 5  |-  ( ph  ->  ( k  e.  NN  |->  ( vol `  ( ( [,)  o.  F ) `
 k ) ) )  =  ( k  e.  NN  |->  ( ( abs  o.  -  ) `  ( F `  k
) ) ) )
9413, 16syl 17 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
95 rr2sscn2 37676 . . . . . . 7  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
9695a1i 11 . . . . . 6  |-  ( ph  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
97 absf 13477 . . . . . . . 8  |-  abs : CC
--> RR
98 subf 9897 . . . . . . . 8  |-  -  :
( CC  X.  CC )
--> CC
99 fco 5751 . . . . . . . 8  |-  ( ( abs : CC --> RR  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
10097, 98, 99mp2an 686 . . . . . . 7  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> RR
101100a1i 11 . . . . . 6  |-  ( ph  ->  ( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
10294, 96, 101fcomptss 37555 . . . . 5  |-  ( ph  ->  ( ( abs  o.  -  )  o.  F
)  =  ( k  e.  NN  |->  ( ( abs  o.  -  ) `  ( F `  k
) ) ) )
10393, 102eqtr4d 2508 . . . 4  |-  ( ph  ->  ( k  e.  NN  |->  ( vol `  ( ( [,)  o.  F ) `
 k ) ) )  =  ( ( abs  o.  -  )  o.  F ) )
104103seqeq3d 12259 . . 3  |-  ( ph  ->  seq 1 (  +  ,  ( k  e.  NN  |->  ( vol `  (
( [,)  o.  F
) `  k )
) ) )  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) ) )
10533, 104eqtr2d 2506 . 2  |-  ( ph  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )  =  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n ) ( vol `  ( ( [,)  o.  F ) `
 k ) ) ) )
106105rneqd 5068 1  |-  ( ph  ->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  F ) )  =  ran  ( n  e.  NN  |->  sum_ k  e.  ( 1 ... n ) ( vol `  (
( [,)  o.  F
) `  k )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ifcif 3872   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   ran crn 4840    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   [,)cico 11662   ...cfz 11810    seqcseq 12251   abscabs 13374   sum_csu 13829   volcvol 22493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-rest 15399  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cmp 20479  df-ovol 22494  df-vol 22496
This theorem is referenced by: (None)
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