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Theorem ovolfs2 19455
Description: Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
ovolfs2.1  |-  G  =  ( ( abs  o.  -  )  o.  F
)
Assertion
Ref Expression
ovolfs2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( ( vol
*  o.  (,) )  o.  F ) )

Proof of Theorem ovolfs2
Dummy variables  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolfcl 19355 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  (
( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) ) )
2 ovolioo 19454 . . . . 5  |-  ( ( ( 1st `  ( F `  n )
)  e.  RR  /\  ( 2nd `  ( F `
 n ) )  e.  RR  /\  ( 1st `  ( F `  n ) )  <_ 
( 2nd `  ( F `  n )
) )  ->  ( vol * `  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) ) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
31, 2syl 16 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( vol * `  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) ) )  =  ( ( 2nd `  ( F `  n )
)  -  ( 1st `  ( F `  n
) ) ) )
4 inss2 3554 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 9121 . . . . . . . . . . 11  |-  RR  C_  RR*
6 xpss12 4973 . . . . . . . . . . 11  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
75, 5, 6mp2an 654 . . . . . . . . . 10  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
84, 7sstri 3349 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 ffvelrn 5860 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
108, 9sseldi 3338 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR*  X.  RR* )
)
11 1st2nd2 6378 . . . . . . . 8  |-  ( ( F `  n )  e.  ( RR*  X.  RR* )  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1210, 11syl 16 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
1312fveq2d 5724 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
14 df-ov 6076 . . . . . 6  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1513, 14syl6eqr 2485 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
1615fveq2d 5724 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( vol * `  ( (,) `  ( F `  n
) ) )  =  ( vol * `  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) ) )
17 ovolfs2.1 . . . . 5  |-  G  =  ( ( abs  o.  -  )  o.  F
)
1817ovolfsval 19359 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
193, 16, 183eqtr4rd 2478 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( vol * `  ( (,) `  ( F `  n )
) ) )
2019mpteq2dva 4287 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
n  e.  NN  |->  ( G `  n ) )  =  ( n  e.  NN  |->  ( vol
* `  ( (,) `  ( F `  n
) ) ) ) )
2117ovolfsf 19360 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) 
+oo ) )
2221feqmptd 5771 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( n  e.  NN  |->  ( G `  n ) ) )
23 id 20 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2423feqmptd 5771 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  =  ( n  e.  NN  |->  ( F `  n ) ) )
25 ioof 10994 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2625a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,) : ( RR*  X.  RR* ) --> ~P RR )
2726ffvelrnda 5862 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  ( RR*  X.  RR* )
)  ->  ( (,) `  x )  e.  ~P RR )
2826feqmptd 5771 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,)  =  ( x  e.  ( RR*  X.  RR* )  |->  ( (,) `  x
) ) )
29 ovolf 19370 . . . . . 6  |-  vol * : ~P RR --> ( 0 [,]  +oo )
3029a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol * : ~P RR --> ( 0 [,]  +oo ) )
3130feqmptd 5771 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol *  =  ( y  e. 
~P RR  |->  ( vol
* `  y )
) )
32 fveq2 5720 . . . 4  |-  ( y  =  ( (,) `  x
)  ->  ( vol * `
 y )  =  ( vol * `  ( (,) `  x ) ) )
3327, 28, 31, 32fmptco 5893 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( vol *  o.  (,) )  =  ( x  e.  ( RR*  X.  RR* )  |->  ( vol * `  ( (,) `  x ) ) ) )
34 fveq2 5720 . . . 4  |-  ( x  =  ( F `  n )  ->  ( (,) `  x )  =  ( (,) `  ( F `  n )
) )
3534fveq2d 5724 . . 3  |-  ( x  =  ( F `  n )  ->  ( vol * `  ( (,) `  x ) )  =  ( vol * `  ( (,) `  ( F `
 n ) ) ) )
3610, 24, 33, 35fmptco 5893 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( vol *  o.  (,) )  o.  F
)  =  ( n  e.  NN  |->  ( vol
* `  ( (,) `  ( F `  n
) ) ) ) )
3720, 22, 363eqtr4d 2477 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( ( vol
*  o.  (,) )  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   <.cop 3809   class class class wbr 4204    e. cmpt 4258    X. cxp 4868    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   RRcr 8981   0cc0 8982    +oocpnf 9109   RR*cxr 9111    <_ cle 9113    - cmin 9283   NNcn 9992   (,)cioo 10908   [,)cico 10910   [,]cicc 10911   abscabs 12031   vol *covol 19351
This theorem is referenced by:  uniioombllem2  19467
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-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  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-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  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-rlim 12275  df-sum 12472  df-rest 13642  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-cmp 17442  df-ovol 19353  df-vol 19354
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