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Theorem ovolfs2 19331
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 19231 . . . . 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 19330 . . . . 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 3506 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 9063 . . . . . . . . . . 11  |-  RR  C_  RR*
6 xpss12 4922 . . . . . . . . . . 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 3301 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 ffvelrn 5808 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
108, 9sseldi 3290 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR*  X.  RR* )
)
11 1st2nd2 6326 . . . . . . . 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 5673 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
14 df-ov 6024 . . . . . 6  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1513, 14syl6eqr 2438 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
1615fveq2d 5673 . . . 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 19235 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
193, 16, 183eqtr4rd 2431 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( vol * `  ( (,) `  ( F `  n )
) ) )
2019mpteq2dva 4237 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
n  e.  NN  |->  ( G `  n ) )  =  ( n  e.  NN  |->  ( vol
* `  ( (,) `  ( F `  n
) ) ) ) )
2117ovolfsf 19236 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) 
+oo ) )
2221feqmptd 5719 . 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 5719 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  =  ( n  e.  NN  |->  ( F `  n ) ) )
25 ioof 10935 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2625a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,) : ( RR*  X.  RR* ) --> ~P RR )
2726ffvelrnda 5810 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  ( RR*  X.  RR* )
)  ->  ( (,) `  x )  e.  ~P RR )
2826feqmptd 5719 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,)  =  ( x  e.  ( RR*  X.  RR* )  |->  ( (,) `  x
) ) )
29 ovolf 19246 . . . . . 6  |-  vol * : ~P RR --> ( 0 [,]  +oo )
3029a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol * : ~P RR --> ( 0 [,]  +oo ) )
3130feqmptd 5719 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol *  =  ( y  e. 
~P RR  |->  ( vol
* `  y )
) )
32 fveq2 5669 . . . 4  |-  ( y  =  ( (,) `  x
)  ->  ( vol * `
 y )  =  ( vol * `  ( (,) `  x ) ) )
3327, 28, 31, 32fmptco 5841 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( vol *  o.  (,) )  =  ( x  e.  ( RR*  X.  RR* )  |->  ( vol * `  ( (,) `  x ) ) ) )
34 fveq2 5669 . . . 4  |-  ( x  =  ( F `  n )  ->  ( (,) `  x )  =  ( (,) `  ( F `  n )
) )
3534fveq2d 5673 . . 3  |-  ( x  =  ( F `  n )  ->  ( vol * `  ( (,) `  x ) )  =  ( vol * `  ( (,) `  ( F `
 n ) ) ) )
3610, 24, 33, 35fmptco 5841 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( vol *  o.  (,) )  o.  F
)  =  ( n  e.  NN  |->  ( vol
* `  ( (,) `  ( F `  n
) ) ) ) )
3720, 22, 363eqtr4d 2430 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 1649    e. wcel 1717    i^i cin 3263    C_ wss 3264   ~Pcpw 3743   <.cop 3761   class class class wbr 4154    e. cmpt 4208    X. cxp 4817    o. ccom 4823   -->wf 5391   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288   RRcr 8923   0cc0 8924    +oocpnf 9051   RR*cxr 9053    <_ cle 9055    - cmin 9224   NNcn 9933   (,)cioo 10849   [,)cico 10851   [,]cicc 10852   abscabs 11967   vol *covol 19227
This theorem is referenced by:  uniioombllem2  19343
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-rest 13578  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-top 16887  df-bases 16889  df-topon 16890  df-cmp 17373  df-ovol 19229  df-vol 19230
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