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Theorem ovolfs2 18926
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 18826 . . . . 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 18925 . . . . 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 15 . . . 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 3390 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 8876 . . . . . . . . . . 11  |-  RR  C_  RR*
6 xpss12 4792 . . . . . . . . . . 11  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
75, 5, 6mp2an 653 . . . . . . . . . 10  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
84, 7sstri 3188 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
108, 9sseldi 3178 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR*  X.  RR* )
)
11 1st2nd2 6159 . . . . . . . 8  |-  ( ( F `  n )  e.  ( RR*  X.  RR* )  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1210, 11syl 15 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
1312fveq2d 5529 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
14 df-ov 5861 . . . . . 6  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1513, 14syl6eqr 2333 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
1615fveq2d 5529 . . . 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 18830 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
193, 16, 183eqtr4rd 2326 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( vol * `  ( (,) `  ( F `  n )
) ) )
2019mpteq2dva 4106 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
n  e.  NN  |->  ( G `  n ) )  =  ( n  e.  NN  |->  ( vol
* `  ( (,) `  ( F `  n
) ) ) ) )
2117ovolfsf 18831 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) 
+oo ) )
2221feqmptd 5575 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( n  e.  NN  |->  ( G `  n ) ) )
23 id 19 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2423feqmptd 5575 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  =  ( n  e.  NN  |->  ( F `  n ) ) )
25 ioof 10741 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2625a1i 10 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,) : ( RR*  X.  RR* ) --> ~P RR )
27 ffvelrn 5663 . . . . 5  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  x  e.  ( RR*  X. 
RR* ) )  -> 
( (,) `  x
)  e.  ~P RR )
2826, 27sylan 457 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  ( RR*  X.  RR* )
)  ->  ( (,) `  x )  e.  ~P RR )
2926feqmptd 5575 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,)  =  ( x  e.  ( RR*  X.  RR* )  |->  ( (,) `  x
) ) )
30 ovolf 18841 . . . . . 6  |-  vol * : ~P RR --> ( 0 [,]  +oo )
3130a1i 10 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol * : ~P RR --> ( 0 [,]  +oo ) )
3231feqmptd 5575 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol *  =  ( y  e. 
~P RR  |->  ( vol
* `  y )
) )
33 fveq2 5525 . . . 4  |-  ( y  =  ( (,) `  x
)  ->  ( vol * `
 y )  =  ( vol * `  ( (,) `  x ) ) )
3428, 29, 32, 33fmptco 5691 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( vol *  o.  (,) )  =  ( x  e.  ( RR*  X.  RR* )  |->  ( vol * `  ( (,) `  x ) ) ) )
35 fveq2 5525 . . . 4  |-  ( x  =  ( F `  n )  ->  ( (,) `  x )  =  ( (,) `  ( F `  n )
) )
3635fveq2d 5529 . . 3  |-  ( x  =  ( F `  n )  ->  ( vol * `  ( (,) `  x ) )  =  ( vol * `  ( (,) `  ( F `
 n ) ) ) )
3710, 24, 34, 36fmptco 5691 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( vol *  o.  (,) )  o.  F
)  =  ( n  e.  NN  |->  ( vol
* `  ( (,) `  ( F `  n
) ) ) ) )
3820, 22, 373eqtr4d 2325 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( ( vol
*  o.  (,) )  o.  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RRcr 8736   0cc0 8737    +oocpnf 8864   RR*cxr 8866    <_ cle 8868    - cmin 9037   NNcn 9746   (,)cioo 10656   [,)cico 10658   [,]cicc 10659   abscabs 11719   vol *covol 18822
This theorem is referenced by:  uniioombllem2  18938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-ovol 18824  df-vol 18825
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