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Theorem ovolfs2 21066
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 20965 . . . . 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 21064 . . . . 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 3586 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 rexpssxrxp 9443 . . . . . . . . . 10  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
64, 5sstri 3380 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 ffvelrn 5856 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
86, 7sseldi 3369 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR*  X.  RR* )
)
9 1st2nd2 6628 . . . . . . . 8  |-  ( ( F `  n )  e.  ( RR*  X.  RR* )  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108, 9syl 16 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
1110fveq2d 5710 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
12 df-ov 6109 . . . . . 6  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1311, 12syl6eqr 2493 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
1413fveq2d 5710 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( vol* `  ( (,) `  ( F `  n
) ) )  =  ( vol* `  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) ) )
15 ovolfs2.1 . . . . 5  |-  G  =  ( ( abs  o.  -  )  o.  F
)
1615ovolfsval 20969 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
173, 14, 163eqtr4rd 2486 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( vol* `  ( (,) `  ( F `  n )
) ) )
1817mpteq2dva 4393 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
n  e.  NN  |->  ( G `  n ) )  =  ( n  e.  NN  |->  ( vol* `  ( (,) `  ( F `  n
) ) ) ) )
1915ovolfsf 20970 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) +oo ) )
2019feqmptd 5759 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( n  e.  NN  |->  ( G `  n ) ) )
21 id 22 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2221feqmptd 5759 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  =  ( n  e.  NN  |->  ( F `  n ) ) )
23 ioof 11402 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2423a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,) : ( RR*  X.  RR* ) --> ~P RR )
2524ffvelrnda 5858 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  ( RR*  X.  RR* )
)  ->  ( (,) `  x )  e.  ~P RR )
2624feqmptd 5759 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,)  =  ( x  e.  ( RR*  X.  RR* )  |->  ( (,) `  x
) ) )
27 ovolf 20980 . . . . . 6  |-  vol* : ~P RR --> ( 0 [,] +oo )
2827a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol* : ~P RR --> ( 0 [,] +oo ) )
2928feqmptd 5759 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol*  =  ( y  e. 
~P RR  |->  ( vol* `  y )
) )
30 fveq2 5706 . . . 4  |-  ( y  =  ( (,) `  x
)  ->  ( vol* `  y )  =  ( vol* `  ( (,) `  x ) ) )
3125, 26, 29, 30fmptco 5891 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( vol*  o.  (,) )  =  ( x  e.  ( RR*  X.  RR* )  |->  ( vol* `  ( (,) `  x ) ) ) )
32 fveq2 5706 . . . 4  |-  ( x  =  ( F `  n )  ->  ( (,) `  x )  =  ( (,) `  ( F `  n )
) )
3332fveq2d 5710 . . 3  |-  ( x  =  ( F `  n )  ->  ( vol* `  ( (,) `  x ) )  =  ( vol* `  ( (,) `  ( F `
 n ) ) ) )
348, 22, 31, 33fmptco 5891 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( vol*  o.  (,) )  o.  F
)  =  ( n  e.  NN  |->  ( vol* `  ( (,) `  ( F `  n
) ) ) ) )
3518, 20, 343eqtr4d 2485 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( ( vol*  o.  (,) )  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3342   ~Pcpw 3875   <.cop 3898   class class class wbr 4307    e. cmpt 4365    X. cxp 4853    o. ccom 4859   -->wf 5429   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591   RRcr 9296   0cc0 9297   +oocpnf 9430   RR*cxr 9432    <_ cle 9434    - cmin 9610   NNcn 10337   (,)cioo 11315   [,)cico 11317   [,]cicc 11318   abscabs 12738   vol*covol 20961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-rlim 12982  df-sum 13179  df-rest 14376  df-topgen 14397  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-top 18518  df-bases 18520  df-topon 18521  df-cmp 19005  df-ovol 20963  df-vol 20964
This theorem is referenced by:  uniioombllem2  21078
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