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Theorem ovolfs2 22272
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 22170 . . . . 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 22270 . . . . 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 17 . . . 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 3660 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 rexpssxrxp 9668 . . . . . . . . . 10  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
64, 5sstri 3451 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 ffvelrn 6007 . . . . . . . . 9  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
86, 7sseldi 3440 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( RR*  X.  RR* )
)
9 1st2nd2 6821 . . . . . . . 8  |-  ( ( F `  n )  e.  ( RR*  X.  RR* )  ->  ( F `  n )  =  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. )
108, 9syl 17 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( F `  n )  =  <. ( 1st `  ( F `  n )
) ,  ( 2nd `  ( F `  n
) ) >. )
1110fveq2d 5853 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( (,) `  <. ( 1st `  ( F `
 n ) ) ,  ( 2nd `  ( F `  n )
) >. ) )
12 df-ov 6281 . . . . . 6  |-  ( ( 1st `  ( F `
 n ) ) (,) ( 2nd `  ( F `  n )
) )  =  ( (,) `  <. ( 1st `  ( F `  n ) ) ,  ( 2nd `  ( F `  n )
) >. )
1311, 12syl6eqr 2461 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( (,) `  ( F `  n ) )  =  ( ( 1st `  ( F `  n )
) (,) ( 2nd `  ( F `  n
) ) ) )
1413fveq2d 5853 . . . 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 22174 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( ( 2nd `  ( F `  n
) )  -  ( 1st `  ( F `  n ) ) ) )
173, 14, 163eqtr4rd 2454 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  n  e.  NN )  ->  ( G `  n )  =  ( vol* `  ( (,) `  ( F `  n )
) ) )
1817mpteq2dva 4481 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
n  e.  NN  |->  ( G `  n ) )  =  ( n  e.  NN  |->  ( vol* `  ( (,) `  ( F `  n
) ) ) ) )
1915ovolfsf 22175 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G : NN --> ( 0 [,) +oo ) )
2019feqmptd 5902 . 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 5902 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  =  ( n  e.  NN  |->  ( F `  n ) ) )
23 ioof 11676 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2423a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,) : ( RR*  X.  RR* ) --> ~P RR )
2524ffvelrnda 6009 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  x  e.  ( RR*  X.  RR* )
)  ->  ( (,) `  x )  e.  ~P RR )
2624feqmptd 5902 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (,)  =  ( x  e.  ( RR*  X.  RR* )  |->  ( (,) `  x
) ) )
27 ovolf 22185 . . . . . 6  |-  vol* : ~P RR --> ( 0 [,] +oo )
2827a1i 11 . . . . 5  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol* : ~P RR --> ( 0 [,] +oo ) )
2928feqmptd 5902 . . . 4  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  vol*  =  ( y  e. 
~P RR  |->  ( vol* `  y )
) )
30 fveq2 5849 . . . 4  |-  ( y  =  ( (,) `  x
)  ->  ( vol* `  y )  =  ( vol* `  ( (,) `  x ) ) )
3125, 26, 29, 30fmptco 6043 . . 3  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  ( vol*  o.  (,) )  =  ( x  e.  ( RR*  X.  RR* )  |->  ( vol* `  ( (,) `  x ) ) ) )
32 fveq2 5849 . . . 4  |-  ( x  =  ( F `  n )  ->  ( (,) `  x )  =  ( (,) `  ( F `  n )
) )
3332fveq2d 5853 . . 3  |-  ( x  =  ( F `  n )  ->  ( vol* `  ( (,) `  x ) )  =  ( vol* `  ( (,) `  ( F `
 n ) ) ) )
348, 22, 31, 33fmptco 6043 . 2  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( vol*  o.  (,) )  o.  F
)  =  ( n  e.  NN  |->  ( vol* `  ( (,) `  ( F `  n
) ) ) ) )
3518, 20, 343eqtr4d 2453 1  |-  ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  G  =  ( ( vol*  o.  (,) )  o.  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3413   ~Pcpw 3955   <.cop 3978   class class class wbr 4395    |-> cmpt 4453    X. cxp 4821    o. ccom 4827   -->wf 5565   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   RRcr 9521   0cc0 9522   +oocpnf 9655   RR*cxr 9657    <_ cle 9659    - cmin 9841   NNcn 10576   (,)cioo 11582   [,)cico 11584   [,]cicc 11585   abscabs 13216   vol*covol 22166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-rest 15037  df-topgen 15058  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-top 19691  df-bases 19693  df-topon 19694  df-cmp 20180  df-ovol 22168  df-vol 22169
This theorem is referenced by:  uniioombllem2  22284
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