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Theorem ovolfsval 21859
Description: The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
Ref Expression
ovolfs.1  |-  G  =  ( ( abs  o.  -  )  o.  F
)
Assertion
Ref Expression
ovolfsval  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( 2nd `  ( F `  N
) )  -  ( 1st `  ( F `  N ) ) ) )

Proof of Theorem ovolfsval
StepHypRef Expression
1 ovolfs.1 . . . 4  |-  G  =  ( ( abs  o.  -  )  o.  F
)
21fveq1i 5857 . . 3  |-  ( G `
 N )  =  ( ( ( abs 
o.  -  )  o.  F ) `  N
)
3 fvco3 5935 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
42, 3syl5eq 2496 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
5 inss2 3704 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ffvelrn 6014 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
75, 6sseldi 3487 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
8 1st2nd2 6822 . . . . . 6  |-  ( ( F `  N )  e.  ( RR  X.  RR )  ->  ( F `
 N )  = 
<. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >. )
97, 8syl 16 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  =  <. ( 1st `  ( F `  N )
) ,  ( 2nd `  ( F `  N
) ) >. )
109fveq2d 5860 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( abs  o.  -  ) `  ( F `  N
) )  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  N
) ) ,  ( 2nd `  ( F `
 N ) )
>. ) )
11 df-ov 6284 . . . 4  |-  ( ( 1st `  ( F `
 N ) ) ( abs  o.  -  ) ( 2nd `  ( F `  N )
) )  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  N
) ) ,  ( 2nd `  ( F `
 N ) )
>. )
1210, 11syl6eqr 2502 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( abs  o.  -  ) `  ( F `  N
) )  =  ( ( 1st `  ( F `  N )
) ( abs  o.  -  ) ( 2nd `  ( F `  N
) ) ) )
13 ovolfcl 21855 . . . . . . 7  |-  ( ( 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 )
) ) )
1413simp1d 1009 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  RR )
1514recnd 9625 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  CC )
1613simp2d 1010 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  RR )
1716recnd 9625 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  CC )
18 eqid 2443 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 21255 . . . . 5  |-  ( ( ( 1st `  ( F `  N )
)  e.  CC  /\  ( 2nd `  ( F `
 N ) )  e.  CC )  -> 
( ( 1st `  ( F `  N )
) ( abs  o.  -  ) ( 2nd `  ( F `  N
) ) )  =  ( abs `  (
( 1st `  ( F `  N )
)  -  ( 2nd `  ( F `  N
) ) ) ) )
2015, 17, 19syl2anc 661 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( 1st `  ( F `  N )
) ( abs  o.  -  ) ( 2nd `  ( F `  N
) ) )  =  ( abs `  (
( 1st `  ( F `  N )
)  -  ( 2nd `  ( F `  N
) ) ) ) )
21 abssuble0 13142 . . . . 5  |-  ( ( ( 1st `  ( F `  N )
)  e.  RR  /\  ( 2nd `  ( F `
 N ) )  e.  RR  /\  ( 1st `  ( F `  N ) )  <_ 
( 2nd `  ( F `  N )
) )  ->  ( abs `  ( ( 1st `  ( F `  N
) )  -  ( 2nd `  ( F `  N ) ) ) )  =  ( ( 2nd `  ( F `
 N ) )  -  ( 1st `  ( F `  N )
) ) )
2213, 21syl 16 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( abs `  ( ( 1st `  ( F `  N
) )  -  ( 2nd `  ( F `  N ) ) ) )  =  ( ( 2nd `  ( F `
 N ) )  -  ( 1st `  ( F `  N )
) ) )
2320, 22eqtrd 2484 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( 1st `  ( F `  N )
) ( abs  o.  -  ) ( 2nd `  ( F `  N
) ) )  =  ( ( 2nd `  ( F `  N )
)  -  ( 1st `  ( F `  N
) ) ) )
2412, 23eqtrd 2484 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( abs  o.  -  ) `  ( F `  N
) )  =  ( ( 2nd `  ( F `  N )
)  -  ( 1st `  ( F `  N
) ) ) )
254, 24eqtrd 2484 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( 2nd `  ( F `  N
) )  -  ( 1st `  ( F `  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    i^i cin 3460   <.cop 4020   class class class wbr 4437    X. cxp 4987    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   CCcc 9493   RRcr 9494    <_ cle 9632    - cmin 9810   NNcn 10543   abscabs 13048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-rp 11231  df-seq 12089  df-exp 12148  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050
This theorem is referenced by:  ovolfsf  21860  ovollb2lem  21876  ovolunlem1a  21884  ovoliunlem1  21890  ovolshftlem1  21897  ovolscalem1  21901  ovolicc1  21904  ovolicc2lem4  21908  ioombl1lem3  21947  ovolfs2  21957  uniioovol  21965  uniioombllem3  21971
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