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Theorem ovolfsval 19359
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 5721 . . 3  |-  ( G `
 N )  =  ( ( ( abs 
o.  -  )  o.  F ) `  N
)
3 fvco3 5792 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
42, 3syl5eq 2479 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
5 inss2 3554 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ffvelrn 5860 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
75, 6sseldi 3338 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
8 1st2nd2 6378 . . . . . 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 5724 . . . 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 6076 . . . 4  |-  ( ( 1st `  ( F `
 N ) ) ( abs  o.  -  ) ( 2nd `  ( F `  N )
) )  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  N
) ) ,  ( 2nd `  ( F `
 N ) )
>. )
1210, 11syl6eqr 2485 . . 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 19355 . . . . . . 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 969 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  RR )
1514recnd 9106 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  CC )
1613simp2d 970 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  RR )
1716recnd 9106 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  CC )
18 eqid 2435 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 18797 . . . . 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 643 . . . 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 12124 . . . . 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 2467 . . 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 2467 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( abs  o.  -  ) `  ( F `  N
) )  =  ( ( 2nd `  ( F `  N )
)  -  ( 1st `  ( F `  N
) ) ) )
254, 24eqtrd 2467 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( 2nd `  ( F `  N
) )  -  ( 1st `  ( F `  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311   <.cop 3809   class class class wbr 4204    X. cxp 4868    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   CCcc 8980   RRcr 8981    <_ cle 9113    - cmin 9283   NNcn 9992   abscabs 12031
This theorem is referenced by:  ovolfsf  19360  ovollb2lem  19376  ovolunlem1a  19384  ovoliunlem1  19390  ovolshftlem1  19397  ovolscalem1  19401  ovolicc1  19404  ovolicc2lem4  19408  ioombl1lem3  19446  ovolfs2  19455  uniioovol  19463  uniioombllem3  19469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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