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Theorem ovolfsval 21610
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 5858 . . 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 2513 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
5 inss2 3712 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ffvelrn 6010 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
75, 6sseldi 3495 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
8 1st2nd2 6811 . . . . . 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 5861 . . . 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 6278 . . . 4  |-  ( ( 1st `  ( F `
 N ) ) ( abs  o.  -  ) ( 2nd `  ( F `  N )
) )  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  N
) ) ,  ( 2nd `  ( F `
 N ) )
>. )
1210, 11syl6eqr 2519 . . 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 21606 . . . . . . 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 1003 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  RR )
1514recnd 9611 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  CC )
1613simp2d 1004 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  RR )
1716recnd 9611 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  CC )
18 eqid 2460 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 21006 . . . . 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 13110 . . . . 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 2501 . . 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 2501 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( abs  o.  -  ) `  ( F `  N
) )  =  ( ( 2nd `  ( F `  N )
)  -  ( 1st `  ( F `  N
) ) ) )
254, 24eqtrd 2501 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 968    = wceq 1374    e. wcel 1762    i^i cin 3468   <.cop 4026   class class class wbr 4440    X. cxp 4990    o. ccom 4996   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   CCcc 9479   RRcr 9480    <_ cle 9618    - cmin 9794   NNcn 10525   abscabs 13017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019
This theorem is referenced by:  ovolfsf  21611  ovollb2lem  21627  ovolunlem1a  21635  ovoliunlem1  21641  ovolshftlem1  21648  ovolscalem1  21652  ovolicc1  21655  ovolicc2lem4  21659  ioombl1lem3  21698  ovolfs2  21708  uniioovol  21716  uniioombllem3  21722
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