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Theorem ovolfsval 20954
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 5692 . . 3  |-  ( G `
 N )  =  ( ( ( abs 
o.  -  )  o.  F ) `  N
)
3 fvco3 5768 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( ( abs  o.  -  )  o.  F
) `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
42, 3syl5eq 2487 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( G `  N )  =  ( ( abs 
o.  -  ) `  ( F `  N ) ) )
5 inss2 3571 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 ffvelrn 5841 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
75, 6sseldi 3354 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( F `  N )  e.  ( RR  X.  RR ) )
8 1st2nd2 6613 . . . . . 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 5695 . . . 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 6094 . . . 4  |-  ( ( 1st `  ( F `
 N ) ) ( abs  o.  -  ) ( 2nd `  ( F `  N )
) )  =  ( ( abs  o.  -  ) `  <. ( 1st `  ( F `  N
) ) ,  ( 2nd `  ( F `
 N ) )
>. )
1210, 11syl6eqr 2493 . . 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 20950 . . . . . . 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 1000 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  RR )
1514recnd 9412 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 1st `  ( F `  N ) )  e.  CC )
1613simp2d 1001 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  ( 2nd `  ( F `  N ) )  e.  RR )
1716recnd 9412 . . . . 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 20350 . . . . 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 12816 . . . . 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 2475 . . 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 2475 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  N  e.  NN )  ->  (
( abs  o.  -  ) `  ( F `  N
) )  =  ( ( 2nd `  ( F `  N )
)  -  ( 1st `  ( F `  N
) ) ) )
254, 24eqtrd 2475 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 965    = wceq 1369    e. wcel 1756    i^i cin 3327   <.cop 3883   class class class wbr 4292    X. cxp 4838    o. ccom 4844   -->wf 5414   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   CCcc 9280   RRcr 9281    <_ cle 9419    - cmin 9595   NNcn 10322   abscabs 12723
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-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725
This theorem is referenced by:  ovolfsf  20955  ovollb2lem  20971  ovolunlem1a  20979  ovoliunlem1  20985  ovolshftlem1  20992  ovolscalem1  20996  ovolicc1  20999  ovolicc2lem4  21003  ioombl1lem3  21041  ovolfs2  21051  uniioovol  21059  uniioombllem3  21065
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