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Theorem ovolfsval 23789
Description: The value of the interval length function. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
Ref Expression
ovolfs.1 𝐺 = ((abs ∘ − ) ∘ 𝐹)
Assertion
Ref Expression
ovolfsval ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))

Proof of Theorem ovolfsval
StepHypRef Expression
1 ovolfs.1 . . . 4 𝐺 = ((abs ∘ − ) ∘ 𝐹)
21fveq1i 6505 . . 3 (𝐺𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3 fvco3 6594 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹𝑁)))
42, 3syl5eq 2828 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((abs ∘ − )‘(𝐹𝑁)))
5 ffvelrn 6680 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
65elin2d 4067 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
7 1st2nd2 7546 . . . . . 6 ((𝐹𝑁) ∈ (ℝ × ℝ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
86, 7syl 17 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
98fveq2d 6508 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩))
10 df-ov 6985 . . . 4 ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
119, 10syl6eqr 2834 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))))
12 ovolfcl 23785 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
1312simp1d 1123 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹𝑁)) ∈ ℝ)
1413recnd 10474 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹𝑁)) ∈ ℂ)
1512simp2d 1124 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹𝑁)) ∈ ℝ)
1615recnd 10474 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹𝑁)) ∈ ℂ)
17 eqid 2780 . . . . . 6 (abs ∘ − ) = (abs ∘ − )
1817cnmetdval 23097 . . . . 5 (((1st ‘(𝐹𝑁)) ∈ ℂ ∧ (2nd ‘(𝐹𝑁)) ∈ ℂ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))))
1914, 16, 18syl2anc 576 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))))
20 abssuble0 14555 . . . . 5 (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))) → (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2112, 20syl 17 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2219, 21eqtrd 2816 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2311, 22eqtrd 2816 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
244, 23eqtrd 2816 1 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1069   = wceq 1508  wcel 2051  cin 3830  cop 4450   class class class wbr 4934   × cxp 5409  ccom 5415  wf 6189  cfv 6193  (class class class)co 6982  1st c1st 7505  2nd c2nd 7506  cc 10339  cr 10340  cle 10481  cmin 10676  cn 11445  abscabs 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-cnex 10397  ax-resscn 10398  ax-1cn 10399  ax-icn 10400  ax-addcl 10401  ax-addrcl 10402  ax-mulcl 10403  ax-mulrcl 10404  ax-mulcom 10405  ax-addass 10406  ax-mulass 10407  ax-distr 10408  ax-i2m1 10409  ax-1ne0 10410  ax-1rid 10411  ax-rnegex 10412  ax-rrecex 10413  ax-cnre 10414  ax-pre-lttri 10415  ax-pre-lttrn 10416  ax-pre-ltadd 10417  ax-pre-mulgt0 10418  ax-pre-sup 10419
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-om 7403  df-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-er 8095  df-en 8313  df-dom 8314  df-sdom 8315  df-sup 8707  df-pnf 10482  df-mnf 10483  df-xr 10484  df-ltxr 10485  df-le 10486  df-sub 10678  df-neg 10679  df-div 11105  df-nn 11446  df-2 11509  df-3 11510  df-n0 11714  df-z 11800  df-uz 12065  df-rp 12211  df-seq 13191  df-exp 13251  df-cj 14325  df-re 14326  df-im 14327  df-sqrt 14461  df-abs 14462
This theorem is referenced by:  ovolfsf  23790  ovollb2lem  23807  ovolunlem1a  23815  ovoliunlem1  23821  ovolshftlem1  23828  ovolscalem1  23832  ovolicc1  23835  ovolicc2lem4  23839  ioombl1lem3  23879  ovolfs2  23890  uniioovol  23898  uniioombllem3  23904
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