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Theorem ovolfsval 24170
 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 6659 . . 3 (𝐺𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3 fvco3 6751 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹𝑁)))
42, 3syl5eq 2805 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((abs ∘ − )‘(𝐹𝑁)))
5 ffvelrn 6840 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
65elin2d 4104 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
7 1st2nd2 7732 . . . . . 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 6662 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩))
10 df-ov 7153 . . . 4 ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
119, 10eqtr4di 2811 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))))
12 ovolfcl 24166 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
1312simp1d 1139 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹𝑁)) ∈ ℝ)
1413recnd 10707 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹𝑁)) ∈ ℂ)
1512simp2d 1140 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹𝑁)) ∈ ℝ)
1615recnd 10707 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹𝑁)) ∈ ℂ)
17 eqid 2758 . . . . . 6 (abs ∘ − ) = (abs ∘ − )
1817cnmetdval 23472 . . . . 5 (((1st ‘(𝐹𝑁)) ∈ ℂ ∧ (2nd ‘(𝐹𝑁)) ∈ ℂ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))))
1914, 16, 18syl2anc 587 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))))
20 abssuble0 14736 . . . . 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 2793 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2311, 22eqtrd 2793 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
244, 23eqtrd 2793 1 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3857  ⟨cop 4528   class class class wbr 5032   × cxp 5522   ∘ ccom 5528  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692  ℂcc 10573  ℝcr 10574   ≤ cle 10714   − cmin 10908  ℕcn 11674  abscabs 14641 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-sup 8939  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-n0 11935  df-z 12021  df-uz 12283  df-rp 12431  df-seq 13419  df-exp 13480  df-cj 14506  df-re 14507  df-im 14508  df-sqrt 14642  df-abs 14643 This theorem is referenced by:  ovolfsf  24171  ovollb2lem  24188  ovolunlem1a  24196  ovoliunlem1  24202  ovolshftlem1  24209  ovolscalem1  24213  ovolicc1  24216  ovolicc2lem4  24220  ioombl1lem3  24260  ovolfs2  24271  uniioovol  24279  uniioombllem3  24285
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