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Theorem ovolfsval 23458
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 6334 . . 3 (𝐺𝑁) = (((abs ∘ − ) ∘ 𝐹)‘𝑁)
3 fvco3 6419 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑁) = ((abs ∘ − )‘(𝐹𝑁)))
42, 3syl5eq 2817 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((abs ∘ − )‘(𝐹𝑁)))
5 inss2 3982 . . . . . . 7 ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
6 ffvelrn 6502 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ ( ≤ ∩ (ℝ × ℝ)))
75, 6sseldi 3750 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) ∈ (ℝ × ℝ))
8 1st2nd2 7358 . . . . . 6 ((𝐹𝑁) ∈ (ℝ × ℝ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
97, 8syl 17 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐹𝑁) = ⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
109fveq2d 6337 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩))
11 df-ov 6799 . . . 4 ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = ((abs ∘ − )‘⟨(1st ‘(𝐹𝑁)), (2nd ‘(𝐹𝑁))⟩)
1210, 11syl6eqr 2823 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))))
13 ovolfcl 23454 . . . . . . 7 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))))
1413simp1d 1136 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹𝑁)) ∈ ℝ)
1514recnd 10274 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (1st ‘(𝐹𝑁)) ∈ ℂ)
1613simp2d 1137 . . . . . 6 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹𝑁)) ∈ ℝ)
1716recnd 10274 . . . . 5 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (2nd ‘(𝐹𝑁)) ∈ ℂ)
18 eqid 2771 . . . . . 6 (abs ∘ − ) = (abs ∘ − )
1918cnmetdval 22794 . . . . 5 (((1st ‘(𝐹𝑁)) ∈ ℂ ∧ (2nd ‘(𝐹𝑁)) ∈ ℂ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))))
2015, 17, 19syl2anc 573 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))))
21 abssuble0 14276 . . . . 5 (((1st ‘(𝐹𝑁)) ∈ ℝ ∧ (2nd ‘(𝐹𝑁)) ∈ ℝ ∧ (1st ‘(𝐹𝑁)) ≤ (2nd ‘(𝐹𝑁))) → (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2213, 21syl 17 . . . 4 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (abs‘((1st ‘(𝐹𝑁)) − (2nd ‘(𝐹𝑁)))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2320, 22eqtrd 2805 . . 3 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((1st ‘(𝐹𝑁))(abs ∘ − )(2nd ‘(𝐹𝑁))) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
2412, 23eqtrd 2805 . 2 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → ((abs ∘ − )‘(𝐹𝑁)) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
254, 24eqtrd 2805 1 ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑁 ∈ ℕ) → (𝐺𝑁) = ((2nd ‘(𝐹𝑁)) − (1st ‘(𝐹𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  cin 3722  cop 4323   class class class wbr 4787   × cxp 5248  ccom 5254  wf 6026  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  cc 10140  cr 10141  cle 10281  cmin 10472  cn 11226  abscabs 14182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-sup 8508  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184
This theorem is referenced by:  ovolfsf  23459  ovollb2lem  23476  ovolunlem1a  23484  ovoliunlem1  23490  ovolshftlem1  23497  ovolscalem1  23501  ovolicc1  23504  ovolicc2lem4  23508  ioombl1lem3  23548  ovolfs2  23559  uniioovol  23567  uniioombllem3  23573
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