Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvvolioof Structured version   Visualization version   GIF version

Theorem fvvolioof 40723
Description: The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
fvvolioof.f (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
fvvolioof.x (𝜑𝑋𝐴)
Assertion
Ref Expression
fvvolioof (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))

Proof of Theorem fvvolioof
StepHypRef Expression
1 fvvolioof.f . . . 4 (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
2 ffun 6188 . . . 4 (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 fvvolioof.x . . . 4 (𝜑𝑋𝐴)
5 fdm 6191 . . . . . 6 (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴)
61, 5syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
76eqcomd 2777 . . . 4 (𝜑𝐴 = dom 𝐹)
84, 7eleqtrd 2852 . . 3 (𝜑𝑋 ∈ dom 𝐹)
9 fvco 6416 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
103, 8, 9syl2anc 573 . 2 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
11 ioof 12477 . . . . 5 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
12 ffun 6188 . . . . 5 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
1311, 12ax-mp 5 . . . 4 Fun (,)
1413a1i 11 . . 3 (𝜑 → Fun (,))
151, 4ffvelrnd 6503 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
1611fdmi 6192 . . . 4 dom (,) = (ℝ* × ℝ*)
1715, 16syl6eleqr 2861 . . 3 (𝜑 → (𝐹𝑋) ∈ dom (,))
18 fvco 6416 . . 3 ((Fun (,) ∧ (𝐹𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
1914, 17, 18syl2anc 573 . 2 (𝜑 → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
20 df-ov 6796 . . . . 5 ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2120a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
22 1st2nd2 7354 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2315, 22syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2423eqcomd 2777 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2524fveq2d 6336 . . . 4 (𝜑 → ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ((,)‘(𝐹𝑋)))
2621, 25eqtr2d 2806 . . 3 (𝜑 → ((,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))))
2726fveq2d 6336 . 2 (𝜑 → (vol‘((,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
2810, 19, 273eqtrd 2809 1 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  𝒫 cpw 4297  cop 4322   × cxp 5247  dom cdm 5249  ccom 5253  Fun wfun 6025  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  cr 10137  *cxr 10275  (,)cioo 12380  volcvol 23451
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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-pre-lttri 10212  ax-pre-lttrn 10213
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-po 5170  df-so 5171  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-ioo 12384
This theorem is referenced by:  volioofmpt  40728  voliooicof  40730
  Copyright terms: Public domain W3C validator