Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvvolicof | Structured version Visualization version GIF version |
Description: The function value of the Lebesgue measure of a left-closed right-open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
fvvolicof.f | ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
fvvolicof.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fvvolicof | ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvvolicof.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) | |
2 | ffun 5961 | . . . 4 ⊢ (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
4 | fvvolicof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
5 | fdm 5964 | . . . . . 6 ⊢ (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴) | |
6 | 1, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
7 | 6 | eqcomd 2616 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
8 | 4, 7 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
9 | fvco 6184 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋))) | |
10 | 3, 8, 9 | syl2anc 691 | . 2 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋))) |
11 | icof 38406 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
12 | ffun 5961 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Fun [,) |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → Fun [,)) |
15 | 1, 4 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
16 | 11 | fdmi 5965 | . . . 4 ⊢ dom [,) = (ℝ* × ℝ*) |
17 | 15, 16 | syl6eleqr 2699 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,)) |
18 | fvco 6184 | . . 3 ⊢ ((Fun [,) ∧ (𝐹‘𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋)))) | |
19 | 14, 17, 18 | syl2anc 691 | . 2 ⊢ (𝜑 → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋)))) |
20 | df-ov 6552 | . . . . 5 ⊢ ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉)) |
22 | 1st2nd2 7096 | . . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
23 | 15, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) |
24 | 23 | eqcomd 2616 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
25 | 24 | fveq2d 6107 | . . . 4 ⊢ (𝜑 → ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ([,)‘(𝐹‘𝑋))) |
26 | 21, 25 | eqtr2d 2645 | . . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))) |
27 | 26 | fveq2d 6107 | . 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
28 | 10, 19, 27 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 𝒫 cpw 4108 〈cop 4131 × cxp 5036 dom cdm 5038 ∘ ccom 5042 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ℝ*cxr 9952 [,)cico 12048 volcvol 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-xr 9957 df-ico 12052 |
This theorem is referenced by: voliooicof 38889 volicofmpt 38890 |
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