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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 | 1 | ffund 6393 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | fvvolicof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | 1 | fdmd 6398 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 4 | eqcomd 2803 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
6 | 3, 5 | eleqtrd 2887 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
7 | fvco 6633 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋))) | |
8 | 2, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋))) |
9 | icof 41043 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
10 | ffun 6392 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun [,) |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → Fun [,)) |
13 | 1, 3 | ffvelrnd 6724 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
14 | 9 | fdmi 6399 | . . . 4 ⊢ dom [,) = (ℝ* × ℝ*) |
15 | 13, 14 | syl6eleqr 2896 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,)) |
16 | fvco 6633 | . . 3 ⊢ ((Fun [,) ∧ (𝐹‘𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋)))) | |
17 | 12, 15, 16 | syl2anc 584 | . 2 ⊢ (𝜑 → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋)))) |
18 | df-ov 7026 | . . . . 5 ⊢ ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉)) |
20 | 1st2nd2 7591 | . . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
21 | 13, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) |
22 | 21 | eqcomd 2803 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
23 | 22 | fveq2d 6549 | . . . 4 ⊢ (𝜑 → ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ([,)‘(𝐹‘𝑋))) |
24 | 19, 23 | eqtr2d 2834 | . . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))) |
25 | 24 | fveq2d 6549 | . 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
26 | 8, 17, 25 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 𝒫 cpw 4459 〈cop 4484 × cxp 5448 dom cdm 5450 ∘ ccom 5454 Fun wfun 6226 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 1st c1st 7550 2nd c2nd 7551 ℝ*cxr 10527 [,)cico 12594 volcvol 23751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 df-xr 10532 df-ico 12598 |
This theorem is referenced by: voliooicof 41845 volicofmpt 41846 |
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