Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfresmf | Structured version Visualization version GIF version |
Description: A Real valued, measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
mbfresmf.1 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
mbfresmf.2 | ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
mbfresmf.3 | ⊢ 𝑆 = dom vol |
Ref | Expression |
---|---|
mbfresmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | mbfresmf.3 | . . . 4 ⊢ 𝑆 = dom vol | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = dom vol) |
4 | dmvolsal 39240 | . . . 4 ⊢ dom vol ∈ SAlg | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom vol ∈ SAlg) |
6 | 3, 5 | eqeltrd 2688 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
7 | mbfresmf.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
8 | mbfdmssre 38893 | . . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
10 | 2 | unieqi 4381 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom vol |
11 | unidmvol 23116 | . . . 4 ⊢ ∪ dom vol = ℝ | |
12 | 10, 11 | eqtri 2632 | . . 3 ⊢ ∪ 𝑆 = ℝ |
13 | 9, 12 | syl6sseqr 3615 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
14 | mbff 23200 | . . . . 5 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | |
15 | ffn 5958 | . . . . 5 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹) | |
16 | 7, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
17 | mbfresmf.2 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) | |
18 | 16, 17 | jca 553 | . . 3 ⊢ (𝜑 → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) |
19 | df-f 5808 | . . 3 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) | |
20 | 18, 19 | sylibr 223 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:dom 𝐹⟶ℝ) |
22 | rexr 9964 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
23 | 22 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
24 | 21, 23 | preimaioomnf 39606 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}) |
25 | 24 | eqcomd 2616 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
26 | 4 | elexi 3186 | . . . . . 6 ⊢ dom vol ∈ V |
27 | 2, 26 | eqeltri 2684 | . . . . 5 ⊢ 𝑆 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ V) |
29 | 7 | dmexd 38417 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
30 | 29 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → dom 𝐹 ∈ V) |
31 | mbfima 23205 | . . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:dom 𝐹⟶ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) | |
32 | 7, 20, 31 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) |
33 | 32, 3 | eleqtrrd 2691 | . . . . 5 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
34 | 33 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
35 | cnvimass 5404 | . . . . 5 ⊢ (◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 | |
36 | dfss 3555 | . . . . . 6 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 ↔ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) | |
37 | 36 | biimpi 205 | . . . . 5 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 → (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) |
38 | 35, 37 | ax-mp 5 | . . . 4 ⊢ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹) |
39 | 28, 30, 34, 38 | elrestd 38322 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
40 | 25, 39 | eqeltrd 2688 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
41 | 1, 6, 13, 20, 40 | issmfd 39621 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 (,)cioo 12046 ↾t crest 15904 volcvol 23039 MblFncmbf 23189 SAlgcsalg 39204 SMblFncsmblfn 39586 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-rest 15906 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 df-mbf 23194 df-salg 39205 df-smblfn 39587 |
This theorem is referenced by: mbfpsssmf 39669 |
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