Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > issmflelem | Structured version Visualization version GIF version |
Description: The predicate "𝐹 is a measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all right closed intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (ii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmflelem.x | ⊢ Ⅎ𝑥𝜑 |
issmflelem.a | ⊢ Ⅎ𝑎𝜑 |
issmflelem.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmflelem.d | ⊢ 𝐷 = dom 𝐹 |
issmflelem.i | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
issmflelem.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
issmflelem.l | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) |
Ref | Expression |
---|---|
issmflelem | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmflelem.i | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
2 | issmflelem.f | . . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
3 | issmflelem.s | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | 3 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝑆 ∈ SAlg) |
5 | simpr 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆) | |
6 | 4, 5 | restuni4 38336 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
7 | 6 | eqcomd 2616 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 = ∪ (𝑆 ↾t 𝐷)) |
8 | 1, 7 | mpdan 699 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ (𝑆 ↾t 𝐷)) |
9 | 8 | rabeqd 38304 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) |
11 | issmflelem.x | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
12 | nfv 1830 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ | |
13 | 11, 12 | nfan 1816 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ) |
14 | issmflelem.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
15 | nfv 1830 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
16 | 14, 15 | nfan 1816 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
17 | 3 | uniexd 38310 | . . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
18 | 17 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V) |
19 | 18, 5 | ssexd 4733 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V) |
20 | eqid 2610 | . . . . . . . . 9 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
21 | 4, 19, 20 | subsalsal 39253 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → (𝑆 ↾t 𝐷) ∈ SAlg) |
22 | 1, 21 | mpdan 699 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
23 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg) |
24 | eqid 2610 | . . . . . 6 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷) | |
25 | simpr 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) | |
26 | 1, 6 | mpdan 699 | . . . . . . . . . . 11 ⊢ (𝜑 → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
27 | 26 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
28 | 25, 27 | eleqtrd 2690 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ 𝐷) |
29 | 2 | ffvelrnda 6267 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ) |
30 | 28, 29 | syldan 486 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ) |
31 | 30 | rexrd 9968 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) |
32 | 31 | adantlr 747 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) |
33 | 26 | rabeqd 38304 | . . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}) |
34 | 33 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}) |
35 | issmflelem.l | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) | |
36 | 34, 35 | eqeltrd 2688 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) |
37 | 36 | adantlr 747 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) |
38 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ) | |
39 | 13, 16, 23, 24, 32, 37, 38 | salpreimalelt 39615 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
40 | 10, 39 | eqeltrd 2688 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
41 | 40 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
42 | 1, 2, 41 | 3jca 1235 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))) |
43 | issmflelem.d | . . 3 ⊢ 𝐷 = dom 𝐹 | |
44 | 3, 43 | issmf 39614 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)))) |
45 | 42, 44 | mpbird 246 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ↾t crest 15904 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-ac2 9168 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-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-iin 4458 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-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 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-card 8648 df-acn 8651 df-ac 8822 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-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ioo 12050 df-ico 12052 df-fl 12455 df-rest 15906 df-salg 39205 df-smblfn 39587 |
This theorem is referenced by: issmfle 39632 |
Copyright terms: Public domain | W3C validator |