Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfgtlem | Structured version Visualization version GIF version |
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above 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 (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmfgtlem.x | ⊢ Ⅎ𝑥𝜑 |
issmfgtlem.a | ⊢ Ⅎ𝑎𝜑 |
issmfgtlem.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmfgtlem.d | ⊢ 𝐷 = dom 𝐹 |
issmfgtlem.i | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
issmfgtlem.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
issmfgtlem.p | ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
Ref | Expression |
---|---|
issmfgtlem | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmfgtlem.i | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
2 | issmfgtlem.f | . . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
3 | issmfgtlem.s | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | 3, 1 | restuni4 39304 | . . . . . . . 8 ⊢ (𝜑 → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
5 | 4 | eqcomd 2628 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ (𝑆 ↾t 𝐷)) |
6 | 5 | rabeqd 39276 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) |
8 | issmfgtlem.x | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
9 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ | |
10 | 8, 9 | nfan 1828 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ) |
11 | issmfgtlem.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
12 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
13 | 11, 12 | nfan 1828 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
14 | 3 | uniexd 39281 | . . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
15 | 14 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V) |
16 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆) | |
17 | 15, 16 | ssexd 4805 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V) |
18 | 1, 17 | mpdan 702 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V) |
19 | eqid 2622 | . . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
20 | 3, 18, 19 | subsalsal 40577 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
21 | 20 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg) |
22 | eqid 2622 | . . . . . 6 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷) | |
23 | 2 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ) |
24 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) | |
25 | 4 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
26 | 24, 25 | eleqtrd 2703 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ 𝐷) |
27 | 23, 26 | ffvelrnd 6360 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ) |
28 | 27 | rexrd 10089 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) |
29 | 28 | adantlr 751 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) |
30 | 4 | rabeqd 39276 | . . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)}) |
31 | 30 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)}) |
32 | issmfgtlem.p | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | |
33 | 32 | r19.21bi 2932 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
34 | 31, 33 | eqeltrd 2701 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
35 | 34 | adantlr 751 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
36 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ) | |
37 | 10, 13, 21, 22, 29, 35, 36 | salpreimagtlt 40939 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
38 | 7, 37 | eqeltrd 2701 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
39 | 38 | ralrimiva 2966 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
40 | 1, 2, 39 | 3jca 1242 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))) |
41 | issmfgtlem.d | . . 3 ⊢ 𝐷 = dom 𝐹 | |
42 | 3, 41 | issmf 40937 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)))) |
43 | 40, 42 | mpbird 247 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∪ cuni 4436 class class class wbr 4653 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 ℝ*cxr 10073 < clt 10074 ↾t crest 16081 SAlgcsalg 40528 SMblFncsmblfn 40909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-acn 8768 df-ac 8939 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ioo 12179 df-ico 12181 df-fl 12593 df-rest 16083 df-salg 40529 df-smblfn 40910 |
This theorem is referenced by: issmfgt 40965 |
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