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 42208 | . . . . . . . 8 ⊢ (𝜑 → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
5 | 4 | eqcomd 2744 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ (𝑆 ↾t 𝐷)) |
6 | 5 | rabeqdv 3386 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) |
8 | issmfgtlem.x | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
9 | nfv 1921 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ | |
10 | 8, 9 | nfan 1906 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ) |
11 | issmfgtlem.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
12 | nfv 1921 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
13 | 11, 12 | nfan 1906 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
14 | 3 | uniexd 7486 | . . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
15 | 14 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V) |
16 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆) | |
17 | 15, 16 | ssexd 5192 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V) |
18 | 1, 17 | mpdan 687 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V) |
19 | eqid 2738 | . . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
20 | 3, 18, 19 | subsalsal 43440 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
21 | 20 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg) |
22 | eqid 2738 | . . . . . 6 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷) | |
23 | 2 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ) |
24 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) | |
25 | 4 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷) |
26 | 24, 25 | eleqtrd 2835 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ 𝐷) |
27 | 23, 26 | ffvelrnd 6862 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ) |
28 | 27 | rexrd 10769 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) |
29 | 28 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) |
30 | 4 | rabeqdv 3386 | . . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)}) |
31 | 30 | adantr 484 | . . . . . . . 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 3121 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
34 | 31, 33 | eqeltrd 2833 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
35 | 34 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
36 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ) | |
37 | 10, 13, 21, 22, 29, 35, 36 | salpreimagtlt 43805 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
38 | 7, 37 | eqeltrd 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
39 | 38 | ralrimiva 3096 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) |
40 | 1, 2, 39 | 3jca 1129 | . 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 43803 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)))) |
43 | 40, 42 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2114 ∀wral 3053 {crab 3057 Vcvv 3398 ⊆ wss 3843 ∪ cuni 4796 class class class wbr 5030 dom cdm 5525 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ℝcr 10614 ℝ*cxr 10752 < clt 10753 ↾t crest 16797 SAlgcsalg 43391 SMblFncsmblfn 43775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cc 9935 ax-ac2 9963 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-card 9441 df-acn 9444 df-ac 9616 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-q 12431 df-rp 12473 df-ioo 12825 df-ico 12827 df-fl 13253 df-rest 16799 df-salg 43392 df-smblfn 43776 |
This theorem is referenced by: issmfgt 43831 |
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