Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfltle | Structured version Visualization version GIF version |
Description: The definition of a measurable function w.r.t. a sigma-algebra, can be stated using less than or equal instead of less than. Proposition 121B (ii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmfltle.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmfltle.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
issmfltle.d | ⊢ 𝐷 = dom 𝐹 |
issmfltle.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
issmfltle | ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | issmfltle.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | issmfltle.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
4 | issmfltle.d | . . . . . 6 ⊢ 𝐷 = dom 𝐹 | |
5 | 2, 3, 4 | smff 39618 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
6 | 5 | frexr 38545 | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ*) |
7 | 6 | ffvelrnda 6267 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐹‘𝑥) ∈ ℝ*) |
8 | issmfltle.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 1, 7, 8 | preimaleiinlt 39608 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝐴} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < (𝐴 + (1 / 𝑛))}) |
10 | 2 | uniexd 38310 | . . . . 5 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
11 | 2, 3, 4 | smfdmss 39619 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
12 | 10, 11 | ssexd 4733 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
13 | eqid 2610 | . . . 4 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
14 | 2, 12, 13 | subsalsal 39253 | . . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
15 | nnct 12642 | . . . 4 ⊢ ℕ ≼ ω | |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
17 | nnn0 38536 | . . . 4 ⊢ ℕ ≠ ∅ | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≠ ∅) |
19 | 2 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ SAlg) |
20 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ (SMblFn‘𝑆)) |
21 | simpl 472 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) | |
22 | nnrecre 10934 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
23 | 22 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
24 | 21, 23 | readdcld 9948 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐴 + (1 / 𝑛)) ∈ ℝ) |
25 | 8, 24 | sylan 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 + (1 / 𝑛)) ∈ ℝ) |
26 | 19, 20, 4, 25 | smfpreimalt 39617 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < (𝐴 + (1 / 𝑛))} ∈ (𝑆 ↾t 𝐷)) |
27 | 14, 16, 18, 26 | saliincl 39221 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < (𝐴 + (1 / 𝑛))} ∈ (𝑆 ↾t 𝐷)) |
28 | 9, 27 | eqeltrd 2688 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 Vcvv 3173 ∅c0 3874 ∪ cuni 4372 ∩ ciin 4456 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 ℝcr 9814 1c1 9816 + caddc 9818 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 ↾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: (None) |
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