Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfsssmf | Structured version Visualization version GIF version |
Description: If a function is measurable w.r.t. to a sigma-algebra, then it is measurable w.r.t. to a larger sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfsssmf.r | ⊢ (𝜑 → 𝑅 ∈ SAlg) |
smfsssmf.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfsssmf.i | ⊢ (𝜑 → 𝑅 ⊆ 𝑆) |
smfsssmf.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑅)) |
Ref | Expression |
---|---|
smfsssmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfsssmf.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smfsssmf.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ SAlg) | |
4 | smfsssmf.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑅)) | |
5 | eqid 2821 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
6 | 3, 4, 5 | smfdmss 43009 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑅) |
7 | smfsssmf.i | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) | |
8 | 7 | unissd 4847 | . . 3 ⊢ (𝜑 → ∪ 𝑅 ⊆ ∪ 𝑆) |
9 | 6, 8 | sstrd 3976 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
10 | 3, 4, 5 | smff 43008 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
11 | ssrest 21783 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝑅 ⊆ 𝑆) → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) | |
12 | 2, 7, 11 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
14 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑅 ∈ SAlg) |
15 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑅)) |
16 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
17 | 14, 15, 5, 16 | smfpreimalt 43007 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑅 ↾t dom 𝐹)) |
18 | 13, 17 | sseldd 3967 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 1, 2, 9, 10, 18 | issmfd 43011 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 ∪ cuni 4837 class class class wbr 5065 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 < clt 10674 ↾t crest 16693 SAlgcsalg 42592 SMblFncsmblfn 42976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-pre-lttri 10610 ax-pre-lttrn 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-ioo 12741 df-ico 12743 df-rest 16695 df-smblfn 42977 |
This theorem is referenced by: bormflebmf 43029 |
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