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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 1874 | . 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 2780 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
6 | 3, 4, 5 | smfdmss 42476 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑅) |
7 | smfsssmf.i | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) | |
8 | 7 | unissd 4741 | . . 3 ⊢ (𝜑 → ∪ 𝑅 ⊆ ∪ 𝑆) |
9 | 6, 8 | sstrd 3870 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
10 | 3, 4, 5 | smff 42475 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
11 | ssrest 21503 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝑅 ⊆ 𝑆) → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) | |
12 | 2, 7, 11 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
13 | 12 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
14 | 3 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑅 ∈ SAlg) |
15 | 4 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑅)) |
16 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
17 | 14, 15, 5, 16 | smfpreimalt 42474 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑅 ↾t dom 𝐹)) |
18 | 13, 17 | sseldd 3861 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 1, 2, 9, 10, 18 | issmfd 42478 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2051 {crab 3094 ⊆ wss 3831 ∪ cuni 4717 class class class wbr 4934 dom cdm 5411 ‘cfv 6193 (class class class)co 6982 ℝcr 10340 < clt 10480 ↾t crest 16556 SAlgcsalg 42059 SMblFncsmblfn 42443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-pre-lttri 10415 ax-pre-lttrn 10416 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-po 5330 df-so 5331 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-ov 6985 df-oprab 6986 df-mpo 6987 df-1st 7507 df-2nd 7508 df-er 8095 df-pm 8215 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-ioo 12564 df-ico 12566 df-rest 16558 df-smblfn 42444 |
This theorem is referenced by: bormflebmf 42496 |
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