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 1909 | . 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 2819 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
6 | 3, 4, 5 | smfdmss 43001 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑅) |
7 | smfsssmf.i | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) | |
8 | 7 | unissd 4854 | . . 3 ⊢ (𝜑 → ∪ 𝑅 ⊆ ∪ 𝑆) |
9 | 6, 8 | sstrd 3975 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
10 | 3, 4, 5 | smff 43000 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
11 | ssrest 21776 | . . . . 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 42999 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑅 ↾t dom 𝐹)) |
18 | 13, 17 | sseldd 3966 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 1, 2, 9, 10, 18 | issmfd 43003 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2108 {crab 3140 ⊆ wss 3934 ∪ cuni 4830 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 < clt 10667 ↾t crest 16686 SAlgcsalg 42584 SMblFncsmblfn 42968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ioo 12734 df-ico 12736 df-rest 16688 df-smblfn 42969 |
This theorem is referenced by: bormflebmf 43021 |
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