![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 1913 | . 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 2740 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
6 | 3, 4, 5 | smfdmss 46656 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑅) |
7 | smfsssmf.i | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) | |
8 | 7 | unissd 4941 | . . 3 ⊢ (𝜑 → ∪ 𝑅 ⊆ ∪ 𝑆) |
9 | 6, 8 | sstrd 4019 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
10 | 3, 4, 5 | smff 46655 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
11 | ssrest 23207 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝑅 ⊆ 𝑆) → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) | |
12 | 2, 7, 11 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑅 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
14 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑅 ∈ SAlg) |
15 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑅)) |
16 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
17 | 14, 15, 5, 16 | smfpreimalt 46654 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑅 ↾t dom 𝐹)) |
18 | 13, 17 | sseldd 4009 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 1, 2, 9, 10, 18 | issmfd 46658 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 {crab 3443 ⊆ wss 3976 ∪ cuni 4931 class class class wbr 5166 dom cdm 5700 ‘cfv 6575 (class class class)co 7450 ℝcr 11185 < clt 11326 ↾t crest 17482 SAlgcsalg 46231 SMblFncsmblfn 46618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-pre-lttri 11260 ax-pre-lttrn 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-er 8765 df-pm 8889 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-ioo 13413 df-ico 13415 df-rest 17484 df-smblfn 46619 |
This theorem is referenced by: bormflebmf 46676 |
Copyright terms: Public domain | W3C validator |