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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 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 2734 | . . . 4 ⊢ dom 𝐹 = dom 𝐹 | |
6 | 3, 4, 5 | smfdmss 46588 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑅) |
7 | smfsssmf.i | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) | |
8 | 7 | unissd 4941 | . . 3 ⊢ (𝜑 → ∪ 𝑅 ⊆ ∪ 𝑆) |
9 | 6, 8 | sstrd 4013 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
10 | 3, 4, 5 | smff 46587 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
11 | ssrest 23198 | . . . . 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 46586 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑅 ↾t dom 𝐹)) |
18 | 13, 17 | sseldd 4003 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
19 | 1, 2, 9, 10, 18 | issmfd 46590 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2103 {crab 3438 ⊆ wss 3970 ∪ cuni 4931 class class class wbr 5169 dom cdm 5699 ‘cfv 6572 (class class class)co 7445 ℝcr 11179 < clt 11320 ↾t crest 17475 SAlgcsalg 46163 SMblFncsmblfn 46550 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-pre-lttri 11254 ax-pre-lttrn 11255 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-1st 8026 df-2nd 8027 df-er 8759 df-pm 8883 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-ioo 13407 df-ico 13409 df-rest 17477 df-smblfn 46551 |
This theorem is referenced by: bormflebmf 46608 |
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