Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimioompt | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfpimioompt.x | ⊢ Ⅎ𝑥𝜑 |
smfpimioompt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpimioompt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
smfpimioompt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
smfpimioompt.m | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
smfpimioompt.l | ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
smfpimioompt.r | ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
Ref | Expression |
---|---|
smfpimioompt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpimioompt.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | smfpimioompt.l | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ*) | |
3 | smfpimioompt.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
4 | smfpimioompt.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
5 | smfpimioompt.m | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
6 | eqid 2821 | . . . . . . 7 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 4, 5, 6 | smff 43099 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℝ) |
8 | eqid 2821 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | smfpimioompt.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) | |
10 | 1, 8, 9 | dmmptdf 41578 | . . . . . . 7 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
11 | 10 | feq2d 6486 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℝ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)) |
12 | 7, 11 | mpbid 234 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
13 | 12 | fvmptelrn 6863 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
14 | 13 | rexrd 10677 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
15 | 1, 2, 3, 14 | pimiooltgt 43079 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵})) |
16 | smfpimioompt.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | eqid 2821 | . . . 4 ⊢ (𝑆 ↾t 𝐴) = (𝑆 ↾t 𝐴) | |
18 | 4, 16, 17 | subsalsal 42732 | . . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐴) ∈ SAlg) |
19 | 1, 4, 9, 5, 3 | smfpimltxrmpt 43125 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴)) |
20 | 1, 4, 9, 5, 2 | smfpimgtxrmpt 43150 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) |
21 | 18, 19, 20 | salincld 42725 | . 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∩ {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) ∈ (𝑆 ↾t 𝐴)) |
22 | 15, 21 | eqeltrd 2913 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (𝐿(,)𝑅)} ∈ (𝑆 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 {crab 3142 ∩ cin 3923 class class class wbr 5052 ↦ cmpt 5132 dom cdm 5541 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 ℝcr 10522 ℝ*cxr 10660 < clt 10661 (,)cioo 12725 ↾t crest 16677 SAlgcsalg 42683 SMblFncsmblfn 43067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cc 9843 ax-ac2 9871 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-card 9354 df-acn 9357 df-ac 9528 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-n0 11885 df-z 11969 df-uz 12231 df-q 12336 df-rp 12377 df-ioo 12729 df-ico 12731 df-fl 13152 df-rest 16679 df-salg 42684 df-smblfn 43068 |
This theorem is referenced by: smfpimioo 43152 smfresal 43153 smfrec 43154 smfmullem4 43159 |
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