Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfconst | Structured version Visualization version GIF version |
Description: Given a sigma-algebra over a base set X, every partial real-valued constant function is measurable. Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfconst.x | ⊢ Ⅎ𝑥𝜑 |
smfconst.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfconst.a | ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
smfconst.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
smfconst.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
smfconst | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfconst.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | nfmpt1 5163 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2975 | . 2 ⊢ Ⅎ𝑥𝐹 |
4 | nfv 1911 | . 2 ⊢ Ⅎ𝑎𝜑 | |
5 | smfconst.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | smfconst.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) | |
7 | smfconst.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
8 | smfconst.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
10 | 7, 9, 1 | fmptdf 6880 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
11 | nfv 1911 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
12 | 7, 11 | nfan 1896 | . . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
13 | nfv 1911 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐵 < 𝑎 | |
14 | 12, 13 | nfan 1896 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) |
15 | 8 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
16 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 < 𝑎) | |
17 | 14, 15, 1, 16 | pimconstlt1 42982 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = 𝐴) |
18 | eqidd 2822 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = 𝐴) | |
19 | sseqin2 4191 | . . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝑆 ↔ (∪ 𝑆 ∩ 𝐴) = 𝐴) | |
20 | 6, 19 | sylib 220 | . . . . . . 7 ⊢ (𝜑 → (∪ 𝑆 ∩ 𝐴) = 𝐴) |
21 | 20 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
22 | 21 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
23 | 17, 18, 22 | 3eqtrd 2860 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (∪ 𝑆 ∩ 𝐴)) |
24 | 5 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝑆 ∈ SAlg) |
25 | 5 | uniexd 7467 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
26 | 25, 6 | ssexd 5227 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
27 | 26 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 ∈ V) |
28 | 24 | salunid 42635 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → ∪ 𝑆 ∈ 𝑆) |
29 | eqid 2821 | . . . . 5 ⊢ (∪ 𝑆 ∩ 𝐴) = (∪ 𝑆 ∩ 𝐴) | |
30 | 24, 27, 28, 29 | elrestd 41372 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → (∪ 𝑆 ∩ 𝐴) ∈ (𝑆 ↾t 𝐴)) |
31 | 23, 30 | eqeltrd 2913 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
32 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐵 < 𝑎 | |
33 | 12, 32 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) |
34 | 8 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
35 | rexr 10686 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | ad2antlr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ*) |
37 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ¬ 𝐵 < 𝑎) | |
38 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ) | |
39 | 38, 34 | lenltd 10785 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → (𝑎 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑎)) |
40 | 37, 39 | mpbird 259 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ≤ 𝐵) |
41 | 33, 34, 1, 36, 40 | pimconstlt0 42981 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = ∅) |
42 | eqid 2821 | . . . . . . 7 ⊢ (𝑆 ↾t 𝐴) = (𝑆 ↾t 𝐴) | |
43 | 5, 26, 42 | subsalsal 42641 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾t 𝐴) ∈ SAlg) |
44 | 43 | 0sald 42632 | . . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐴)) |
45 | 44 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ∅ ∈ (𝑆 ↾t 𝐴)) |
46 | 41, 45 | eqeltrd 2913 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
47 | 31, 46 | pm2.61dan 811 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
48 | 3, 4, 5, 6, 10, 47 | issmfdf 43013 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ∪ cuni 4837 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 ↾t crest 16693 SAlgcsalg 42592 SMblFncsmblfn 42976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cc 9856 ax-ac2 9884 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-acn 9370 df-ac 9541 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-ioo 12741 df-ico 12743 df-rest 16695 df-salg 42593 df-smblfn 42977 |
This theorem is referenced by: smfmbfcex 43035 smfmulc1 43070 |
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