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 5155 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2972 | . 2 ⊢ Ⅎ𝑥𝐹 |
4 | nfv 1906 | . 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 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
10 | 7, 9, 1 | fmptdf 6873 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
11 | nfv 1906 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
12 | 7, 11 | nfan 1891 | . . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
13 | nfv 1906 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐵 < 𝑎 | |
14 | 12, 13 | nfan 1891 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) |
15 | 8 | ad2antrr 722 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
16 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 < 𝑎) | |
17 | 14, 15, 1, 16 | pimconstlt1 42860 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = 𝐴) |
18 | eqidd 2819 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = 𝐴) | |
19 | sseqin2 4189 | . . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝑆 ↔ (∪ 𝑆 ∩ 𝐴) = 𝐴) | |
20 | 6, 19 | sylib 219 | . . . . . . 7 ⊢ (𝜑 → (∪ 𝑆 ∩ 𝐴) = 𝐴) |
21 | 20 | eqcomd 2824 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
22 | 21 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
23 | 17, 18, 22 | 3eqtrd 2857 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (∪ 𝑆 ∩ 𝐴)) |
24 | 5 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝑆 ∈ SAlg) |
25 | 5 | uniexd 7457 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
26 | 25, 6 | ssexd 5219 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
27 | 26 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 ∈ V) |
28 | 24 | salunid 42513 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → ∪ 𝑆 ∈ 𝑆) |
29 | eqid 2818 | . . . . 5 ⊢ (∪ 𝑆 ∩ 𝐴) = (∪ 𝑆 ∩ 𝐴) | |
30 | 24, 27, 28, 29 | elrestd 41251 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → (∪ 𝑆 ∩ 𝐴) ∈ (𝑆 ↾t 𝐴)) |
31 | 23, 30 | eqeltrd 2910 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
32 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐵 < 𝑎 | |
33 | 12, 32 | nfan 1891 | . . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) |
34 | 8 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
35 | rexr 10675 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | ad2antlr 723 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ*) |
37 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ¬ 𝐵 < 𝑎) | |
38 | simplr 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ) | |
39 | 38, 34 | lenltd 10774 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → (𝑎 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑎)) |
40 | 37, 39 | mpbird 258 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ≤ 𝐵) |
41 | 33, 34, 1, 36, 40 | pimconstlt0 42859 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = ∅) |
42 | eqid 2818 | . . . . . . 7 ⊢ (𝑆 ↾t 𝐴) = (𝑆 ↾t 𝐴) | |
43 | 5, 26, 42 | subsalsal 42519 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾t 𝐴) ∈ SAlg) |
44 | 43 | 0sald 42510 | . . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐴)) |
45 | 44 | ad2antrr 722 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ∅ ∈ (𝑆 ↾t 𝐴)) |
46 | 41, 45 | eqeltrd 2910 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
47 | 31, 46 | pm2.61dan 809 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
48 | 3, 4, 5, 6, 10, 47 | issmfdf 42891 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 ∪ cuni 4830 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 ↾t crest 16682 SAlgcsalg 42470 SMblFncsmblfn 42854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-ioo 12730 df-ico 12732 df-rest 16684 df-salg 42471 df-smblfn 42855 |
This theorem is referenced by: smfmbfcex 42913 smfmulc1 42948 |
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