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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 5128 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2953 | . 2 ⊢ Ⅎ𝑥𝐹 |
4 | nfv 1915 | . 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 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
10 | 7, 9, 1 | fmptdf 6858 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
11 | nfv 1915 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
12 | 7, 11 | nfan 1900 | . . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
13 | nfv 1915 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐵 < 𝑎 | |
14 | 12, 13 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) |
15 | 8 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
16 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 < 𝑎) | |
17 | 14, 15, 1, 16 | pimconstlt1 43340 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = 𝐴) |
18 | eqidd 2799 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = 𝐴) | |
19 | sseqin2 4142 | . . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝑆 ↔ (∪ 𝑆 ∩ 𝐴) = 𝐴) | |
20 | 6, 19 | sylib 221 | . . . . . . 7 ⊢ (𝜑 → (∪ 𝑆 ∩ 𝐴) = 𝐴) |
21 | 20 | eqcomd 2804 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
22 | 21 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
23 | 17, 18, 22 | 3eqtrd 2837 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (∪ 𝑆 ∩ 𝐴)) |
24 | 5 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝑆 ∈ SAlg) |
25 | 5 | uniexd 7448 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
26 | 25, 6 | ssexd 5192 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
27 | 26 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 ∈ V) |
28 | 24 | salunid 42993 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → ∪ 𝑆 ∈ 𝑆) |
29 | eqid 2798 | . . . . 5 ⊢ (∪ 𝑆 ∩ 𝐴) = (∪ 𝑆 ∩ 𝐴) | |
30 | 24, 27, 28, 29 | elrestd 41744 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → (∪ 𝑆 ∩ 𝐴) ∈ (𝑆 ↾t 𝐴)) |
31 | 23, 30 | eqeltrd 2890 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
32 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐵 < 𝑎 | |
33 | 12, 32 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) |
34 | 8 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
35 | rexr 10676 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | ad2antlr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ*) |
37 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ¬ 𝐵 < 𝑎) | |
38 | simplr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ) | |
39 | 38, 34 | lenltd 10775 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → (𝑎 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑎)) |
40 | 37, 39 | mpbird 260 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ≤ 𝐵) |
41 | 33, 34, 1, 36, 40 | pimconstlt0 43339 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = ∅) |
42 | eqid 2798 | . . . . . . 7 ⊢ (𝑆 ↾t 𝐴) = (𝑆 ↾t 𝐴) | |
43 | 5, 26, 42 | subsalsal 42999 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾t 𝐴) ∈ SAlg) |
44 | 43 | 0sald 42990 | . . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐴)) |
45 | 44 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ∅ ∈ (𝑆 ↾t 𝐴)) |
46 | 41, 45 | eqeltrd 2890 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
47 | 31, 46 | pm2.61dan 812 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
48 | 3, 4, 5, 6, 10, 47 | issmfdf 43371 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ↾t crest 16686 SAlgcsalg 42950 SMblFncsmblfn 43334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-ioo 12730 df-ico 12732 df-rest 16688 df-salg 42951 df-smblfn 43335 |
This theorem is referenced by: smfmbfcex 43393 smfmulc1 43428 |
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