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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 5277 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2902 | . 2 ⊢ Ⅎ𝑥𝐹 |
4 | nfv 1913 | . 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
10 | 7, 9, 1 | fmptdf 7149 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
11 | nfv 1913 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
12 | 7, 11 | nfan 1898 | . . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
13 | nfv 1913 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐵 < 𝑎 | |
14 | 12, 13 | nfan 1898 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) |
15 | 8 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
16 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 < 𝑎) | |
17 | 14, 15, 1, 16 | pimconstlt1 46557 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = 𝐴) |
18 | eqidd 2735 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = 𝐴) | |
19 | sseqin2 4238 | . . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝑆 ↔ (∪ 𝑆 ∩ 𝐴) = 𝐴) | |
20 | 6, 19 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → (∪ 𝑆 ∩ 𝐴) = 𝐴) |
21 | 20 | eqcomd 2740 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
22 | 21 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
23 | 17, 18, 22 | 3eqtrd 2778 | . . . 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 7773 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
26 | 25, 6 | ssexd 5345 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
27 | 26 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 ∈ V) |
28 | 24 | salunid 46208 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → ∪ 𝑆 ∈ 𝑆) |
29 | eqid 2734 | . . . . 5 ⊢ (∪ 𝑆 ∩ 𝐴) = (∪ 𝑆 ∩ 𝐴) | |
30 | 24, 27, 28, 29 | elrestd 44944 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → (∪ 𝑆 ∩ 𝐴) ∈ (𝑆 ↾t 𝐴)) |
31 | 23, 30 | eqeltrd 2838 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
32 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐵 < 𝑎 | |
33 | 12, 32 | nfan 1898 | . . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) |
34 | 8 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
35 | rexr 11332 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | ad2antlr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ*) |
37 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ¬ 𝐵 < 𝑎) | |
38 | simplr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ) | |
39 | 38, 34 | lenltd 11432 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → (𝑎 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑎)) |
40 | 37, 39 | mpbird 257 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ≤ 𝐵) |
41 | 33, 34, 1, 36, 40 | pimconstlt0 46556 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = ∅) |
42 | eqid 2734 | . . . . . . 7 ⊢ (𝑆 ↾t 𝐴) = (𝑆 ↾t 𝐴) | |
43 | 5, 26, 42 | subsalsal 46214 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾t 𝐴) ∈ SAlg) |
44 | 43 | 0sald 46205 | . . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐴)) |
45 | 44 | ad2antrr 725 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ∅ ∈ (𝑆 ↾t 𝐴)) |
46 | 41, 45 | eqeltrd 2838 | . . 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 46592 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1781 ∈ wcel 2103 {crab 3438 Vcvv 3482 ∩ cin 3969 ⊆ wss 3970 ∅c0 4347 ∪ cuni 4931 class class class wbr 5169 ↦ cmpt 5252 ‘cfv 6572 (class class class)co 7445 ℝcr 11179 ℝ*cxr 11319 < clt 11320 ≤ cle 11321 ↾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-inf2 9706 ax-cc 10500 ax-ac2 10528 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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-rmo 3383 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-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 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-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 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-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-pm 8883 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-card 10004 df-acn 10007 df-ac 10181 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-n0 12550 df-z 12636 df-uz 12900 df-ioo 13407 df-ico 13409 df-rest 17477 df-salg 46164 df-smblfn 46551 |
This theorem is referenced by: smfmbfcex 46615 smfmulc1 46651 |
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