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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfmbfcex | Structured version Visualization version GIF version |
Description: A constant function, with non-lebesgue-measurable domain is a sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) but it is not a measurable functions ( w.r.t. to df-mbf 23903). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfmbfcex.s | ⊢ 𝑆 = dom vol |
smfmbfcex.x | ⊢ (𝜑 → 𝑋 ⊆ ℝ) |
smfmbfcex.n | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) |
smfmbfcex.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) |
Ref | Expression |
---|---|
smfmbfcex | ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1892 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | smfmbfcex.s | . . . . 5 ⊢ 𝑆 = dom vol | |
3 | dmvolsal 42171 | . . . . 5 ⊢ dom vol ∈ SAlg | |
4 | 2, 3 | eqeltri 2879 | . . . 4 ⊢ 𝑆 ∈ SAlg |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | smfmbfcex.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℝ) | |
7 | 2 | unieqi 4754 | . . . . 5 ⊢ ∪ 𝑆 = ∪ dom vol |
8 | unidmvol 23825 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
9 | 7, 8 | eqtri 2819 | . . . 4 ⊢ ∪ 𝑆 = ℝ |
10 | 6, 9 | syl6sseqr 3939 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝑆) |
11 | 0red 10490 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | smfmbfcex.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) | |
13 | 1, 5, 10, 11, 12 | smfconst 42568 | . 2 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
14 | smfmbfcex.n | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) | |
15 | 0red 10490 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) | |
16 | 15, 12 | fmptd 6741 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
17 | 16 | fdmd 6391 | . . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
18 | 2 | eqcomi 2804 | . . . . . . 7 ⊢ dom vol = 𝑆 |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝜑 → dom vol = 𝑆) |
20 | 17, 19 | eleq12d 2877 | . . . . 5 ⊢ (𝜑 → (dom 𝐹 ∈ dom vol ↔ 𝑋 ∈ 𝑆)) |
21 | 20 | notbid 319 | . . . 4 ⊢ (𝜑 → (¬ dom 𝐹 ∈ dom vol ↔ ¬ 𝑋 ∈ 𝑆)) |
22 | 14, 21 | mpbird 258 | . . 3 ⊢ (𝜑 → ¬ dom 𝐹 ∈ dom vol) |
23 | mbfdm 23910 | . . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | |
24 | 23 | con3i 157 | . . 3 ⊢ (¬ dom 𝐹 ∈ dom vol → ¬ 𝐹 ∈ MblFn) |
25 | 22, 24 | syl 17 | . 2 ⊢ (𝜑 → ¬ 𝐹 ∈ MblFn) |
26 | 13, 25 | jca 512 | 1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ⊆ wss 3859 ∪ cuni 4745 ↦ cmpt 5041 dom cdm 5443 ‘cfv 6225 ℝcr 10382 0cc0 10383 volcvol 23747 MblFncmbf 23898 SAlgcsalg 42135 SMblFncsmblfn 42519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cc 9703 ax-ac2 9731 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-disj 4931 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-oi 8820 df-dju 9176 df-card 9214 df-acn 9217 df-ac 9388 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-rp 12240 df-xadd 12358 df-ioo 12592 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-rlim 14680 df-sum 14877 df-rest 16525 df-xmet 20220 df-met 20221 df-ovol 23748 df-vol 23749 df-mbf 23903 df-salg 42136 df-smblfn 42520 |
This theorem is referenced by: nsssmfmbflem 42596 |
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