![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 25566). (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 1909 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | smfmbfcex.s | . . . . 5 ⊢ 𝑆 = dom vol | |
3 | dmvolsal 45797 | . . . . 5 ⊢ dom vol ∈ SAlg | |
4 | 2, 3 | eqeltri 2821 | . . . 4 ⊢ 𝑆 ∈ SAlg |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | smfmbfcex.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℝ) | |
7 | 2 | unieqi 4915 | . . . . 5 ⊢ ∪ 𝑆 = ∪ dom vol |
8 | unidmvol 25488 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
9 | 7, 8 | eqtri 2753 | . . . 4 ⊢ ∪ 𝑆 = ℝ |
10 | 6, 9 | sseqtrrdi 4024 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝑆) |
11 | 0red 11247 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | smfmbfcex.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) | |
13 | 1, 5, 10, 11, 12 | smfconst 46200 | . 2 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
14 | smfmbfcex.n | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) | |
15 | 0red 11247 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) | |
16 | 12, 15 | dmmptd 6695 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
17 | 2 | eqcomi 2734 | . . . . . 6 ⊢ dom vol = 𝑆 |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → dom vol = 𝑆) |
19 | 16, 18 | eleq12d 2819 | . . . 4 ⊢ (𝜑 → (dom 𝐹 ∈ dom vol ↔ 𝑋 ∈ 𝑆)) |
20 | 14, 19 | mtbird 324 | . . 3 ⊢ (𝜑 → ¬ dom 𝐹 ∈ dom vol) |
21 | mbfdm 25573 | . . 3 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | |
22 | 20, 21 | nsyl 140 | . 2 ⊢ (𝜑 → ¬ 𝐹 ∈ MblFn) |
23 | 13, 22 | jca 510 | 1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3939 ∪ cuni 4903 ↦ cmpt 5226 dom cdm 5672 ‘cfv 6543 ℝcr 11137 0cc0 11138 volcvol 25410 MblFncmbf 25561 SAlgcsalg 45759 SMblFncsmblfn 46146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cc 10458 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xadd 13125 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-rest 17403 df-xmet 21276 df-met 21277 df-ovol 25411 df-vol 25412 df-mbf 25566 df-salg 45760 df-smblfn 46147 |
This theorem is referenced by: nsssmfmbflem 46229 |
Copyright terms: Public domain | W3C validator |