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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 25136). (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 1918 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | smfmbfcex.s | . . . . 5 ⊢ 𝑆 = dom vol | |
3 | dmvolsal 45062 | . . . . 5 ⊢ dom vol ∈ SAlg | |
4 | 2, 3 | eqeltri 2830 | . . . 4 ⊢ 𝑆 ∈ SAlg |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
6 | smfmbfcex.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℝ) | |
7 | 2 | unieqi 4922 | . . . . 5 ⊢ ∪ 𝑆 = ∪ dom vol |
8 | unidmvol 25058 | . . . . 5 ⊢ ∪ dom vol = ℝ | |
9 | 7, 8 | eqtri 2761 | . . . 4 ⊢ ∪ 𝑆 = ℝ |
10 | 6, 9 | sseqtrrdi 4034 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝑆) |
11 | 0red 11217 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | smfmbfcex.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) | |
13 | 1, 5, 10, 11, 12 | smfconst 45465 | . 2 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
14 | smfmbfcex.n | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) | |
15 | 0red 11217 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) | |
16 | 12, 15 | dmmptd 6696 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝑋) |
17 | 2 | eqcomi 2742 | . . . . . 6 ⊢ dom vol = 𝑆 |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → dom vol = 𝑆) |
19 | 16, 18 | eleq12d 2828 | . . . 4 ⊢ (𝜑 → (dom 𝐹 ∈ dom vol ↔ 𝑋 ∈ 𝑆)) |
20 | 14, 19 | mtbird 325 | . . 3 ⊢ (𝜑 → ¬ dom 𝐹 ∈ dom vol) |
21 | mbfdm 25143 | . . 3 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | |
22 | 20, 21 | nsyl 140 | . 2 ⊢ (𝜑 → ¬ 𝐹 ∈ MblFn) |
23 | 13, 22 | jca 513 | 1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 ∪ cuni 4909 ↦ cmpt 5232 dom cdm 5677 ‘cfv 6544 ℝcr 11109 0cc0 11110 volcvol 24980 MblFncmbf 25131 SAlgcsalg 45024 SMblFncsmblfn 45411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xadd 13093 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-rest 17368 df-xmet 20937 df-met 20938 df-ovol 24981 df-vol 24982 df-mbf 25136 df-salg 45025 df-smblfn 45412 |
This theorem is referenced by: nsssmfmbflem 45494 |
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