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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfpsssmf | Structured version Visualization version GIF version |
Description: Real-valued measurable functions are a proper subset of sigma-measurable functions (w.r.t. the Lebesgue measure on the reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
mbfpsssmf.1 | ⊢ 𝑆 = dom vol |
Ref | Expression |
---|---|
mbfpsssmf | ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel1 3997 | . . . . 5 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ MblFn) | |
2 | elinel2 3998 | . . . . . 6 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ (ℝ ↑pm ℝ)) | |
3 | elpmrn 40166 | . . . . . 6 ⊢ (𝑓 ∈ (ℝ ↑pm ℝ) → ran 𝑓 ⊆ ℝ) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → ran 𝑓 ⊆ ℝ) |
5 | mbfpsssmf.1 | . . . . 5 ⊢ 𝑆 = dom vol | |
6 | 1, 4, 5 | mbfresmf 41694 | . . . 4 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ (SMblFn‘𝑆)) |
7 | 6 | ssriv 3802 | . . 3 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) |
8 | 5 | nsssmfmbf 41733 | . . . 4 ⊢ ¬ (SMblFn‘𝑆) ⊆ MblFn |
9 | 1 | ssriv 3802 | . . . 4 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ MblFn |
10 | nsstr 40032 | . . . 4 ⊢ ((¬ (SMblFn‘𝑆) ⊆ MblFn ∧ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ MblFn) → ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ))) | |
11 | 8, 9, 10 | mp2an 684 | . . 3 ⊢ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ)) |
12 | 7, 11 | pm3.2i 463 | . 2 ⊢ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) ∧ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ))) |
13 | dfpss3 3890 | . 2 ⊢ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) ↔ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) ∧ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ)))) | |
14 | 12, 13 | mpbir 223 | 1 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∩ cin 3768 ⊆ wss 3769 ⊊ wpss 3770 dom cdm 5312 ran crn 5313 ‘cfv 6101 (class class class)co 6878 ↑pm cpm 8096 ℝcr 10223 volcvol 23571 MblFncmbf 23722 SMblFncsmblfn 41655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cc 9545 ax-ac2 9573 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-disj 4812 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-omul 7804 df-er 7982 df-ec 7984 df-qs 7988 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-acn 9054 df-ac 9225 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-rlim 14561 df-sum 14758 df-rest 16398 df-topgen 16419 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-top 21027 df-topon 21044 df-bases 21079 df-cmp 21519 df-ovol 23572 df-vol 23573 df-mbf 23727 df-salg 41272 df-smblfn 41656 |
This theorem is referenced by: (None) |
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