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 4129 | . . . . 5 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ MblFn) | |
2 | elinel2 4130 | . . . . . 6 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ (ℝ ↑pm ℝ)) | |
3 | elpmrn 42719 | . . . . . 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 44231 | . . . 4 ⊢ (𝑓 ∈ (MblFn ∩ (ℝ ↑pm ℝ)) → 𝑓 ∈ (SMblFn‘𝑆)) |
7 | 6 | ssriv 3925 | . . 3 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) |
8 | 5 | nsssmfmbf 44270 | . . . 4 ⊢ ¬ (SMblFn‘𝑆) ⊆ MblFn |
9 | 1 | ssriv 3925 | . . . 4 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ MblFn |
10 | nsstr 42604 | . . . 4 ⊢ ((¬ (SMblFn‘𝑆) ⊆ MblFn ∧ (MblFn ∩ (ℝ ↑pm ℝ)) ⊆ MblFn) → ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ))) | |
11 | 8, 9, 10 | mp2an 689 | . . 3 ⊢ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ)) |
12 | 7, 11 | pm3.2i 471 | . 2 ⊢ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) ∧ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ))) |
13 | dfpss3 4021 | . 2 ⊢ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) ↔ ((MblFn ∩ (ℝ ↑pm ℝ)) ⊆ (SMblFn‘𝑆) ∧ ¬ (SMblFn‘𝑆) ⊆ (MblFn ∩ (ℝ ↑pm ℝ)))) | |
14 | 12, 13 | mpbir 230 | 1 ⊢ (MblFn ∩ (ℝ ↑pm ℝ)) ⊊ (SMblFn‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 ⊊ wpss 3888 dom cdm 5585 ran crn 5586 ‘cfv 6427 (class class class)co 7268 ↑pm cpm 8604 ℝcr 10858 volcvol 24615 MblFncmbf 24766 SMblFncsmblfn 44192 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cc 10179 ax-ac2 10207 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-oadd 8289 df-omul 8290 df-er 8486 df-ec 8488 df-qs 8492 df-map 8605 df-pm 8606 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-dju 9647 df-card 9685 df-acn 9688 df-ac 9860 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ioo 13071 df-ico 13073 df-icc 13074 df-fz 13228 df-fzo 13371 df-fl 13500 df-seq 13710 df-exp 13771 df-hash 14033 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-clim 15185 df-rlim 15186 df-sum 15386 df-rest 17121 df-topgen 17142 df-psmet 20577 df-xmet 20578 df-met 20579 df-bl 20580 df-mopn 20581 df-top 22031 df-topon 22048 df-bases 22084 df-cmp 22526 df-ovol 24616 df-vol 24617 df-mbf 24771 df-salg 43809 df-smblfn 44193 |
This theorem is referenced by: (None) |
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