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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nsssmfmbflem | Structured version Visualization version GIF version |
Description: The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
nsssmfmbflem.s | ⊢ 𝑆 = dom vol |
nsssmfmbflem.x | ⊢ (𝜑 → 𝑋 ⊆ ℝ) |
nsssmfmbflem.n | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) |
nsssmfmbflem.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) |
Ref | Expression |
---|---|
nsssmfmbflem | ⊢ (𝜑 → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 11289 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℝ) | |
2 | nsssmfmbflem.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 0) | |
3 | 1, 2 | fmptd 7146 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
4 | reex 11271 | . . . . 5 ⊢ ℝ ∈ V | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ V) |
6 | nsssmfmbflem.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℝ) | |
7 | 5, 6 | ssexd 5345 | . . 3 ⊢ (𝜑 → 𝑋 ∈ V) |
8 | 3, 7 | fexd 7262 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
9 | nsssmfmbflem.s | . . 3 ⊢ 𝑆 = dom vol | |
10 | nsssmfmbflem.n | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑆) | |
11 | 9, 6, 10, 2 | smfmbfcex 46616 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn)) |
12 | eleq1 2826 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (SMblFn‘𝑆) ↔ 𝐹 ∈ (SMblFn‘𝑆))) | |
13 | eleq1 2826 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ MblFn ↔ 𝐹 ∈ MblFn)) | |
14 | 13 | notbid 318 | . . . 4 ⊢ (𝑓 = 𝐹 → (¬ 𝑓 ∈ MblFn ↔ ¬ 𝐹 ∈ MblFn)) |
15 | 12, 14 | anbi12d 631 | . . 3 ⊢ (𝑓 = 𝐹 → ((𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn) ↔ (𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn))) |
16 | 15 | spcegv 3606 | . 2 ⊢ (𝐹 ∈ V → ((𝐹 ∈ (SMblFn‘𝑆) ∧ ¬ 𝐹 ∈ MblFn) → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn))) |
17 | 8, 11, 16 | sylc 65 | 1 ⊢ (𝜑 → ∃𝑓(𝑓 ∈ (SMblFn‘𝑆) ∧ ¬ 𝑓 ∈ MblFn)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2103 Vcvv 3482 ⊆ wss 3970 ↦ cmpt 5252 dom cdm 5699 ‘cfv 6572 ℝcr 11179 0cc0 11180 volcvol 25510 MblFncmbf 25661 SMblFncsmblfn 46551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cc 10500 ax-ac2 10528 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-disj 5137 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-inf 9508 df-oi 9575 df-dju 9966 df-card 10004 df-acn 10007 df-ac 10181 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-z 12636 df-uz 12900 df-q 13010 df-rp 13054 df-xadd 13172 df-ioo 13407 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-clim 15530 df-rlim 15531 df-sum 15731 df-rest 17477 df-xmet 21375 df-met 21376 df-ovol 25511 df-vol 25512 df-mbf 25666 df-salg 46165 df-smblfn 46552 |
This theorem is referenced by: nsssmfmbf 46635 |
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