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Mirrors > Home > MPE Home > Th. List > ismbf2d | Structured version Visualization version GIF version |
Description: Deduction to prove measurability of a real function. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
ismbf2d.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
ismbf2d.2 | ⊢ (𝜑 → 𝐴 ∈ dom vol) |
ismbf2d.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
ismbf2d.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
Ref | Expression |
---|---|
ismbf2d | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbf2d.1 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
2 | elxr 13093 | . . 3 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) | |
3 | ismbf2d.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) | |
4 | oveq1 7408 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = (+∞(,)+∞)) | |
5 | iooid 13349 | . . . . . . . 8 ⊢ (+∞(,)+∞) = ∅ | |
6 | 4, 5 | eqtrdi 2780 | . . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅) |
7 | 6 | imaeq2d 6049 | . . . . . 6 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ∅)) |
8 | ima0 6066 | . . . . . . 7 ⊢ (◡𝐹 “ ∅) = ∅ | |
9 | 0mbl 25390 | . . . . . . 7 ⊢ ∅ ∈ dom vol | |
10 | 8, 9 | eqeltri 2821 | . . . . . 6 ⊢ (◡𝐹 “ ∅) ∈ dom vol |
11 | 7, 10 | eqeltrdi 2833 | . . . . 5 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
13 | fimacnv 6729 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → (◡𝐹 “ ℝ) = 𝐴) | |
14 | 1, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ ℝ) = 𝐴) |
15 | ismbf2d.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom vol) | |
16 | 14, 15 | eqeltrd 2825 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ ℝ) ∈ dom vol) |
17 | oveq1 7408 | . . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞)) | |
18 | ioomax 13396 | . . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ | |
19 | 17, 18 | eqtrdi 2780 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ) |
20 | 19 | imaeq2d 6049 | . . . . . . 7 ⊢ (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ℝ)) |
21 | 20 | eleq1d 2810 | . . . . . 6 ⊢ (𝑥 = -∞ → ((◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ↔ (◡𝐹 “ ℝ) ∈ dom vol)) |
22 | 16, 21 | syl5ibrcom 246 | . . . . 5 ⊢ (𝜑 → (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)) |
23 | 22 | imp 406 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
24 | 3, 12, 23 | 3jaodan 1427 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
25 | 2, 24 | sylan2b 593 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
26 | ismbf2d.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) | |
27 | oveq2 7409 | . . . . . . . . 9 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = (-∞(,)+∞)) | |
28 | 27, 18 | eqtrdi 2780 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ) |
29 | 28 | imaeq2d 6049 | . . . . . . 7 ⊢ (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ℝ)) |
30 | 29 | eleq1d 2810 | . . . . . 6 ⊢ (𝑥 = +∞ → ((◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (◡𝐹 “ ℝ) ∈ dom vol)) |
31 | 16, 30 | syl5ibrcom 246 | . . . . 5 ⊢ (𝜑 → (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)) |
32 | 31 | imp 406 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
33 | oveq2 7409 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞)) | |
34 | iooid 13349 | . . . . . . . 8 ⊢ (-∞(,)-∞) = ∅ | |
35 | 33, 34 | eqtrdi 2780 | . . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅) |
36 | 35 | imaeq2d 6049 | . . . . . 6 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ∅)) |
37 | 36, 10 | eqeltrdi 2833 | . . . . 5 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
38 | 37 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
39 | 26, 32, 38 | 3jaodan 1427 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
40 | 2, 39 | sylan2b 593 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
41 | 1, 25, 40 | ismbfd 25490 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1083 = wceq 1533 ∈ wcel 2098 ∅c0 4314 ◡ccnv 5665 dom cdm 5666 “ cima 5669 ⟶wf 6529 (class class class)co 7401 ℝcr 11105 +∞cpnf 11242 -∞cmnf 11243 ℝ*cxr 11244 (,)cioo 13321 volcvol 25314 MblFncmbf 25465 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xadd 13090 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-sum 15630 df-xmet 21221 df-met 21222 df-ovol 25315 df-vol 25316 df-mbf 25470 |
This theorem is referenced by: mbfres 25495 mbfmulc2lem 25498 mbfposr 25503 ismbf3d 25505 iblabsnclem 37041 ftc1anclem1 37051 ftc1anclem6 37056 |
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