![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12156 | . . 3 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) | |
3 | ismbf2d.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) | |
4 | oveq1 6801 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = (+∞(,)+∞)) | |
5 | iooid 12409 | . . . . . . . 8 ⊢ (+∞(,)+∞) = ∅ | |
6 | 4, 5 | syl6eq 2821 | . . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅) |
7 | 6 | imaeq2d 5608 | . . . . . 6 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ∅)) |
8 | ima0 5623 | . . . . . . 7 ⊢ (◡𝐹 “ ∅) = ∅ | |
9 | 0mbl 23528 | . . . . . . 7 ⊢ ∅ ∈ dom vol | |
10 | 8, 9 | eqeltri 2846 | . . . . . 6 ⊢ (◡𝐹 “ ∅) ∈ dom vol |
11 | 7, 10 | syl6eqel 2858 | . . . . 5 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
12 | 11 | adantl 467 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
13 | fimacnv 6491 | . . . . . . . 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 2850 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ ℝ) ∈ dom vol) |
17 | oveq1 6801 | . . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞)) | |
18 | ioomax 12454 | . . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ | |
19 | 17, 18 | syl6eq 2821 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ) |
20 | 19 | imaeq2d 5608 | . . . . . . 7 ⊢ (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ℝ)) |
21 | 20 | eleq1d 2835 | . . . . . 6 ⊢ (𝑥 = -∞ → ((◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ↔ (◡𝐹 “ ℝ) ∈ dom vol)) |
22 | 16, 21 | syl5ibrcom 237 | . . . . 5 ⊢ (𝜑 → (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)) |
23 | 22 | imp 393 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
24 | 3, 12, 23 | 3jaodan 1542 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
25 | 2, 24 | sylan2b 575 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
26 | ismbf2d.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) | |
27 | oveq2 6802 | . . . . . . . . 9 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = (-∞(,)+∞)) | |
28 | 27, 18 | syl6eq 2821 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ) |
29 | 28 | imaeq2d 5608 | . . . . . . 7 ⊢ (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ℝ)) |
30 | 29 | eleq1d 2835 | . . . . . 6 ⊢ (𝑥 = +∞ → ((◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (◡𝐹 “ ℝ) ∈ dom vol)) |
31 | 16, 30 | syl5ibrcom 237 | . . . . 5 ⊢ (𝜑 → (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)) |
32 | 31 | imp 393 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
33 | oveq2 6802 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞)) | |
34 | iooid 12409 | . . . . . . . 8 ⊢ (-∞(,)-∞) = ∅ | |
35 | 33, 34 | syl6eq 2821 | . . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅) |
36 | 35 | imaeq2d 5608 | . . . . . 6 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ∅)) |
37 | 36, 10 | syl6eqel 2858 | . . . . 5 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
38 | 37 | adantl 467 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
39 | 26, 32, 38 | 3jaodan 1542 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
40 | 2, 39 | sylan2b 575 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
41 | 1, 25, 40 | ismbfd 23628 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∨ w3o 1070 = wceq 1631 ∈ wcel 2145 ∅c0 4064 ◡ccnv 5249 dom cdm 5250 “ cima 5253 ⟶wf 6028 (class class class)co 6794 ℝcr 10138 +∞cpnf 10274 -∞cmnf 10275 ℝ*cxr 10276 (,)cioo 12381 volcvol 23452 MblFncmbf 23603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-inf2 8703 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 ax-pre-sup 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-isom 6041 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-of 7045 df-om 7214 df-1st 7316 df-2nd 7317 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-2o 7715 df-oadd 7718 df-er 7897 df-map 8012 df-pm 8013 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-sup 8505 df-inf 8506 df-oi 8572 df-card 8966 df-cda 9193 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-n0 11496 df-z 11581 df-uz 11890 df-q 11993 df-rp 12037 df-xadd 12153 df-ioo 12385 df-ico 12387 df-icc 12388 df-fz 12535 df-fzo 12675 df-fl 12802 df-seq 13010 df-exp 13069 df-hash 13323 df-cj 14048 df-re 14049 df-im 14050 df-sqrt 14184 df-abs 14185 df-clim 14428 df-sum 14626 df-xmet 19955 df-met 19956 df-ovol 23453 df-vol 23454 df-mbf 23608 |
This theorem is referenced by: mbfres 23632 mbfmulc2lem 23635 mbfposr 23640 ismbf3d 23642 iblabsnclem 33806 ftc1anclem1 33818 ftc1anclem6 33823 |
Copyright terms: Public domain | W3C validator |