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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 12834 | . . 3 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) | |
3 | ismbf2d.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) | |
4 | oveq1 7275 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = (+∞(,)+∞)) | |
5 | iooid 13089 | . . . . . . . 8 ⊢ (+∞(,)+∞) = ∅ | |
6 | 4, 5 | eqtrdi 2795 | . . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅) |
7 | 6 | imaeq2d 5966 | . . . . . 6 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ∅)) |
8 | ima0 5982 | . . . . . . 7 ⊢ (◡𝐹 “ ∅) = ∅ | |
9 | 0mbl 24684 | . . . . . . 7 ⊢ ∅ ∈ dom vol | |
10 | 8, 9 | eqeltri 2836 | . . . . . 6 ⊢ (◡𝐹 “ ∅) ∈ dom vol |
11 | 7, 10 | eqeltrdi 2848 | . . . . 5 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
13 | fimacnv 6618 | . . . . . . . 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 2840 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ ℝ) ∈ dom vol) |
17 | oveq1 7275 | . . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞)) | |
18 | ioomax 13136 | . . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ | |
19 | 17, 18 | eqtrdi 2795 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ) |
20 | 19 | imaeq2d 5966 | . . . . . . 7 ⊢ (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ℝ)) |
21 | 20 | eleq1d 2824 | . . . . . 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 1428 | . . 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 7276 | . . . . . . . . 9 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = (-∞(,)+∞)) | |
28 | 27, 18 | eqtrdi 2795 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ) |
29 | 28 | imaeq2d 5966 | . . . . . . 7 ⊢ (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ℝ)) |
30 | 29 | eleq1d 2824 | . . . . . 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 7276 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞)) | |
34 | iooid 13089 | . . . . . . . 8 ⊢ (-∞(,)-∞) = ∅ | |
35 | 33, 34 | eqtrdi 2795 | . . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅) |
36 | 35 | imaeq2d 5966 | . . . . . 6 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ∅)) |
37 | 36, 10 | eqeltrdi 2848 | . . . . 5 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
38 | 37 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
39 | 26, 32, 38 | 3jaodan 1428 | . . 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 24784 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1084 = wceq 1541 ∈ wcel 2109 ∅c0 4261 ◡ccnv 5587 dom cdm 5588 “ cima 5591 ⟶wf 6426 (class class class)co 7268 ℝcr 10854 +∞cpnf 10990 -∞cmnf 10991 ℝ*cxr 10992 (,)cioo 13061 volcvol 24608 MblFncmbf 24759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-map 8591 df-pm 8592 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-oi 9230 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-q 12671 df-rp 12713 df-xadd 12831 df-ioo 13065 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-fl 13493 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-sum 15379 df-xmet 20571 df-met 20572 df-ovol 24609 df-vol 24610 df-mbf 24764 |
This theorem is referenced by: mbfres 24789 mbfmulc2lem 24792 mbfposr 24797 ismbf3d 24799 iblabsnclem 35819 ftc1anclem1 35829 ftc1anclem6 35834 |
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