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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 12514 | . . 3 ⊢ (𝑥 ∈ ℝ* ↔ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) | |
3 | ismbf2d.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) | |
4 | oveq1 7165 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = (+∞(,)+∞)) | |
5 | iooid 12769 | . . . . . . . 8 ⊢ (+∞(,)+∞) = ∅ | |
6 | 4, 5 | syl6eq 2874 | . . . . . . 7 ⊢ (𝑥 = +∞ → (𝑥(,)+∞) = ∅) |
7 | 6 | imaeq2d 5931 | . . . . . 6 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ∅)) |
8 | ima0 5947 | . . . . . . 7 ⊢ (◡𝐹 “ ∅) = ∅ | |
9 | 0mbl 24142 | . . . . . . 7 ⊢ ∅ ∈ dom vol | |
10 | 8, 9 | eqeltri 2911 | . . . . . 6 ⊢ (◡𝐹 “ ∅) ∈ dom vol |
11 | 7, 10 | eqeltrdi 2923 | . . . . 5 ⊢ (𝑥 = +∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
12 | 11 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
13 | fimacnv 6841 | . . . . . . . 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 2915 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ ℝ) ∈ dom vol) |
17 | oveq1 7165 | . . . . . . . . 9 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = (-∞(,)+∞)) | |
18 | ioomax 12814 | . . . . . . . . 9 ⊢ (-∞(,)+∞) = ℝ | |
19 | 17, 18 | syl6eq 2874 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (𝑥(,)+∞) = ℝ) |
20 | 19 | imaeq2d 5931 | . . . . . . 7 ⊢ (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) = (◡𝐹 “ ℝ)) |
21 | 20 | eleq1d 2899 | . . . . . 6 ⊢ (𝑥 = -∞ → ((◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ↔ (◡𝐹 “ ℝ) ∈ dom vol)) |
22 | 16, 21 | syl5ibrcom 249 | . . . . 5 ⊢ (𝜑 → (𝑥 = -∞ → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)) |
23 | 22 | imp 409 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
24 | 3, 12, 23 | 3jaodan 1426 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
25 | 2, 24 | sylan2b 595 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol) |
26 | ismbf2d.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) | |
27 | oveq2 7166 | . . . . . . . . 9 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = (-∞(,)+∞)) | |
28 | 27, 18 | syl6eq 2874 | . . . . . . . 8 ⊢ (𝑥 = +∞ → (-∞(,)𝑥) = ℝ) |
29 | 28 | imaeq2d 5931 | . . . . . . 7 ⊢ (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ℝ)) |
30 | 29 | eleq1d 2899 | . . . . . 6 ⊢ (𝑥 = +∞ → ((◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol ↔ (◡𝐹 “ ℝ) ∈ dom vol)) |
31 | 16, 30 | syl5ibrcom 249 | . . . . 5 ⊢ (𝜑 → (𝑥 = +∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)) |
32 | 31 | imp 409 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = +∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
33 | oveq2 7166 | . . . . . . . 8 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = (-∞(,)-∞)) | |
34 | iooid 12769 | . . . . . . . 8 ⊢ (-∞(,)-∞) = ∅ | |
35 | 33, 34 | syl6eq 2874 | . . . . . . 7 ⊢ (𝑥 = -∞ → (-∞(,)𝑥) = ∅) |
36 | 35 | imaeq2d 5931 | . . . . . 6 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) = (◡𝐹 “ ∅)) |
37 | 36, 10 | eqeltrdi 2923 | . . . . 5 ⊢ (𝑥 = -∞ → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
38 | 37 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = -∞) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
39 | 26, 32, 38 | 3jaodan 1426 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∨ 𝑥 = +∞ ∨ 𝑥 = -∞)) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
40 | 2, 39 | sylan2b 595 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ*) → (◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol) |
41 | 1, 25, 40 | ismbfd 24242 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ∅c0 4293 ◡ccnv 5556 dom cdm 5557 “ cima 5560 ⟶wf 6353 (class class class)co 7158 ℝcr 10538 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 (,)cioo 12741 volcvol 24066 MblFncmbf 24217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-xmet 20540 df-met 20541 df-ovol 24067 df-vol 24068 df-mbf 24222 |
This theorem is referenced by: mbfres 24247 mbfmulc2lem 24250 mbfposr 24255 ismbf3d 24257 iblabsnclem 34957 ftc1anclem1 34969 ftc1anclem6 34974 |
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