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Mirrors > Home > MPE Home > Th. List > isi1f | Structured version Visualization version GIF version |
Description: The predicate "𝐹 is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom 𝐹 ∈ dom ∫1 to represent this concept because ∫1 is the first preparation function for our final definition ∫ (see df-itg 24227); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
isi1f | ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1 6468 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑔:ℝ⟶ℝ ↔ 𝐹:ℝ⟶ℝ)) | |
2 | rneq 5770 | . . . 4 ⊢ (𝑔 = 𝐹 → ran 𝑔 = ran 𝐹) | |
3 | 2 | eleq1d 2874 | . . 3 ⊢ (𝑔 = 𝐹 → (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin)) |
4 | cnveq 5708 | . . . . . 6 ⊢ (𝑔 = 𝐹 → ◡𝑔 = ◡𝐹) | |
5 | 4 | imaeq1d 5895 | . . . . 5 ⊢ (𝑔 = 𝐹 → (◡𝑔 “ (ℝ ∖ {0})) = (◡𝐹 “ (ℝ ∖ {0}))) |
6 | 5 | fveq2d 6649 | . . . 4 ⊢ (𝑔 = 𝐹 → (vol‘(◡𝑔 “ (ℝ ∖ {0}))) = (vol‘(◡𝐹 “ (ℝ ∖ {0})))) |
7 | 6 | eleq1d 2874 | . . 3 ⊢ (𝑔 = 𝐹 → ((vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ ↔ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)) |
8 | 1, 3, 7 | 3anbi123d 1433 | . 2 ⊢ (𝑔 = 𝐹 → ((𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ) ↔ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) |
9 | sumex 15036 | . . 3 ⊢ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥}))) ∈ V | |
10 | df-itg1 24224 | . . 3 ⊢ ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} ↦ Σ𝑥 ∈ (ran 𝑓 ∖ {0})(𝑥 · (vol‘(◡𝑓 “ {𝑥})))) | |
11 | 9, 10 | dmmpti 6464 | . 2 ⊢ dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧ (vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈ ℝ)} |
12 | 8, 11 | elrab2 3631 | 1 ⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 ∖ cdif 3878 {csn 4525 ◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℝcr 10525 0cc0 10526 · cmul 10531 Σcsu 15034 volcvol 24067 MblFncmbf 24218 ∫1citg1 24219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-sum 15035 df-itg1 24224 |
This theorem is referenced by: i1fmbf 24279 i1ff 24280 i1frn 24281 i1fima2 24283 i1fd 24285 |
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