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Mirrors > Home > MPE Home > Th. List > itg2val | Structured version Visualization version GIF version |
Description: Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg2val.1 | ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} |
Ref | Expression |
---|---|
itg2val | ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 12528 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | supex 8921 | . 2 ⊢ sup(𝐿, ℝ*, < ) ∈ V |
3 | reex 10622 | . 2 ⊢ ℝ ∈ V | |
4 | ovex 7183 | . 2 ⊢ (0[,]+∞) ∈ V | |
5 | breq2 5063 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑔 ∘r ≤ 𝑓 ↔ 𝑔 ∘r ≤ 𝐹)) | |
6 | 5 | anbi1d 631 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)))) |
7 | 6 | rexbidv 3297 | . . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)))) |
8 | 7 | abbidv 2885 | . . . 4 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}) |
9 | itg2val.1 | . . . 4 ⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} | |
10 | 8, 9 | syl6eqr 2874 | . . 3 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = 𝐿) |
11 | 10 | supeq1d 8904 | . 2 ⊢ (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < )) |
12 | df-itg2 24216 | . 2 ⊢ ∫2 = (𝑓 ∈ ((0[,]+∞) ↑m ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < )) | |
13 | 2, 3, 4, 11, 12 | fvmptmap 8439 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 {cab 2799 ∃wrex 3139 class class class wbr 5059 dom cdm 5550 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∘r cofr 7402 supcsup 8898 ℝcr 10530 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,]cicc 12735 ∫1citg1 24210 ∫2citg2 24211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-itg2 24216 |
This theorem is referenced by: itg2cl 24327 itg2ub 24328 itg2leub 24329 itg2addnclem 34937 |
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