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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 13119 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | supex 9457 | . 2 ⊢ sup(𝐿, ℝ*, < ) ∈ V |
3 | reex 11200 | . 2 ⊢ ℝ ∈ V | |
4 | ovex 7441 | . 2 ⊢ (0[,]+∞) ∈ V | |
5 | breq2 5152 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑔 ∘r ≤ 𝑓 ↔ 𝑔 ∘r ≤ 𝐹)) | |
6 | 5 | anbi1d 630 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)))) |
7 | 6 | rexbidv 3178 | . . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)))) |
8 | 7 | abbidv 2801 | . . . 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 | eqtr4di 2790 | . . 3 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = 𝐿) |
11 | 10 | supeq1d 9440 | . 2 ⊢ (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < )) |
12 | df-itg2 25137 | . 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 8874 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 {cab 2709 ∃wrex 3070 class class class wbr 5148 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ∘r cofr 7668 supcsup 9434 ℝcr 11108 0cc0 11109 +∞cpnf 11244 ℝ*cxr 11246 < clt 11247 ≤ cle 11248 [,]cicc 13326 ∫1citg1 25131 ∫2citg2 25132 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-itg2 25137 |
This theorem is referenced by: itg2cl 25249 itg2ub 25250 itg2leub 25251 itg2addnclem 36534 |
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