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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 12351 | . . 3 ⊢ < Or ℝ* | |
2 | 1 | supex 8722 | . 2 ⊢ sup(𝐿, ℝ*, < ) ∈ V |
3 | reex 10426 | . 2 ⊢ ℝ ∈ V | |
4 | ovex 7008 | . 2 ⊢ (0[,]+∞) ∈ V | |
5 | breq2 4933 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑔 ∘𝑟 ≤ 𝑓 ↔ 𝑔 ∘𝑟 ≤ 𝐹)) | |
6 | 5 | anbi1d 620 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ (𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)))) |
7 | 6 | rexbidv 3242 | . . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔)) ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔)))) |
8 | 7 | abbidv 2843 | . . . 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 2832 | . . 3 ⊢ (𝑓 = 𝐹 → {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))} = 𝐿) |
11 | 10 | supeq1d 8705 | . 2 ⊢ (𝑓 = 𝐹 → sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟 ≤ 𝑓 ∧ 𝑥 = (∫1‘𝑔))}, ℝ*, < ) = sup(𝐿, ℝ*, < )) |
12 | df-itg2 23925 | . 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 8244 | 1 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 {cab 2758 ∃wrex 3089 class class class wbr 4929 dom cdm 5407 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∘𝑟 cofr 7226 supcsup 8699 ℝcr 10334 0cc0 10335 +∞cpnf 10471 ℝ*cxr 10473 < clt 10474 ≤ cle 10475 [,]cicc 12557 ∫1citg1 23919 ∫2citg2 23920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-pre-lttri 10409 ax-pre-lttrn 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-itg2 23925 |
This theorem is referenced by: itg2cl 24036 itg2ub 24037 itg2leub 24038 itg2addnclem 34390 |
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