| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0reval | Structured version Visualization version GIF version | ||
| Description: Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| sge0reval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| sge0reval.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) |
| Ref | Expression |
|---|---|
| sge0reval | ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sge0reval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 2 | sge0reval.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) | |
| 3 | 2 | fge0icoicc 46380 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
| 4 | 1, 3 | sge0vald 46384 | . 2 ⊢ (𝜑 → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))) |
| 5 | 2 | fge0npnf 46382 | . . 3 ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹) |
| 6 | 5 | iffalsed 4536 | . 2 ⊢ (𝜑 → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )) |
| 7 | 4, 6 | eqtrd 2777 | 1 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ifcif 4525 𝒫 cpw 4600 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 supcsup 9480 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 [,)cico 13389 Σcsu 15722 Σ^csumge0 46377 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-icc 13394 df-seq 14043 df-sum 15723 df-sumge0 46378 |
| This theorem is referenced by: sge0z 46390 sge00 46391 fsumlesge0 46392 sge0revalmpt 46393 sge0sn 46394 sge0tsms 46395 sge0cl 46396 sge0fsum 46402 sge0supre 46404 sge0sup 46406 sge0less 46407 sge0resplit 46421 sge0split 46424 |
| Copyright terms: Public domain | W3C validator |