Mathbox for Glauco Siliprandi |
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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 42667 | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
4 | 1, 3 | sge0vald 42671 | . 2 ⊢ (𝜑 → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))) |
5 | 2 | fge0npnf 42669 | . . 3 ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹) |
6 | 5 | iffalsed 4478 | . 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 2856 | 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 1537 ∈ wcel 2114 ∩ cin 3935 ifcif 4467 𝒫 cpw 4539 ↦ cmpt 5146 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 supcsup 8904 0cc0 10537 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 [,)cico 12741 Σcsu 15042 Σ^csumge0 42664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-ico 12745 df-icc 12746 df-fz 12894 df-seq 13371 df-sum 15043 df-sumge0 42665 |
This theorem is referenced by: sge0z 42677 sge00 42678 fsumlesge0 42679 sge0revalmpt 42680 sge0sn 42681 sge0tsms 42682 sge0cl 42683 sge0fsum 42689 sge0supre 42691 sge0sup 42693 sge0less 42694 sge0resplit 42708 sge0split 42711 |
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