Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0pnfval | Structured version Visualization version GIF version |
Description: If a term in the sum of nonnegative extended reals is +∞, then the value of the sum is +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0pnfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
sge0pnfval.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
sge0pnfval.pnf | ⊢ (𝜑 → +∞ ∈ ran 𝐹) |
Ref | Expression |
---|---|
sge0pnfval | ⊢ (𝜑 → (Σ^‘𝐹) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0pnfval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | sge0pnfval.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
3 | 1, 2 | sge0vald 40586 | . 2 ⊢ (𝜑 → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))) |
4 | sge0pnfval.pnf | . . 3 ⊢ (𝜑 → +∞ ∈ ran 𝐹) | |
5 | 4 | iftrued 4094 | . 2 ⊢ (𝜑 → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )) = +∞) |
6 | 3, 5 | eqtrd 2656 | 1 ⊢ (𝜑 → (Σ^‘𝐹) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ifcif 4086 𝒫 cpw 4158 ↦ cmpt 4729 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 supcsup 8346 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 [,]cicc 12178 Σcsu 14416 Σ^csumge0 40579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-sum 14417 df-sumge0 40580 |
This theorem is referenced by: sge0sn 40596 sge0tsms 40597 sge0cl 40598 sge0f1o 40599 sge0rern 40605 sge0supre 40606 sge0sup 40608 sge0pr 40611 sge0le 40624 sge0split 40626 sge0iunmpt 40635 sge0pnfmpt 40662 |
Copyright terms: Public domain | W3C validator |