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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0xrcl | Structured version Visualization version GIF version |
Description: The arbitrary sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0xrcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
sge0xrcl.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
Ref | Expression |
---|---|
sge0xrcl | ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12817 | . 2 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | sge0xrcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | sge0xrcl.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
4 | 2, 3 | sge0cl 42737 | . 2 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞)) |
5 | 1, 4 | sseldi 3962 | 1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ⟶wf 6348 ‘cfv 6352 (class class class)co 7153 0cc0 10534 +∞cpnf 10669 ℝ*cxr 10671 [,]cicc 12739 Σ^csumge0 42718 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-inf2 9101 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 ax-pre-sup 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-sup 8903 df-oi 8971 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-div 11295 df-nn 11636 df-2 11698 df-3 11699 df-n0 11896 df-z 11980 df-uz 12242 df-rp 12388 df-ico 12742 df-icc 12743 df-fz 12891 df-fzo 13032 df-seq 13368 df-exp 13428 df-hash 13689 df-cj 14454 df-re 14455 df-im 14456 df-sqrt 14590 df-abs 14591 df-clim 14841 df-sum 15039 df-sumge0 42719 |
This theorem is referenced by: sge0repnf 42742 sge0fsum 42743 sge0sup 42747 sge0less 42748 sge0gerp 42751 sge0pnffigt 42752 sge0ssre 42753 sge0lefi 42754 sge0le 42763 sge0split 42765 sge0ss 42768 sge0iunmptlemre 42771 sge0iunmpt 42774 sge0rpcpnf 42777 sge0isum 42783 sge0xadd 42791 sge0seq 42802 ismeannd 42823 omeunle 42872 omeiunle 42873 omeiunltfirp 42875 caratheodorylem2 42883 isomenndlem 42886 hoicvrrex 42912 ovnlecvr 42914 ovnsubadd 42928 sge0hsphoire 42945 hoidmv1lelem2 42948 hoidmv1lelem3 42949 hoidmvlelem1 42951 hoidmvlelem5 42955 ovolval5lem2 43009 |
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