sge0xrcl - Mathbox for Glauco Siliprandi

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Theorem sge0xrcl 45835

Description: The arbitrary sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
sge0xrcl.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0xrcl.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

**Assertion**
Ref	Expression
sge0xrcl	⊢ (𝜑 → (Σ^{^}‘𝐹) ∈ ℝ^*)

**Proof of Theorem sge0xrcl**
Step	Hyp	Ref	Expression
1		iccssxr 13437	. 2 ⊢ (0[,]+∞) ⊆ ℝ^*
2		sge0xrcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		sge0xrcl.f	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
4	2, 3	sge0cl 45831	. 2 ⊢ (𝜑 → (Σ^{^}‘𝐹) ∈ (0[,]+∞))
5	1, 4	sselid 3970	1 ⊢ (𝜑 → (Σ^{^}‘𝐹) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2098 ⟶wf 6538 ‘cfv 6542 (class class class)co 7415 0cc0 11136 +∞cpnf 11273 ℝ^*cxr 11275 [,]cicc 13357 Σ^{^}csumge0 45812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-sumge0 45813

This theorem is referenced by: sge0repnf 45836 sge0fsum 45837 sge0sup 45841 sge0less 45842 sge0gerp 45845 sge0pnffigt 45846 sge0ssre 45847 sge0lefi 45848 sge0le 45857 sge0split 45859 sge0ss 45862 sge0iunmptlemre 45865 sge0iunmpt 45868 sge0rpcpnf 45871 sge0isum 45877 sge0xadd 45885 sge0seq 45896 ismeannd 45917 omeunle 45966 omeiunle 45967 omeiunltfirp 45969 caratheodorylem2 45977 isomenndlem 45980 hoicvrrex 46006 ovnlecvr 46008 ovnsubadd 46022 sge0hsphoire 46039 hoidmv1lelem2 46042 hoidmv1lelem3 46043 hoidmvlelem1 46045 hoidmvlelem5 46049 ovolval5lem2 46103

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