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Theorem sge0pnfval 46329
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.)
Hypotheses
Ref Expression
sge0pnfval.x (𝜑𝑋𝑉)
sge0pnfval.f (𝜑𝐹:𝑋⟶(0[,]+∞))
sge0pnfval.pnf (𝜑 → +∞ ∈ ran 𝐹)
Assertion
Ref Expression
sge0pnfval (𝜑 → (Σ^𝐹) = +∞)

Proof of Theorem sge0pnfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0pnfval.x . . 3 (𝜑𝑋𝑉)
2 sge0pnfval.f . . 3 (𝜑𝐹:𝑋⟶(0[,]+∞))
31, 2sge0vald 46325 . 2 (𝜑 → (Σ^𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < )))
4 sge0pnfval.pnf . . 3 (𝜑 → +∞ ∈ ran 𝐹)
54iftrued 4539 . 2 (𝜑 → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < )) = +∞)
63, 5eqtrd 2775 1 (𝜑 → (Σ^𝐹) = +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cin 3962  ifcif 4531  𝒫 cpw 4605  cmpt 5231  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  supcsup 9478  0cc0 11153  +∞cpnf 11290  *cxr 11292   < clt 11293  [,]cicc 13387  Σcsu 15719  Σ^csumge0 46318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-seq 14040  df-sum 15720  df-sumge0 46319
This theorem is referenced by:  sge0sn  46335  sge0tsms  46336  sge0cl  46337  sge0f1o  46338  sge0rern  46344  sge0supre  46345  sge0sup  46347  sge0pr  46350  sge0le  46363  sge0split  46365  sge0iunmpt  46374  sge0pnfmpt  46401
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