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Theorem sge0val 46321
Description: The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
sge0val ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → (Σ^𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < )))
Distinct variable groups:   𝑤,𝐹,𝑦   𝑦,𝑋
Allowed substitution hints:   𝑉(𝑦,𝑤)   𝑋(𝑤)

Proof of Theorem sge0val
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-sumge0 46318 . . 3 Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)), ℝ*, < )))
21a1i 11 . 2 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)), ℝ*, < ))))
3 rneq 5949 . . . . 5 (𝑥 = 𝐹 → ran 𝑥 = ran 𝐹)
43eleq2d 2824 . . . 4 (𝑥 = 𝐹 → (+∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹))
54adantl 481 . . 3 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (+∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹))
6 dmeq 5916 . . . . . . . . . . . 12 (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
76adantl 481 . . . . . . . . . . 11 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = dom 𝐹)
8 fdm 6745 . . . . . . . . . . . 12 (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
98adantr 480 . . . . . . . . . . 11 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝐹 = 𝑋)
107, 9eqtrd 2774 . . . . . . . . . 10 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = 𝑋)
1110pweqd 4621 . . . . . . . . 9 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → 𝒫 dom 𝑥 = 𝒫 𝑋)
1211ineq1d 4226 . . . . . . . 8 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝒫 dom 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
1312mpteq1d 5242 . . . . . . 7 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)))
1413adantll 714 . . . . . 6 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)))
15 fveq1 6905 . . . . . . . . 9 (𝑥 = 𝐹 → (𝑥𝑤) = (𝐹𝑤))
1615sumeq2sdv 15735 . . . . . . . 8 (𝑥 = 𝐹 → Σ𝑤𝑦 (𝑥𝑤) = Σ𝑤𝑦 (𝐹𝑤))
1716mpteq2dv 5249 . . . . . . 7 (𝑥 = 𝐹 → (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)))
1817adantl 481 . . . . . 6 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)))
1914, 18eqtrd 2774 . . . . 5 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)))
2019rneqd 5951 . . . 4 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)))
2120supeq1d 9483 . . 3 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)), ℝ*, < ) = sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < ))
225, 21ifbieq2d 4556 . 2 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)), ℝ*, < )) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < )))
23 simpr 484 . . 3 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → 𝐹:𝑋⟶(0[,]+∞))
24 simpl 482 . . 3 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → 𝑋𝑉)
2523, 24fexd 7246 . 2 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → 𝐹 ∈ V)
26 pnfxr 11312 . . . 4 +∞ ∈ ℝ*
2726a1i 11 . . 3 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → +∞ ∈ ℝ*)
28 xrltso 13179 . . . . 5 < Or ℝ*
2928supex 9500 . . . 4 sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < ) ∈ V
3029a1i 11 . . 3 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < ) ∈ V)
3127, 30ifexd 4578 . 2 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < )) ∈ V)
322, 22, 25, 31fvmptd 7022 1 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → (Σ^𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  ifcif 4530  𝒫 cpw 4604  cmpt 5230  dom cdm 5688  ran crn 5689  wf 6558  cfv 6562  (class class class)co 7430  Fincfn 8983  supcsup 9477  0cc0 11152  +∞cpnf 11289  *cxr 11291   < clt 11292  [,]cicc 13386  Σcsu 15718  Σ^csumge0 46317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-seq 14039  df-sum 15719  df-sumge0 46318
This theorem is referenced by:  sge0vald  46324
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