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Theorem sge0val 46381
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 46378 . . 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 5947 . . . . 5 (𝑥 = 𝐹 → ran 𝑥 = ran 𝐹)
43eleq2d 2827 . . . 4 (𝑥 = 𝐹 → (+∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹))
54adantl 481 . . 3 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (+∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹))
6 dmeq 5914 . . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = 𝑋)
1110pweqd 4617 . . . . . . . . 9 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → 𝒫 dom 𝑥 = 𝒫 𝑋)
1211ineq1d 4219 . . . . . . . 8 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝒫 dom 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
1312mpteq1d 5237 . . . . . . 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 15739 . . . . . . . 8 (𝑥 = 𝐹 → Σ𝑤𝑦 (𝑥𝑤) = Σ𝑤𝑦 (𝐹𝑤))
1716mpteq2dv 5244 . . . . . . 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 2777 . . . . 5 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)))
2019rneqd 5949 . . . 4 (((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)) = ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)))
2120supeq1d 9486 . . 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 4552 . 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 7247 . 2 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → 𝐹 ∈ V)
26 pnfxr 11315 . . . 4 +∞ ∈ ℝ*
2726a1i 11 . . 3 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → +∞ ∈ ℝ*)
28 xrltso 13183 . . . . 5 < Or ℝ*
2928supex 9503 . . . 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 4574 . 2 ((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < )) ∈ V)
322, 22, 25, 31fvmptd 7023 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 1540  wcel 2108  Vcvv 3480  cin 3950  ifcif 4525  𝒫 cpw 4600  cmpt 5225  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  supcsup 9480  0cc0 11155  +∞cpnf 11292  *cxr 11294   < clt 11295  [,]cicc 13390  Σcsu 15722  Σ^csumge0 46377
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-seq 14043  df-sum 15723  df-sumge0 46378
This theorem is referenced by:  sge0vald  46384
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