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Theorem sge0sup 39902
Description: The arbitrary sum of nonnegative extended reals is the supremum of finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0sup.x (𝜑𝑋𝑉)
sge0sup.f (𝜑𝐹:𝑋⟶(0[,]+∞))
Assertion
Ref Expression
sge0sup (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑋   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sge0sup
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2627 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ = +∞)
2 sge0sup.x . . . . 5 (𝜑𝑋𝑉)
32adantr 481 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
4 sge0sup.f . . . . 5 (𝜑𝐹:𝑋⟶(0[,]+∞))
54adantr 481 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
6 simpr 477 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
73, 5, 6sge0pnfval 39884 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
8 vex 3194 . . . . . . . . 9 𝑥 ∈ V
98a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ V)
104adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
11 elinel1 3782 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
12 elpwi 4145 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
1311, 12syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
1413adantl 482 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
1510, 14fssresd 6030 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
169, 15sge0xrcl 39896 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1716adantlr 750 . . . . . 6 (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1817ralrimiva 2965 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹𝑥)) ∈ ℝ*)
19 eqid 2626 . . . . . 6 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))
2019rnmptss 6348 . . . . 5 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) ⊆ ℝ*)
2118, 20syl 17 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) ⊆ ℝ*)
22 ffn 6004 . . . . . . . . 9 (𝐹:𝑋⟶(0[,]+∞) → 𝐹 Fn 𝑋)
234, 22syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
24 fvelrnb 6201 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2523, 24syl 17 . . . . . . 7 (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2625adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
276, 26mpbid 222 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
28 snelpwi 4878 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
29 snfi 7983 . . . . . . . . . . . . 13 {𝑦} ∈ Fin
3029a1i 11 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ Fin)
3128, 30elind 3781 . . . . . . . . . . 11 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
32313ad2ant2 1081 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
33 simp2 1060 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
3443ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
3533snssd 4314 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ⊆ 𝑋)
3634, 35fssresd 6030 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
3733, 36sge0sn 39890 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (Σ^‘(𝐹 ↾ {𝑦})) = ((𝐹 ↾ {𝑦})‘𝑦))
38 vsnid 4185 . . . . . . . . . . . . 13 𝑦 ∈ {𝑦}
39 fvres 6165 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹𝑦))
4038, 39ax-mp 5 . . . . . . . . . . . 12 ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹𝑦)
4140a1i 11 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹𝑦))
42 simp3 1061 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
4337, 41, 423eqtrrd 2665 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (Σ^‘(𝐹 ↾ {𝑦})))
44 reseq2 5355 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
4544fveq2d 6154 . . . . . . . . . . . 12 (𝑥 = {𝑦} → (Σ^‘(𝐹𝑥)) = (Σ^‘(𝐹 ↾ {𝑦})))
4645eqeq2d 2636 . . . . . . . . . . 11 (𝑥 = {𝑦} → (+∞ = (Σ^‘(𝐹𝑥)) ↔ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))))
4746rspcev 3300 . . . . . . . . . 10 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
4832, 43, 47syl2anc 692 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
49 pnfex 10038 . . . . . . . . . 10 +∞ ∈ V
5049a1i 11 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ V)
5119, 48, 50elrnmptd 38826 . . . . . . . 8 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))
52513exp 1261 . . . . . . 7 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))))
5352rexlimdv 3028 . . . . . 6 (𝜑 → (∃𝑦𝑋 (𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))))
5453adantr 481 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦𝑋 (𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))))
5527, 54mpd 15 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))
56 supxrpnf 12088 . . . 4 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = +∞)
5721, 55, 56syl2anc 692 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = +∞)
581, 7, 573eqtr4d 2670 . 2 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
592adantr 481 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
604adantr 481 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
61 simpr 477 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
6260, 61fge0iccico 39881 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
6359, 62sge0reval 39883 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
64 elinel2 3783 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
6564adantl 482 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
6615adantlr 750 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
67 nelrnres 38834 . . . . . . . . . 10 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝑥))
6867ad2antlr 762 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝑥))
6966, 68fge0iccico 39881 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
7065, 69sge0fsum 39898 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
71 simpr 477 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
72 fvres 6165 . . . . . . . . . 10 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7371, 72syl 17 . . . . . . . . 9 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7473sumeq2dv 14362 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7574adantl 482 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7670, 75eqtrd 2660 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 (𝐹𝑦))
7776mpteq2dva 4709 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7877rneqd 5317 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7978supeq1d 8297 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
8063, 79eqtr4d 2663 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
8158, 80pm2.61dan 831 1 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  cin 3559  wss 3560  𝒫 cpw 4135  {csn 4153  cmpt 4678  ran crn 5080  cres 5081   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  Fincfn 7900  supcsup 8291  0cc0 9881  +∞cpnf 10016  *cxr 10018   < clt 10019  [,]cicc 12117  Σcsu 14345  Σ^csumge0 39873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-ico 12120  df-icc 12121  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-sum 14346  df-sumge0 39874
This theorem is referenced by:  sge0gerp  39906  sge0pnffigt  39907  sge0lefi  39909
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