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Theorem sge0sup 40926
 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 2652 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ = +∞)
2 sge0sup.x . . . . 5 (𝜑𝑋𝑉)
32adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
4 sge0sup.f . . . . 5 (𝜑𝐹:𝑋⟶(0[,]+∞))
54adantr 480 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
6 simpr 476 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
73, 5, 6sge0pnfval 40908 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
8 vex 3234 . . . . . . . . 9 𝑥 ∈ V
98a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ V)
104adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
11 elinel1 3832 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
12 elpwi 4201 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
1311, 12syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
1413adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
1510, 14fssresd 6109 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
169, 15sge0xrcl 40920 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1716adantlr 751 . . . . . 6 (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1817ralrimiva 2995 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹𝑥)) ∈ ℝ*)
19 eqid 2651 . . . . . 6 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))
2019rnmptss 6432 . . . . 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 6083 . . . . . . . . 9 (𝐹:𝑋⟶(0[,]+∞) → 𝐹 Fn 𝑋)
234, 22syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
24 fvelrnb 6282 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2523, 24syl 17 . . . . . . 7 (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2625adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
276, 26mpbid 222 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
28 snelpwi 4942 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
29 snfi 8079 . . . . . . . . . . . . 13 {𝑦} ∈ Fin
3029a1i 11 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ Fin)
3128, 30elind 3831 . . . . . . . . . . 11 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
32313ad2ant2 1103 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
33 simp2 1082 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
3443ad2ant1 1102 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
3533snssd 4372 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ⊆ 𝑋)
3634, 35fssresd 6109 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
3733, 36sge0sn 40914 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (Σ^‘(𝐹 ↾ {𝑦})) = ((𝐹 ↾ {𝑦})‘𝑦))
38 vsnid 4242 . . . . . . . . . . . . 13 𝑦 ∈ {𝑦}
39 fvres 6245 . . . . . . . . . . . . 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 1083 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
4337, 41, 423eqtrrd 2690 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (Σ^‘(𝐹 ↾ {𝑦})))
44 reseq2 5423 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
4544fveq2d 6233 . . . . . . . . . . . 12 (𝑥 = {𝑦} → (Σ^‘(𝐹𝑥)) = (Σ^‘(𝐹 ↾ {𝑦})))
4645eqeq2d 2661 . . . . . . . . . . 11 (𝑥 = {𝑦} → (+∞ = (Σ^‘(𝐹𝑥)) ↔ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))))
4746rspcev 3340 . . . . . . . . . 10 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
4832, 43, 47syl2anc 694 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
49 pnfex 10131 . . . . . . . . . 10 +∞ ∈ V
5049a1i 11 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ V)
5119, 48, 50elrnmptd 39680 . . . . . . . 8 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))
52513exp 1283 . . . . . . 7 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))))
5352rexlimdv 3059 . . . . . 6 (𝜑 → (∃𝑦𝑋 (𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))))
5453adantr 480 . . . . 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 12186 . . . 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 694 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = +∞)
581, 7, 573eqtr4d 2695 . 2 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
592adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
604adantr 480 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
61 simpr 476 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
6260, 61fge0iccico 40905 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
6359, 62sge0reval 40907 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
64 elinel2 3833 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
6564adantl 481 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
6615adantlr 751 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
67 nelrnres 39688 . . . . . . . . . 10 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝑥))
6867ad2antlr 763 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝑥))
6966, 68fge0iccico 40905 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
7065, 69sge0fsum 40922 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
71 simpr 476 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
72 fvres 6245 . . . . . . . . . 10 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7371, 72syl 17 . . . . . . . . 9 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7473sumeq2dv 14477 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7574adantl 481 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7670, 75eqtrd 2685 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 (𝐹𝑦))
7776mpteq2dva 4777 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7877rneqd 5385 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7978supeq1d 8393 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
8063, 79eqtr4d 2688 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
8158, 80pm2.61dan 849 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  𝒫 cpw 4191  {csn 4210   ↦ cmpt 4762  ran crn 5144   ↾ cres 5145   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  supcsup 8387  0cc0 9974  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112  [,]cicc 12216  Σcsu 14460  Σ^csumge0 40897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-sumge0 40898 This theorem is referenced by:  sge0gerp  40930  sge0pnffigt  40931  sge0lefi  40933
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