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Theorem sge0sup 46346
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 2735 . . 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 484 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
73, 5, 6sge0pnfval 46328 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
8 vex 3481 . . . . . . . . 9 𝑥 ∈ V
98a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ V)
104adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
11 elinel1 4210 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
12 elpwi 4611 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
1311, 12syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
1413adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
1510, 14fssresd 6775 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
169, 15sge0xrcl 46340 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1716adantlr 715 . . . . . 6 (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1817ralrimiva 3143 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹𝑥)) ∈ ℝ*)
19 eqid 2734 . . . . . 6 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))
2019rnmptss 7142 . . . . 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*)
224ffnd 6737 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
23 fvelrnb 6968 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2422, 23syl 17 . . . . . . 7 (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2524adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
266, 25mpbid 232 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
27 snelpwi 5453 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
28 snfi 9081 . . . . . . . . . . . . 13 {𝑦} ∈ Fin
2928a1i 11 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ Fin)
3027, 29elind 4209 . . . . . . . . . . 11 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
31303ad2ant2 1133 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
32 simp2 1136 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
3343ad2ant1 1132 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
3432snssd 4813 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ⊆ 𝑋)
3533, 34fssresd 6775 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
3632, 35sge0sn 46334 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (Σ^‘(𝐹 ↾ {𝑦})) = ((𝐹 ↾ {𝑦})‘𝑦))
37 vsnid 4667 . . . . . . . . . . . . 13 𝑦 ∈ {𝑦}
38 fvres 6925 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹𝑦))
3937, 38ax-mp 5 . . . . . . . . . . . 12 ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹𝑦)
4039a1i 11 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹𝑦))
41 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
4236, 40, 413eqtrrd 2779 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (Σ^‘(𝐹 ↾ {𝑦})))
43 reseq2 5994 . . . . . . . . . . . 12 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
4443fveq2d 6910 . . . . . . . . . . 11 (𝑥 = {𝑦} → (Σ^‘(𝐹𝑥)) = (Σ^‘(𝐹 ↾ {𝑦})))
4544rspceeqv 3644 . . . . . . . . . 10 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
4631, 42, 45syl2anc 584 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
47 pnfex 11311 . . . . . . . . . 10 +∞ ∈ V
4847a1i 11 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ V)
4919, 46, 48elrnmptd 5976 . . . . . . . 8 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))
50493exp 1118 . . . . . . 7 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))))
5150rexlimdv 3150 . . . . . 6 (𝜑 → (∃𝑦𝑋 (𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))))
5251adantr 480 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦𝑋 (𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))))
5326, 52mpd 15 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))
54 supxrpnf 13356 . . . 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*, < ) = +∞)
5521, 53, 54syl2anc 584 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = +∞)
561, 7, 553eqtr4d 2784 . 2 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
572adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
584adantr 480 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
59 simpr 484 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
6058, 59fge0iccico 46325 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
6157, 60sge0reval 46327 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
62 elinel2 4211 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
6362adantl 481 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
6415adantlr 715 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
65 nelrnres 45129 . . . . . . . . . 10 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝑥))
6665ad2antlr 727 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝑥))
6764, 66fge0iccico 46325 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6863, 67sge0fsum 46342 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
69 simpr 484 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
70 fvres 6925 . . . . . . . . . 10 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7169, 70syl 17 . . . . . . . . 9 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7271sumeq2dv 15734 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7372adantl 481 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7468, 73eqtrd 2774 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 (𝐹𝑦))
7574mpteq2dva 5247 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7675rneqd 5951 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7776supeq1d 9483 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
7861, 77eqtr4d 2777 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
7956, 78pm2.61dan 813 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cin 3961  wss 3962  𝒫 cpw 4604  {csn 4630  cmpt 5230  ran crn 5689  cres 5690   Fn wfn 6557  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-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
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-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  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-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  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-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-sumge0 46318
This theorem is referenced by:  sge0gerp  46350  sge0pnffigt  46351  sge0lefi  46353
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