Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0sup Structured version   Visualization version   GIF version

Theorem sge0sup 41084
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 2806 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ = +∞)
2 sge0sup.x . . . . 5 (𝜑𝑋𝑉)
32adantr 468 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
4 sge0sup.f . . . . 5 (𝜑𝐹:𝑋⟶(0[,]+∞))
54adantr 468 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
6 simpr 473 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
73, 5, 6sge0pnfval 41066 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
8 vex 3393 . . . . . . . . 9 𝑥 ∈ V
98a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ V)
104adantr 468 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
11 elinel1 3995 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
12 elpwi 4358 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
1311, 12syl 17 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
1413adantl 469 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
1510, 14fssresd 6283 . . . . . . . 8 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
169, 15sge0xrcl 41078 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1716adantlr 697 . . . . . 6 (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) ∈ ℝ*)
1817ralrimiva 3153 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹𝑥)) ∈ ℝ*)
19 eqid 2805 . . . . . 6 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))
2019rnmptss 6611 . . . . 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 6254 . . . . . . . 8 (𝜑𝐹 Fn 𝑋)
23 fvelrnb 6461 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2422, 23syl 17 . . . . . . 7 (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2524adantr 468 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
266, 25mpbid 223 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
27 snelpwi 5099 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
28 snfi 8274 . . . . . . . . . . . . 13 {𝑦} ∈ Fin
2928a1i 11 . . . . . . . . . . . 12 (𝑦𝑋 → {𝑦} ∈ Fin)
3027, 29elind 3994 . . . . . . . . . . 11 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
31303ad2ant2 1157 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
32 simp2 1160 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
3343ad2ant1 1156 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
3432snssd 4527 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ⊆ 𝑋)
3533, 34fssresd 6283 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
3632, 35sge0sn 41072 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (Σ^‘(𝐹 ↾ {𝑦})) = ((𝐹 ↾ {𝑦})‘𝑦))
37 vsnid 4400 . . . . . . . . . . . . 13 𝑦 ∈ {𝑦}
38 fvres 6424 . . . . . . . . . . . . 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 1161 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
4236, 40, 413eqtrrd 2844 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (Σ^‘(𝐹 ↾ {𝑦})))
43 reseq2 5589 . . . . . . . . . . . 12 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
4443fveq2d 6409 . . . . . . . . . . 11 (𝑥 = {𝑦} → (Σ^‘(𝐹𝑥)) = (Σ^‘(𝐹 ↾ {𝑦})))
4544rspceeqv 3519 . . . . . . . . . 10 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
4631, 42, 45syl2anc 575 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹𝑥)))
47 pnfex 10375 . . . . . . . . . 10 +∞ ∈ V
4847a1i 11 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ V)
4919, 46, 48elrnmptd 39852 . . . . . . . 8 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))
50493exp 1141 . . . . . . 7 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))))))
5150rexlimdv 3217 . . . . . 6 (𝜑 → (∃𝑦𝑋 (𝐹𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥)))))
5251adantr 468 . . . . 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 12362 . . . 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 575 . . 3 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = +∞)
561, 7, 553eqtr4d 2849 . 2 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
572adantr 468 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
584adantr 468 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
59 simpr 473 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
6058, 59fge0iccico 41063 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
6157, 60sge0reval 41065 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
62 elinel2 3996 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
6362adantl 469 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
6415adantlr 697 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
65 nelrnres 39860 . . . . . . . . . 10 (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹𝑥))
6665ad2antlr 709 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹𝑥))
6764, 66fge0iccico 41063 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,)+∞))
6863, 67sge0fsum 41080 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 ((𝐹𝑥)‘𝑦))
69 simpr 473 . . . . . . . . . 10 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑥)
70 fvres 6424 . . . . . . . . . 10 (𝑦𝑥 → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7169, 70syl 17 . . . . . . . . 9 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → ((𝐹𝑥)‘𝑦) = (𝐹𝑦))
7271sumeq2dv 14652 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7372adantl 469 . . . . . . 7 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 ((𝐹𝑥)‘𝑦) = Σ𝑦𝑥 (𝐹𝑦))
7468, 73eqtrd 2839 . . . . . 6 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹𝑥)) = Σ𝑦𝑥 (𝐹𝑦))
7574mpteq2dva 4934 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7675rneqd 5551 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
7776supeq1d 8588 . . 3 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
7861, 77eqtr4d 2842 . 2 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))
7956, 78pm2.61dan 838 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2158  wral 3095  wrex 3096  Vcvv 3390  cin 3765  wss 3766  𝒫 cpw 4348  {csn 4367  cmpt 4919  ran crn 5309  cres 5310   Fn wfn 6093  wf 6094  cfv 6098  (class class class)co 6871  Fincfn 8189  supcsup 8582  0cc0 10218  +∞cpnf 10353  *cxr 10355   < clt 10356  [,]cicc 12392  Σcsu 14635  Σ^csumge0 41055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-inf2 8782  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295  ax-pre-sup 10296
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-se 5268  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-isom 6107  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-sup 8584  df-oi 8651  df-card 9045  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-div 10967  df-nn 11303  df-2 11360  df-3 11361  df-n0 11556  df-z 11640  df-uz 11901  df-rp 12043  df-ico 12395  df-icc 12396  df-fz 12546  df-fzo 12686  df-seq 13021  df-exp 13080  df-hash 13334  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-sum 14636  df-sumge0 41056
This theorem is referenced by:  sge0gerp  41088  sge0pnffigt  41089  sge0lefi  41091
  Copyright terms: Public domain W3C validator