Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0supre Structured version   Visualization version   GIF version

Theorem sge0supre 45749
Description: If the arbitrary sum of nonnegative extended reals is real, then it is the supremum (in the real numbers) of finite subsums. Similar to sge0sup 45751, but here we can use sup with respect to instead of *. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0supre.x (𝜑𝑋𝑉)
sge0supre.f (𝜑𝐹:𝑋⟶(0[,]+∞))
sge0supre.re (𝜑 → (Σ^𝐹) ∈ ℝ)
Assertion
Ref Expression
sge0supre (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem sge0supre
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sge0supre.x . . 3 (𝜑𝑋𝑉)
2 sge0supre.f . . . 4 (𝜑𝐹:𝑋⟶(0[,]+∞))
31adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
42adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
5 simpr 484 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
63, 4, 5sge0pnfval 45733 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
7 sge0supre.re . . . . . . 7 (𝜑 → (Σ^𝐹) ∈ ℝ)
81, 2sge0repnf 45746 . . . . . . 7 (𝜑 → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))
97, 8mpbid 231 . . . . . 6 (𝜑 → ¬ (Σ^𝐹) = +∞)
109adantr 480 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ¬ (Σ^𝐹) = +∞)
116, 10pm2.65da 816 . . . 4 (𝜑 → ¬ +∞ ∈ ran 𝐹)
122, 11fge0iccico 45730 . . 3 (𝜑𝐹:𝑋⟶(0[,)+∞))
131, 12sge0reval 45732 . 2 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
1412sge0rnre 45724 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)
15 sge0rnn0 45728 . . . 4 ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅
1615a1i 11 . . 3 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅)
17 simpr 484 . . . . . . 7 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
18 eqid 2727 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))
1918elrnmpt 5952 . . . . . . . 8 (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) → (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦)))
2019adantl 481 . . . . . . 7 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦)))
2117, 20mpbid 231 . . . . . 6 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦))
22 simp3 1136 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 = Σ𝑦𝑥 (𝐹𝑦))
23 ressxr 11282 . . . . . . . . . . . . . . . 16 ℝ ⊆ ℝ*
2423a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → ℝ ⊆ ℝ*)
2514, 24sstrd 3988 . . . . . . . . . . . . . 14 (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
2625adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ*)
27 id 22 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
28 sumex 15660 . . . . . . . . . . . . . . . 16 Σ𝑦𝑥 (𝐹𝑦) ∈ V
2928a1i 11 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦𝑥 (𝐹𝑦) ∈ V)
3018elrnmpt1 5954 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ Σ𝑦𝑥 (𝐹𝑦) ∈ V) → Σ𝑦𝑥 (𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
3127, 29, 30syl2anc 583 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦𝑥 (𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
3231adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)))
33 supxrub 13329 . . . . . . . . . . . . 13 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ* ∧ Σ𝑦𝑥 (𝐹𝑦) ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → Σ𝑦𝑥 (𝐹𝑦) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
3426, 32, 33syl2anc 583 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ≤ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
3513eqcomd 2733 . . . . . . . . . . . . 13 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
3635adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = (Σ^𝐹))
3734, 36breqtrd 5168 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) ≤ (Σ^𝐹))
38373adant3 1130 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → Σ𝑦𝑥 (𝐹𝑦) ≤ (Σ^𝐹))
3922, 38eqbrtrd 5164 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑤 = Σ𝑦𝑥 (𝐹𝑦)) → 𝑤 ≤ (Σ^𝐹))
40393exp 1117 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → (𝑤 = Σ𝑦𝑥 (𝐹𝑦) → 𝑤 ≤ (Σ^𝐹))))
4140rexlimdv 3148 . . . . . . 7 (𝜑 → (∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦) → 𝑤 ≤ (Σ^𝐹)))
4241adantr 480 . . . . . 6 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → (∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑤 = Σ𝑦𝑥 (𝐹𝑦) → 𝑤 ≤ (Σ^𝐹)))
4321, 42mpd 15 . . . . 5 ((𝜑𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))) → 𝑤 ≤ (Σ^𝐹))
4443ralrimiva 3141 . . . 4 (𝜑 → ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤 ≤ (Σ^𝐹))
45 brralrspcev 5202 . . . 4 (((Σ^𝐹) ∈ ℝ ∧ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤 ≤ (Σ^𝐹)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
467, 44, 45syl2anc 583 . . 3 (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)
47 supxrre 13332 . . 3 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ ∧ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4814, 16, 46, 47syl3anc 1369 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
4913, 48eqtrd 2767 1 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  Vcvv 3469  cin 3943  wss 3944  c0 4318  𝒫 cpw 4598   class class class wbr 5142  cmpt 5225  ran crn 5673  wf 6538  cfv 6542  (class class class)co 7414  Fincfn 8957  supcsup 9457  cr 11131  0cc0 11132  +∞cpnf 11269  *cxr 11271   < clt 11272  cle 11273  [,]cicc 13353  Σcsu 15658  Σ^csumge0 45722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-oi 9527  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-rp 13001  df-ico 13356  df-icc 13357  df-fz 13511  df-fzo 13654  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15458  df-sum 15659  df-sumge0 45723
This theorem is referenced by:  sge0ltfirp  45760  sge0resplit  45766
  Copyright terms: Public domain W3C validator