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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0pnffsumgt | Structured version Visualization version GIF version |
Description: If the sum of nonnegative extended reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
sge0pnffsumgt.k | ⊢ Ⅎ𝑘𝜑 |
sge0pnffsumgt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0pnffsumgt.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
sge0pnffsumgt.p | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞) |
sge0pnffsumgt.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
Ref | Expression |
---|---|
sge0pnffsumgt | ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘 ∈ 𝑥 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0pnffsumgt.k | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | sge0pnffsumgt.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | icossicc 13413 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | sge0pnffsumgt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
5 | 3, 4 | sselid 3981 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
6 | sge0pnffsumgt.p | . . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞) | |
7 | sge0pnffsumgt.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
8 | 1, 2, 5, 6, 7 | sge0pnffigtmpt 45156 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) |
9 | simpr 486 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) → 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) | |
10 | nfv 1918 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ (𝒫 𝐴 ∩ Fin) | |
11 | 1, 10 | nfan 1903 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) |
12 | elinel2 4197 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin) | |
13 | 12 | adantl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ Fin) |
14 | simpll 766 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝜑) | |
15 | elpwinss 43736 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
16 | 15 | sselda 3983 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴) |
17 | 16 | adantll 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴) |
18 | 14, 17, 4 | syl2anc 585 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ (0[,)+∞)) |
19 | 11, 13, 18 | sge0fsummptf 45152 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) = Σ𝑘 ∈ 𝑥 𝐵) |
20 | 19 | adantr 482 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) → (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) = Σ𝑘 ∈ 𝑥 𝐵) |
21 | 9, 20 | breqtrd 5175 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) → 𝑌 < Σ𝑘 ∈ 𝑥 𝐵) |
22 | 21 | ex 414 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) → 𝑌 < Σ𝑘 ∈ 𝑥 𝐵)) |
23 | 22 | reximdva 3169 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘 ∈ 𝑥 𝐵)) |
24 | 8, 23 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘 ∈ 𝑥 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ∃wrex 3071 ∩ cin 3948 𝒫 cpw 4603 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℝcr 11109 0cc0 11110 +∞cpnf 11245 < clt 11248 [,)cico 13326 [,]cicc 13327 Σcsu 15632 Σ^csumge0 45078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-sumge0 45079 |
This theorem is referenced by: sge0gtfsumgt 45159 |
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