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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 13359 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | sge0pnffsumgt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
5 | 3, 4 | sselid 3943 | . . 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 44767 | . 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 4157 | . . . . . . . 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 43345 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
16 | 15 | sselda 3945 | . . . . . . . . 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 44763 | . . . . . 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 5132 | . . . 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 3162 | . 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 3070 ∩ cin 3910 𝒫 cpw 4561 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 Fincfn 8886 ℝcr 11055 0cc0 11056 +∞cpnf 11191 < clt 11194 [,)cico 13272 [,]cicc 13273 Σcsu 15576 Σ^csumge0 44689 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 df-sumge0 44690 |
This theorem is referenced by: sge0gtfsumgt 44770 |
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