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 12910 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | sge0pnffsumgt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
5 | 3, 4 | sseldi 3875 | . . 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 43520 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) |
9 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) → 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) | |
10 | nfv 1921 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝑥 ∈ (𝒫 𝐴 ∩ Fin) | |
11 | 1, 10 | nfan 1906 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) |
12 | elinel2 4086 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin) | |
13 | 12 | adantl 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ Fin) |
14 | simpll 767 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝜑) | |
15 | elpwinss 42135 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
16 | 15 | sselda 3877 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴) |
17 | 16 | adantll 714 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝑘 ∈ 𝐴) |
18 | 14, 17, 4 | syl2anc 587 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑘 ∈ 𝑥) → 𝐵 ∈ (0[,)+∞)) |
19 | 11, 13, 18 | sge0fsummptf 43516 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) = Σ𝑘 ∈ 𝑥 𝐵) |
20 | 19 | adantr 484 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) → (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) = Σ𝑘 ∈ 𝑥 𝐵) |
21 | 9, 20 | breqtrd 5056 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) → 𝑌 < Σ𝑘 ∈ 𝑥 𝐵) |
22 | 21 | ex 416 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)) → 𝑌 < Σ𝑘 ∈ 𝑥 𝐵)) |
23 | 22 | reximdva 3184 | . 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 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2114 ∃wrex 3054 ∩ cin 3842 𝒫 cpw 4488 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7170 Fincfn 8555 ℝcr 10614 0cc0 10615 +∞cpnf 10750 < clt 10753 [,)cico 12823 [,]cicc 12824 Σcsu 15135 Σ^csumge0 43442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-sum 15136 df-sumge0 43443 |
This theorem is referenced by: sge0gtfsumgt 43523 |
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