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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0pnffigtmpt | Structured version Visualization version GIF version |
Description: If the generalized sum of nonnegative reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
sge0pnffigtmpt.k | ⊢ Ⅎ𝑘𝜑 |
sge0pnffigtmpt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0pnffigtmpt.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
sge0pnffigtmpt.p | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞) |
sge0pnffigtmpt.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
Ref | Expression |
---|---|
sge0pnffigtmpt | ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0pnffigtmpt.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sge0pnffigtmpt.k | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
3 | sge0pnffigtmpt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
4 | eqid 2824 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | fmptdf 6884 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
6 | sge0pnffigtmpt.p | . . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞) | |
7 | sge0pnffigtmpt.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
8 | 1, 5, 6, 7 | sge0pnffigt 42685 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) |
9 | simpr 487 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) → 𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) | |
10 | elpwinss 41317 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴) | |
11 | 10 | adantr 483 | . . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) → 𝑥 ⊆ 𝐴) |
12 | 11 | resmptd 5911 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵)) |
13 | 12 | fveq2d 6677 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) → (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) = (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) |
14 | 9, 13 | breqtrd 5095 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) → 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) |
15 | 14 | ex 415 | . . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → (𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) → 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)))) |
16 | 15 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) → 𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)))) |
17 | 16 | reximdva 3277 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)))) |
18 | 8, 17 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 ∃wrex 3142 ∩ cin 3938 ⊆ wss 3939 𝒫 cpw 4542 class class class wbr 5069 ↦ cmpt 5149 ↾ cres 5560 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 ℝcr 10539 0cc0 10540 +∞cpnf 10675 < clt 10678 [,]cicc 12744 Σ^csumge0 42651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-sumge0 42652 |
This theorem is referenced by: sge0pnffsumgt 42731 |
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