Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0gerp | Structured version Visualization version GIF version |
Description: The arbitrary sum of nonnegative extended reals is greater than or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0gerp.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
sge0gerp.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
sge0gerp.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
sge0gerp.z | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝑋 ∩ Fin)𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) |
Ref | Expression |
---|---|
sge0gerp | ⊢ (𝜑 → 𝐴 ≤ (Σ^‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) | |
3 | sge0gerp.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
4 | 3 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞)) |
5 | elinel1 4172 | . . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝑋 ∩ Fin) → 𝑧 ∈ 𝒫 𝑋) | |
6 | elpwi 4551 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝒫 𝑋 → 𝑧 ⊆ 𝑋) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝑋 ∩ Fin) → 𝑧 ⊆ 𝑋) |
8 | 7 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑧 ⊆ 𝑋) |
9 | 4, 8 | fssresd 6540 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑧):𝑧⟶(0[,]+∞)) |
10 | 2, 9 | sge0xrcl 42660 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑧)) ∈ ℝ*) |
11 | 10 | ralrimiva 3182 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑧)) ∈ ℝ*) |
12 | eqid 2821 | . . . . 5 ⊢ (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) = (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) | |
13 | 12 | rnmptss 6881 | . . . 4 ⊢ (∀𝑧 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑧)) ∈ ℝ* → ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) ⊆ ℝ*) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) ⊆ ℝ*) |
15 | sge0gerp.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
16 | sge0gerp.z | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝑋 ∩ Fin)𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) | |
17 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑥 ∈ ℝ+) | |
18 | nfmpt1 5157 | . . . . . . 7 ⊢ Ⅎ𝑧(𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) | |
19 | 18 | nfrn 5819 | . . . . . 6 ⊢ Ⅎ𝑧ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) |
20 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑧 𝐴 ≤ (𝑦 +𝑒 𝑥) | |
21 | 19, 20 | nfrex 3309 | . . . . 5 ⊢ Ⅎ𝑧∃𝑦 ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))𝐴 ≤ (𝑦 +𝑒 𝑥) |
22 | id 22 | . . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝑋 ∩ Fin) → 𝑧 ∈ (𝒫 𝑋 ∩ Fin)) | |
23 | fvexd 6680 | . . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝑋 ∩ Fin) → (Σ^‘(𝐹 ↾ 𝑧)) ∈ V) | |
24 | 12 | elrnmpt1 5825 | . . . . . . . . 9 ⊢ ((𝑧 ∈ (𝒫 𝑋 ∩ Fin) ∧ (Σ^‘(𝐹 ↾ 𝑧)) ∈ V) → (Σ^‘(𝐹 ↾ 𝑧)) ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))) |
25 | 22, 23, 24 | syl2anc 586 | . . . . . . . 8 ⊢ (𝑧 ∈ (𝒫 𝑋 ∩ Fin) → (Σ^‘(𝐹 ↾ 𝑧)) ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))) |
26 | 25 | 3ad2ant2 1130 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) → (Σ^‘(𝐹 ↾ 𝑧)) ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))) |
27 | simp3 1134 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) → 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) | |
28 | nfv 1911 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥) | |
29 | oveq1 7157 | . . . . . . . . 9 ⊢ (𝑦 = (Σ^‘(𝐹 ↾ 𝑧)) → (𝑦 +𝑒 𝑥) = ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) | |
30 | 29 | breq2d 5071 | . . . . . . . 8 ⊢ (𝑦 = (Σ^‘(𝐹 ↾ 𝑧)) → (𝐴 ≤ (𝑦 +𝑒 𝑥) ↔ 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥))) |
31 | 28, 30 | rspce 3612 | . . . . . . 7 ⊢ (((Σ^‘(𝐹 ↾ 𝑧)) ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))) ∧ 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) → ∃𝑦 ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))𝐴 ≤ (𝑦 +𝑒 𝑥)) |
32 | 26, 27, 31 | syl2anc 586 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑧 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥)) → ∃𝑦 ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))𝐴 ≤ (𝑦 +𝑒 𝑥)) |
33 | 32 | 3exp 1115 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑧 ∈ (𝒫 𝑋 ∩ Fin) → (𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥) → ∃𝑦 ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))𝐴 ≤ (𝑦 +𝑒 𝑥)))) |
34 | 17, 21, 33 | rexlimd 3317 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (∃𝑧 ∈ (𝒫 𝑋 ∩ Fin)𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥) → ∃𝑦 ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))𝐴 ≤ (𝑦 +𝑒 𝑥))) |
35 | 16, 34 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧)))𝐴 ≤ (𝑦 +𝑒 𝑥)) |
36 | 1, 14, 15, 35 | supxrge 41598 | . 2 ⊢ (𝜑 → 𝐴 ≤ sup(ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))), ℝ*, < )) |
37 | sge0gerp.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
38 | 37, 3 | sge0sup 42666 | . . 3 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))), ℝ*, < )) |
39 | 38 | eqcomd 2827 | . 2 ⊢ (𝜑 → sup(ran (𝑧 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑧))), ℝ*, < ) = (Σ^‘𝐹)) |
40 | 36, 39 | breqtrd 5085 | 1 ⊢ (𝜑 → 𝐴 ≤ (Σ^‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 class class class wbr 5059 ↦ cmpt 5139 ran crn 5551 ↾ cres 5552 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 supcsup 8898 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 ℝ+crp 12383 +𝑒 cxad 12499 [,]cicc 12735 Σ^csumge0 42637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-xadd 12502 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-sumge0 42638 |
This theorem is referenced by: sge0gerpmpt 42677 |
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