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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumrp0cl | Structured version Visualization version GIF version |
Description: Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.) |
Ref | Expression |
---|---|
fsumrp0cl.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumrp0cl.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
fsumrp0cl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 13009 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | ax-resscn 10751 | . . . 4 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3896 | . . 3 ⊢ (0[,)+∞) ⊆ ℂ |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
5 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ (0[,)+∞)) | |
6 | 1, 5 | sseldi 3885 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ ℝ) |
7 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ (0[,)+∞)) | |
8 | 1, 7 | sseldi 3885 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ ℝ) |
9 | 6, 8 | readdcld 10827 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ ℝ) |
10 | 9 | rexrd 10848 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ ℝ*) |
11 | 0xr 10845 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
12 | pnfxr 10852 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
13 | elico1 12943 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))) | |
14 | 11, 12, 13 | mp2an 692 | . . . . . 6 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) |
15 | 14 | simp2bi 1148 | . . . . 5 ⊢ (𝑥 ∈ (0[,)+∞) → 0 ≤ 𝑥) |
16 | 5, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 0 ≤ 𝑥) |
17 | elico1 12943 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))) | |
18 | 11, 12, 17 | mp2an 692 | . . . . . 6 ⊢ (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)) |
19 | 18 | simp2bi 1148 | . . . . 5 ⊢ (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦) |
20 | 7, 19 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 0 ≤ 𝑦) |
21 | 6, 8, 16, 20 | addge0d 11373 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 0 ≤ (𝑥 + 𝑦)) |
22 | ltpnf 12677 | . . . 4 ⊢ ((𝑥 + 𝑦) ∈ ℝ → (𝑥 + 𝑦) < +∞) | |
23 | 9, 22 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) < +∞) |
24 | elico1 12943 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑥 + 𝑦) ∈ (0[,)+∞) ↔ ((𝑥 + 𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥 + 𝑦) ∧ (𝑥 + 𝑦) < +∞))) | |
25 | 11, 12, 24 | mp2an 692 | . . 3 ⊢ ((𝑥 + 𝑦) ∈ (0[,)+∞) ↔ ((𝑥 + 𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥 + 𝑦) ∧ (𝑥 + 𝑦) < +∞)) |
26 | 10, 21, 23, 25 | syl3anbrc 1345 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞)) |
27 | fsumrp0cl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
28 | fsumrp0cl.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
29 | 0e0icopnf 13011 | . . 3 ⊢ 0 ∈ (0[,)+∞) | |
30 | 29 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
31 | 4, 26, 27, 28, 30 | fsumcllem 15261 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 ⊆ wss 3853 class class class wbr 5039 (class class class)co 7191 Fincfn 8604 ℂcc 10692 ℝcr 10693 0cc0 10694 + caddc 10697 +∞cpnf 10829 ℝ*cxr 10831 < clt 10832 ≤ cle 10833 [,)cico 12902 Σcsu 15214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 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 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-ico 12906 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 |
This theorem is referenced by: esumcvg 31720 |
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