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Mathbox for Thierry Arnoux |
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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 12658 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | ax-resscn 10390 | . . . 4 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3860 | . . 3 ⊢ (0[,)+∞) ⊆ ℂ |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
5 | simprl 759 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ (0[,)+∞)) | |
6 | 1, 5 | sseldi 3849 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑥 ∈ ℝ) |
7 | simprr 761 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ (0[,)+∞)) | |
8 | 1, 7 | sseldi 3849 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 𝑦 ∈ ℝ) |
9 | 6, 8 | readdcld 10467 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ ℝ) |
10 | 9 | rexrd 10488 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ ℝ*) |
11 | 0xr 10485 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
12 | pnfxr 10492 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
13 | elico1 12595 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞))) | |
14 | 11, 12, 13 | mp2an 680 | . . . . . 6 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞)) |
15 | 14 | simp2bi 1127 | . . . . 5 ⊢ (𝑥 ∈ (0[,)+∞) → 0 ≤ 𝑥) |
16 | 5, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 0 ≤ 𝑥) |
17 | elico1 12595 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞))) | |
18 | 11, 12, 17 | mp2an 680 | . . . . . 6 ⊢ (𝑦 ∈ (0[,)+∞) ↔ (𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞)) |
19 | 18 | simp2bi 1127 | . . . . 5 ⊢ (𝑦 ∈ (0[,)+∞) → 0 ≤ 𝑦) |
20 | 7, 19 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 0 ≤ 𝑦) |
21 | 6, 8, 16, 20 | addge0d 11015 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → 0 ≤ (𝑥 + 𝑦)) |
22 | ltpnf 12330 | . . . 4 ⊢ ((𝑥 + 𝑦) ∈ ℝ → (𝑥 + 𝑦) < +∞) | |
23 | 9, 22 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) < +∞) |
24 | elico1 12595 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝑥 + 𝑦) ∈ (0[,)+∞) ↔ ((𝑥 + 𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥 + 𝑦) ∧ (𝑥 + 𝑦) < +∞))) | |
25 | 11, 12, 24 | mp2an 680 | . . 3 ⊢ ((𝑥 + 𝑦) ∈ (0[,)+∞) ↔ ((𝑥 + 𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥 + 𝑦) ∧ (𝑥 + 𝑦) < +∞)) |
26 | 10, 21, 23, 25 | syl3anbrc 1324 | . 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 12660 | . . 3 ⊢ 0 ∈ (0[,)+∞) | |
30 | 29 | a1i 11 | . 2 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
31 | 4, 26, 27, 28, 30 | fsumcllem 14947 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 ∈ wcel 2051 ⊆ wss 3822 class class class wbr 4925 (class class class)co 6974 Fincfn 8304 ℂcc 10331 ℝcr 10332 0cc0 10333 + caddc 10336 +∞cpnf 10469 ℝ*cxr 10471 < clt 10472 ≤ cle 10473 [,)cico 12554 Σcsu 14901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-ico 12558 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-sum 14902 |
This theorem is referenced by: esumcvg 31021 |
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