![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsumrpcl | Structured version Visualization version GIF version |
Description: Closure of a finite sum of positive reals. (Contributed by Mario Carneiro, 3-Jun-2014.) |
Ref | Expression |
---|---|
fsumcl.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumrpcl.2 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
fsumrpcl.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
fsumrpcl | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpssre 12144 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
2 | ax-resscn 10329 | . . . 4 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3829 | . . 3 ⊢ ℝ+ ⊆ ℂ |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → ℝ+ ⊆ ℂ) |
5 | rpaddcl 12161 | . . 3 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 + 𝑦) ∈ ℝ+) | |
6 | 5 | adantl 475 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) → (𝑥 + 𝑦) ∈ ℝ+) |
7 | fsumcl.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
8 | fsumrpcl.3 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+) | |
9 | fsumrpcl.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
10 | 4, 6, 7, 8, 9 | fsumcl2lem 14869 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ≠ wne 2968 ⊆ wss 3791 ∅c0 4140 (class class class)co 6922 Fincfn 8241 ℂcc 10270 ℝcr 10271 + caddc 10275 ℝ+crp 12137 Σcsu 14824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 |
This theorem is referenced by: fsumlt 14936 isumltss 14984 amgmlem 25168 sgmnncl 25325 amgmwlem 43647 |
Copyright terms: Public domain | W3C validator |