| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcl | Structured version Visualization version GIF version | ||
| Description: Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| esumcl.1 | ⊢ Ⅎ𝑘𝐴 |
| Ref | Expression |
|---|---|
| esumcl | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrge0base 33016 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 2 | xrge0cmn 21426 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
| 3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
| 4 | xrge0tps 33941 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
| 5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
| 6 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → 𝐴 ∈ 𝑉) | |
| 7 | esumcl.1 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
| 8 | 7 | nfel1 2922 | . . . . 5 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉 |
| 9 | nfra1 3284 | . . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞) | |
| 10 | 8, 9 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑘(𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
| 11 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
| 12 | simpr 484 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) | |
| 13 | 12 | r19.21bi 3251 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
| 14 | eqid 2737 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
| 15 | 10, 7, 11, 13, 14 | fmptdF 32666 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
| 16 | 1, 3, 5, 6, 15 | tsmscl 24143 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ⊆ (0[,]+∞)) |
| 17 | df-esum 34029 | . . 3 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
| 18 | eqid 2737 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
| 19 | 18, 6, 15 | xrge0tsmsbi 33066 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ↔ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
| 20 | 17, 19 | mpbiri 258 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
| 21 | 16, 20 | sseldd 3984 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∪ cuni 4907 ↦ cmpt 5225 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 ↾s cress 17274 ℝ*𝑠cxrs 17545 CMndccmn 19798 TopSpctps 22938 tsums ctsu 24134 Σ*cesum 34028 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-xadd 13155 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-ordt 17546 df-xrs 17547 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-cntz 19335 df-cmn 19800 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-ntr 23028 df-nei 23106 df-cn 23235 df-haus 23323 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tsms 24135 df-esum 34029 |
| This theorem is referenced by: esumel 34048 esummono 34055 esumpad 34056 esumpad2 34057 esumle 34059 esumlef 34063 esumrnmpt2 34069 esumfsup 34071 esumpinfval 34074 esumpinfsum 34078 esumpmono 34080 esummulc1 34082 esummulc2 34083 esumdivc 34084 hasheuni 34086 esumcvg 34087 esumgect 34091 esum2dlem 34093 esum2d 34094 measiun 34219 omscl 34297 oms0 34299 omsmon 34300 omssubadd 34302 carsggect 34320 carsgclctunlem2 34321 omsmeas 34325 |
| Copyright terms: Public domain | W3C validator |