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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpfinval | Structured version Visualization version GIF version |
Description: The value of the extended sum of a finite set of nonnegative finite terms. (Contributed by Thierry Arnoux, 28-Jun-2017.) (Proof shortened by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
esumpfinval.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
esumpfinval.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
esumpfinval | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 33489 | . . . 4 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | xrge0base 32618 | . . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
3 | xrge00 32619 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
4 | xrge0cmn 21275 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
6 | xrge0tps 33385 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
8 | esumpfinval.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
9 | icossicc 13420 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
10 | esumpfinval.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
11 | 9, 10 | sselid 3980 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
12 | 11 | fmpttd 7116 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
13 | eqid 2731 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
14 | c0ex 11215 | . . . . . . . 8 ⊢ 0 ∈ V | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
16 | 13, 8, 10, 15 | fsuppmptdm 9380 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
17 | xrge0topn 33386 | . . . . . . 7 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
18 | 17 | eqcomi 2740 | . . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
19 | xrhaus 23208 | . . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus | |
20 | ovex 7445 | . . . . . . . 8 ⊢ (0[,]+∞) ∈ V | |
21 | resthaus 23191 | . . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) | |
22 | 19, 20, 21 | mp2an 689 | . . . . . . 7 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) |
24 | 2, 3, 5, 7, 8, 12, 16, 18, 23 | haustsmsid 23964 | . . . . 5 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
25 | 24 | unieqd 4922 | . . . 4 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
26 | 1, 25 | eqtrid 2783 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
27 | ovex 7445 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
28 | 27 | unisn 4930 | . . 3 ⊢ ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) |
29 | 26, 28 | eqtrdi 2787 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
30 | 10 | fmpttd 7116 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
31 | esumpfinvallem 33535 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) | |
32 | 8, 30, 31 | syl2anc 583 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
33 | rge0ssre 13440 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
34 | ax-resscn 11173 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
35 | 33, 34 | sstri 3991 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
36 | 35, 10 | sselid 3980 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
37 | 8, 36 | gsumfsum 21300 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
38 | 29, 32, 37 | 3eqtr2d 2777 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4628 ∪ cuni 4908 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 Fincfn 8945 ℂcc 11114 ℝcr 11115 0cc0 11116 +∞cpnf 11252 ≤ cle 11256 [,)cico 13333 [,]cicc 13334 Σcsu 15639 ↾s cress 17180 ↾t crest 17373 TopOpenctopn 17374 Σg cgsu 17393 ordTopcordt 17452 ℝ*𝑠cxrs 17453 CMndccmn 19696 ℂfldccnfld 21232 TopSpctps 22753 Hauscha 23131 tsums ctsu 23949 Σ*cesum 33488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-xadd 13100 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-ordt 17454 df-xrs 17455 df-ps 18529 df-tsr 18530 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-ur 20083 df-ring 20136 df-cring 20137 df-fbas 21229 df-fg 21230 df-cnfld 21233 df-top 22715 df-topon 22732 df-topsp 22754 df-bases 22768 df-cld 22842 df-ntr 22843 df-cls 22844 df-nei 22921 df-cn 23050 df-haus 23138 df-fil 23669 df-fm 23761 df-flim 23762 df-flf 23763 df-tsms 23950 df-esum 33489 |
This theorem is referenced by: hasheuni 33546 esumcvg 33547 sibfof 33802 |
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