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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 31287 | . . . 4 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | xrge0base 30672 | . . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
3 | xrge00 30673 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
4 | xrge0cmn 20587 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
6 | xrge0tps 31185 | . . . . . . 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 12825 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
10 | esumpfinval.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
11 | 9, 10 | sseldi 3965 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
12 | 11 | fmpttd 6879 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
13 | eqid 2821 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
14 | c0ex 10635 | . . . . . . . 8 ⊢ 0 ∈ V | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
16 | 13, 8, 10, 15 | fsuppmptdm 8844 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
17 | xrge0topn 31186 | . . . . . . 7 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
18 | 17 | eqcomi 2830 | . . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
19 | xrhaus 21993 | . . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus | |
20 | ovex 7189 | . . . . . . . 8 ⊢ (0[,]+∞) ∈ V | |
21 | resthaus 21976 | . . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) | |
22 | 19, 20, 21 | mp2an 690 | . . . . . . 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 22749 | . . . . 5 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
25 | 24 | unieqd 4852 | . . . 4 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
26 | 1, 25 | syl5eq 2868 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
27 | ovex 7189 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
28 | 27 | unisn 4858 | . . 3 ⊢ ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) |
29 | 26, 28 | syl6eq 2872 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
30 | 10 | fmpttd 6879 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
31 | esumpfinvallem 31333 | . . 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 586 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
33 | rge0ssre 12845 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
34 | ax-resscn 10594 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
35 | 33, 34 | sstri 3976 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
36 | 35, 10 | sseldi 3965 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
37 | 8, 36 | gsumfsum 20612 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
38 | 29, 32, 37 | 3eqtr2d 2862 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ∪ cuni 4838 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℂcc 10535 ℝcr 10536 0cc0 10537 +∞cpnf 10672 ≤ cle 10676 [,)cico 12741 [,]cicc 12742 Σcsu 15042 ↾s cress 16484 ↾t crest 16694 TopOpenctopn 16695 Σg cgsu 16714 ordTopcordt 16772 ℝ*𝑠cxrs 16773 CMndccmn 18906 ℂfldccnfld 20545 TopSpctps 21540 Hauscha 21916 tsums ctsu 22734 Σ*cesum 31286 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-xadd 12509 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-ordt 16774 df-xrs 16775 df-ps 17810 df-tsr 17811 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-cn 21835 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-esum 31287 |
This theorem is referenced by: hasheuni 31344 esumcvg 31345 sibfof 31598 |
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