esumpfinval - Mathbox for Thierry Arnoux

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Theorem esumpfinval 30137

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.)

**Hypotheses**
Ref	Expression
esumpfinval.a	⊢ (𝜑 → 𝐴 ∈ Fin)
esumpfinval.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))

**Assertion**
Ref	Expression
esumpfinval	⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Distinct variable groups: 𝐴,𝑘 𝜑,𝑘 Allowed substitution hint: 𝐵(𝑘)

**Proof of Theorem esumpfinval**
Step	Hyp	Ref	Expression
1		df-esum 30090	. . . 4 ⊢ Σ^𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
2		xrge0base 29685	. . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
3		xrge00 29686	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
4		xrge0cmn 19788	. . . . . . 7 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd)
6		xrge0tps 29988	. . . . . . 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 12260	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
10		esumpfinval.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
11	9, 10	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
12		eqid 2622	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
13	11, 12	fmptd 6385	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
14		c0ex 10034	. . . . . . . 8 ⊢ 0 ∈ V
15	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
16	12, 8, 10, 15	fsuppmptdm 8286	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0)
17		xrge0topn 29989	. . . . . . 7 ⊢ (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
18	17	eqcomi 2631	. . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
19		xrhaus 29535	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus
20		ovex 6678	. . . . . . . 8 ⊢ (0[,]+∞) ∈ V
21		resthaus 21172	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
22	19, 20, 21	mp2an 708	. . . . . . 7 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ Haus)
24	2, 3, 5, 7, 8, 13, 16, 18, 23	haustsmsid 21944	. . . . 5 ⊢ (𝜑 → ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
25	24	unieqd 4446	. . . 4 ⊢ (𝜑 → ∪ ((ℝ^_𝑠 ↾_s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
26	1, 25	syl5eq 2668	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))})
27		ovex 6678	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V
28	27	unisn 4451	. . 3 ⊢ ∪ {((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵))
29	26, 28	syl6eq 2672	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
30	10, 12	fmptd 6385	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞))
31		esumpfinvallem 30136	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
32	8, 30, 31	syl2anc 693	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)))
33		rge0ssre 12280	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
34		ax-resscn 9993	. . . . 5 ⊢ ℝ ⊆ ℂ
35	33, 34	sstri 3612	. . . 4 ⊢ (0[,)+∞) ⊆ ℂ
36	35, 10	sseldi 3601	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
37	8, 36	gsumfsum 19813	. 2 ⊢ (𝜑 → (ℂ_fld Σ_g (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
38	29, 32, 37	3eqtr2d 2662	1 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 ∪ cuni 4436 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ≤ cle 10075 [,)cico 12177 [,]cicc 12178 Σcsu 14416 ↾_s cress 15858 ↾_t crest 16081 TopOpenctopn 16082 Σ_g cgsu 16101 ordTopcordt 16159 ℝ^*_𝑠cxrs 16160 CMndccmn 18193 ℂ_fldccnfld 19746 TopSpctps 20736 Hauscha 21112 tsums ctsu 21929 Σ^*cesum 30089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-xadd 11947 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-esum 30090

This theorem is referenced by: hasheuni 30147 esumcvg 30148 sibfof 30402

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