esum0 - Mathbox for Thierry Arnoux

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Theorem esum0 31308

Description: Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)

**Hypothesis**
Ref	Expression
esum0.k	⊢ Ⅎ𝑘𝐴

**Assertion**
Ref	Expression
esum0	⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Distinct variable group: 𝑘,𝑉 Allowed substitution hint: 𝐴(𝑘)

Proof of Theorem esum0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		esum0.k	. . . 4 ⊢ Ⅎ𝑘𝐴
2	1	nfel1 2994	. . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉
3		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0e0iccpnf 12846	. . . 4 ⊢ 0 ∈ (0[,]+∞)
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞))
6		xrge0cmn 20586	. . . . . 6 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
7		cmnmnd 18921	. . . . . 6 ⊢ ((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ CMnd → (ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd)
8	6, 7	ax-mp 5	. . . . 5 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ Mnd
9		vex 3497	. . . . 5 ⊢ 𝑥 ∈ V
10		xrge00 30673	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11	10	gsumz 17999	. . . . 5 ⊢ (((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
12	8, 9, 11	mp2an 690	. . . 4 ⊢ ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0
13	12	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ^*_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
14	2, 1, 3, 5, 13	esumval 31305	. 2 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ))
15		fconstmpt 5613	. . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
16	15	eqcomi 2830	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0})
17		0xr 10687	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
18	17	rgenw 3150	. . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^*
19		eqid 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
20	19	fnmpt 6487	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^* → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin))
21	18, 20	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin)
22		0elpw 5255	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴
23		0fin 8745	. . . . . . . . 9 ⊢ ∅ ∈ Fin
24		elin 4168	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin))
25	22, 23, 24	mpbir2an 709	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
26	25	ne0ii 4302	. . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
27		fconst5 6967	. . . . . . 7 ⊢ (((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) Fn (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≠ ∅) → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}))
28	21, 26, 27	mp2an 690	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})
29	16, 28	mpbi 232	. . . . 5 ⊢ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0}
30	29	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})
31	30	supeq1d 8909	. . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < ))
32		xrltso 12533	. . . 4 ⊢ < Or ℝ^*
33		supsn 8935	. . . 4 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
34	32, 17, 33	mp2an 690	. . 3 ⊢ sup({0}, ℝ^*, < ) = 0
35	31, 34	syl6eq 2872	. 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^*, < ) = 0)
36	14, 35	eqtrd 2856	1 ⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ∅c0 4290 𝒫 cpw 4538 {csn 4566 ↦ cmpt 5145 Or wor 5472 × cxp 5552 ran crn 5555 Fn wfn 6349 (class class class)co 7155 Fincfn 8508 supcsup 8903 0cc0 10536 +∞cpnf 10671 ℝ^*cxr 10673 < clt 10674 [,]cicc 12740 ↾_s cress 16483 Σ_g cgsu 16713 ℝ^*_𝑠cxrs 16772 Mndcmnd 17910 CMndccmn 18905 Σ^*cesum 31286

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-xadd 12507 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-tset 16583 df-ple 16584 df-ds 16586 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-ordt 16773 df-xrs 16774 df-mre 16856 df-mrc 16857 df-acs 16859 df-ps 17809 df-tsr 17810 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-cntz 18446 df-cmn 18907 df-fbas 20541 df-fg 20542 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-ntr 21627 df-nei 21705 df-cn 21834 df-haus 21922 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-tsms 22734 df-esum 31287

This theorem is referenced by: esumpad 31314 esumrnmpt2 31327 measvunilem0 31472 ddemeas 31495

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