esum0 - Mathbox for Thierry Arnoux

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Theorem esum0 31917

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 2922	. . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉
3		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0e0iccpnf 13120	. . . 4 ⊢ 0 ∈ (0[,]+∞)
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞))
6		xrge0cmn 20552	. . . . . 6 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
7		cmnmnd 19317	. . . . . 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 3426	. . . . 5 ⊢ 𝑥 ∈ V
10		xrge00 31197	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11	10	gsumz 18389	. . . . 5 ⊢ (((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
12	8, 9, 11	mp2an 688	. . . 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 31914	. 2 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ))
15		fconstmpt 5640	. . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
16	15	eqcomi 2747	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0})
17		0xr 10953	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
18	17	rgenw 3075	. . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^*
19		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
20	19	fnmpt 6557	. . . . . . . 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 5273	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴
23		0fin 8916	. . . . . . . . 9 ⊢ ∅ ∈ Fin
24		elin 3899	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin))
25	22, 23, 24	mpbir2an 707	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
26	25	ne0ii 4268	. . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
27		fconst5 7063	. . . . . . 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 688	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})
29	16, 28	mpbi 229	. . . . 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 9135	. . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < ))
32		xrltso 12804	. . . 4 ⊢ < Or ℝ^*
33		supsn 9161	. . . 4 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
34	32, 17, 33	mp2an 688	. . 3 ⊢ sup({0}, ℝ^*, < ) = 0
35	31, 34	eqtrdi 2795	. 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^*, < ) = 0)
36	14, 35	eqtrd 2778	1 ⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Ⅎwnfc 2886 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ∩ cin 3882 ∅c0 4253 𝒫 cpw 4530 {csn 4558 ↦ cmpt 5153 Or wor 5493 × cxp 5578 ran crn 5581 Fn wfn 6413 (class class class)co 7255 Fincfn 8691 supcsup 9129 0cc0 10802 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 [,]cicc 13011 ↾_s cress 16867 Σ_g cgsu 17068 ℝ^*_𝑠cxrs 17128 Mndcmnd 18300 CMndccmn 19301 Σ^*cesum 31895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-xadd 12778 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-tset 16907 df-ple 16908 df-ds 16910 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-ordt 17129 df-xrs 17130 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-cntz 18838 df-cmn 19303 df-fbas 20507 df-fg 20508 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-ntr 22079 df-nei 22157 df-cn 22286 df-haus 22374 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tsms 23186 df-esum 31896

This theorem is referenced by: esumpad 31923 esumrnmpt2 31936 measvunilem0 32081 ddemeas 32104

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