esum0 - Mathbox for Thierry Arnoux

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Theorem esum0 30341

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 2881	. . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉
3		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0e0iccpnf 12397	. . . 4 ⊢ 0 ∈ (0[,]+∞)
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞))
6		xrge0cmn 19911	. . . . . 6 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
7		cmnmnd 18329	. . . . . 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 3307	. . . . 5 ⊢ 𝑥 ∈ V
10		xrge00 29916	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11	10	gsumz 17496	. . . . 5 ⊢ (((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
12	8, 9, 11	mp2an 710	. . . 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 30338	. 2 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ))
15		fconstmpt 5272	. . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
16	15	eqcomi 2733	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0})
17		0xr 10199	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
18	17	rgenw 3026	. . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^*
19		eqid 2724	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
20	19	fnmpt 6133	. . . . . . . 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 4939	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴
23		0fin 8304	. . . . . . . . 9 ⊢ ∅ ∈ Fin
24		elin 3904	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin))
25	22, 23, 24	mpbir2an 993	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
26	25	ne0ii 4031	. . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
27		fconst5 6587	. . . . . . 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 710	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0}) ↔ ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = {0})
29	16, 28	mpbi 220	. . . . 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 8468	. . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < ))
32		xrltso 12088	. . . 4 ⊢ < Or ℝ^*
33		supsn 8494	. . . 4 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
34	32, 17, 33	mp2an 710	. . 3 ⊢ sup({0}, ℝ^*, < ) = 0
35	31, 34	syl6eq 2774	. 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^*, < ) = 0)
36	14, 35	eqtrd 2758	1 ⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1596 ∈ wcel 2103 Ⅎwnfc 2853 ≠ wne 2896 ∀wral 3014 Vcvv 3304 ∩ cin 3679 ∅c0 4023 𝒫 cpw 4266 {csn 4285 ↦ cmpt 4837 Or wor 5138 × cxp 5216 ran crn 5219 Fn wfn 5996 (class class class)co 6765 Fincfn 8072 supcsup 8462 0cc0 10049 +∞cpnf 10184 ℝ^*cxr 10186 < clt 10187 [,]cicc 12292 ↾_s cress 15981 Σ_g cgsu 16224 ℝ^*_𝑠cxrs 16283 Mndcmnd 17416 CMndccmn 18314 Σ^*cesum 30319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-fal 1602 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-iin 4631 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-se 5178 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-isom 6010 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-of 7014 df-om 7183 df-1st 7285 df-2nd 7286 df-supp 7416 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-oadd 7684 df-er 7862 df-map 7976 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-fsupp 8392 df-fi 8433 df-sup 8464 df-inf 8465 df-oi 8531 df-card 8878 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-4 11194 df-5 11195 df-6 11196 df-7 11197 df-8 11198 df-9 11199 df-n0 11406 df-z 11491 df-dec 11607 df-uz 11801 df-q 11903 df-xadd 12061 df-ioo 12293 df-ioc 12294 df-ico 12295 df-icc 12296 df-fz 12441 df-fzo 12581 df-seq 12917 df-hash 13233 df-struct 15982 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-ress 15988 df-plusg 16077 df-mulr 16078 df-tset 16083 df-ple 16084 df-ds 16087 df-rest 16206 df-topn 16207 df-0g 16225 df-gsum 16226 df-topgen 16227 df-ordt 16284 df-xrs 16285 df-mre 16369 df-mrc 16370 df-acs 16372 df-ps 17322 df-tsr 17323 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-submnd 17458 df-cntz 17871 df-cmn 18316 df-fbas 19866 df-fg 19867 df-top 20822 df-topon 20839 df-topsp 20860 df-bases 20873 df-ntr 20947 df-nei 21025 df-cn 21154 df-haus 21242 df-fil 21772 df-fm 21864 df-flim 21865 df-flf 21866 df-tsms 22052 df-esum 30320

This theorem is referenced by: esumpad 30347 esumrnmpt2 30360 measvunilem0 30506 ddemeas 30529

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