esum0 - Mathbox for Thierry Arnoux

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Theorem esum0 30451

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 2928	. . 3 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉
3		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0e0iccpnf 12490	. . . 4 ⊢ 0 ∈ (0[,]+∞)
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑘 ∈ 𝐴) → 0 ∈ (0[,]+∞))
6		xrge0cmn 20003	. . . . . 6 ⊢ (ℝ^*_𝑠 ↾_s (0[,]+∞)) ∈ CMnd
7		cmnmnd 18415	. . . . . 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 3354	. . . . 5 ⊢ 𝑥 ∈ V
10		xrge00 30026	. . . . . 6 ⊢ 0 = (0_g‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
11	10	gsumz 17582	. . . . 5 ⊢ (((ℝ^_𝑠 ↾_s (0[,]+∞)) ∈ Mnd ∧ 𝑥 ∈ V) → ((ℝ^_𝑠 ↾_s (0[,]+∞)) Σ_g (𝑘 ∈ 𝑥 ↦ 0)) = 0)
12	8, 9, 11	mp2an 672	. . . 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 30448	. 2 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑘 ∈ 𝐴0 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ))
15		fconstmpt 5303	. . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) × {0}) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
16	15	eqcomi 2780	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = ((𝒫 𝐴 ∩ Fin) × {0})
17		0xr 10288	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
18	17	rgenw 3073	. . . . . . . 8 ⊢ ∀𝑥 ∈ (𝒫 𝐴 ∩ Fin)0 ∈ ℝ^*
19		eqid 2771	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0) = (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0)
20	19	fnmpt 6160	. . . . . . . 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 4965	. . . . . . . . 9 ⊢ ∅ ∈ 𝒫 𝐴
23		0fin 8344	. . . . . . . . 9 ⊢ ∅ ∈ Fin
24		elin 3947	. . . . . . . . 9 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) ↔ (∅ ∈ 𝒫 𝐴 ∧ ∅ ∈ Fin))
25	22, 23, 24	mpbir2an 690	. . . . . . . 8 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
26	25	ne0ii 4071	. . . . . . 7 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
27		fconst5 6615	. . . . . . 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 672	. . . . . 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 8508	. . 3 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^, < ) = sup({0}, ℝ^, < ))
32		xrltso 12179	. . . 4 ⊢ < Or ℝ^*
33		supsn 8534	. . . 4 ⊢ (( < Or ℝ^* ∧ 0 ∈ ℝ^) → sup({0}, ℝ^, < ) = 0)
34	32, 17, 33	mp2an 672	. . 3 ⊢ sup({0}, ℝ^*, < ) = 0
35	31, 34	syl6eq 2821	. 2 ⊢ (𝐴 ∈ 𝑉 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 0), ℝ^*, < ) = 0)
36	14, 35	eqtrd 2805	1 ⊢ (𝐴 ∈ 𝑉 → Σ^*𝑘 ∈ 𝐴0 = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Ⅎwnfc 2900 ≠ wne 2943 ∀wral 3061 Vcvv 3351 ∩ cin 3722 ∅c0 4063 𝒫 cpw 4297 {csn 4316 ↦ cmpt 4863 Or wor 5169 × cxp 5247 ran crn 5250 Fn wfn 6026 (class class class)co 6793 Fincfn 8109 supcsup 8502 0cc0 10138 +∞cpnf 10273 ℝ^*cxr 10275 < clt 10276 [,]cicc 12383 ↾_s cress 16065 Σ_g cgsu 16309 ℝ^*_𝑠cxrs 16368 Mndcmnd 17502 CMndccmn 18400 Σ^*cesum 30429

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-xadd 12152 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-seq 13009 df-hash 13322 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-tset 16168 df-ple 16169 df-ds 16172 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-ordt 16369 df-xrs 16370 df-mre 16454 df-mrc 16455 df-acs 16457 df-ps 17408 df-tsr 17409 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-cntz 17957 df-cmn 18402 df-fbas 19958 df-fg 19959 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-ntr 21045 df-nei 21123 df-cn 21252 df-haus 21340 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-tsms 22150 df-esum 30430

This theorem is referenced by: esumpad 30457 esumrnmpt2 30470 measvunilem0 30616 ddemeas 30639

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