esumpinfval - Mathbox for Thierry Arnoux

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Theorem esumpinfval 34053

Description: The value of the extended sum of nonnegative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.)

**Hypotheses**
Ref	Expression
esumpinfval.0	⊢ Ⅎ𝑘𝜑
esumpinfval.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumpinfval.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
esumpinfval.3	⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞)

**Assertion**
Ref	Expression
esumpinfval	⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = +∞)

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

**Proof of Theorem esumpinfval**
Step	Hyp	Ref	Expression
1		iccssxr 13466	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
2		esumpinfval.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		esumpinfval.0	. . . . 5 ⊢ Ⅎ𝑘𝜑
4		esumpinfval.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
5	4	ex 412	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞)))
6	3, 5	ralrimi 3254	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 34010	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 584	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sselid 3992	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		nfrab1 3453	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}
12		ssrab2 4089	. . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴)
14		0xr 11305	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		pnfxr 11312	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
16		0lepnf 13171	. . . . . . . 8 ⊢ 0 ≤ +∞
17		ubicc2 13501	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
18	14, 15, 16, 17	mp3an 1460	. . . . . . 7 ⊢ +∞ ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞))
20		0e0iccpnf 13495	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
21	20	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞))
22	19, 21	ifclda 4565	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞))
23		eldif 3972	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
24		rabid 3454	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = +∞))
25	24	simplbi2 500	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (𝐵 = +∞ → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
26	25	con3dimp 408	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞)
27	23, 26	sylbi 217	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞)
28	27	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞)
29	28	iffalsed 4541	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0)
30	3, 11, 7, 13, 2, 22, 29	esumss 34052	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0))
31		eqidd 2735	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})
32	24	simprbi 496	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞)
33	32	iftrued 4538	. . . . . . 7 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → if(𝐵 = +∞, +∞, 0) = +∞)
34	33	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → if(𝐵 = +∞, +∞, 0) = +∞)
35	3, 31, 34	esumeq12dvaf 34011	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞)
36	2, 13	ssexd 5329	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V)
37		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑘+∞
38	11, 37	esumcst 34043	. . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞))
39	36, 18, 38	sylancl 586	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞))
40		hashxrcl 14392	. . . . . . 7 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V → (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ^*)
41	36, 40	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ^*)
42		esumpinfval.3	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞)
43		rabn0 4394	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞)
44	42, 43	sylibr 234	. . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅)
45		hashgt0 14423	. . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
46	36, 44, 45	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
47		xmulpnf1 13312	. . . . . 6 ⊢ (((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ^* ∧ 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞) = +∞)
48	41, 46, 47	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞) = +∞)
49	35, 39, 48	3eqtrd 2778	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞)
50	30, 49	eqtr3d 2776	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞)
51		breq1 5150	. . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
52		breq1 5150	. . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
53		pnfge 13169	. . . . . . . 8 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
54	15, 53	ax-mp 5	. . . . . . 7 ⊢ +∞ ≤ +∞
55		breq2 5151	. . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞))
56	54, 55	mpbiri 258	. . . . . 6 ⊢ (𝐵 = +∞ → +∞ ≤ 𝐵)
57	56	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐵)
58	4	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ (0[,]+∞))
59		iccgelb 13439	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵)
60	14, 15, 59	mp3an12 1450	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵)
61	58, 60	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵)
62	51, 52, 57, 61	ifbothda 4568	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵)
63	3, 7, 2, 22, 4, 62	esumlef 34042	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ^𝑘 ∈ 𝐴𝐵)
64	50, 63	eqbrtrrd 5171	. 2 ⊢ (𝜑 → +∞ ≤ Σ^*𝑘 ∈ 𝐴𝐵)
65		xgepnf 13203	. . 3 ⊢ (Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^ → (+∞ ≤ Σ^𝑘 ∈ 𝐴𝐵 ↔ Σ^𝑘 ∈ 𝐴𝐵 = +∞))
66	65	biimpd 229	. 2 ⊢ (Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^ → (+∞ ≤ Σ^𝑘 ∈ 𝐴𝐵 → Σ^𝑘 ∈ 𝐴𝐵 = +∞))
67	10, 64, 66	sylc 65	1 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3432 Vcvv 3477 ∖ cdif 3959 ⊆ wss 3962 ∅c0 4338 ifcif 4530 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 0cc0 11152 +∞cpnf 11289 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 ·_e cxmu 13150 [,]cicc 13386 ♯chash 14365 Σ^*cesum 34007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-ordt 17547 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-ps 18623 df-tsr 18624 df-plusf 18664 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-abv 20826 df-lmod 20876 df-scaf 20877 df-sra 21189 df-rgmod 21190 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-tmd 24095 df-tgp 24096 df-tsms 24150 df-trg 24183 df-xms 24345 df-ms 24346 df-tms 24347 df-nm 24610 df-ngp 24611 df-nrg 24613 df-nlm 24614 df-ii 24916 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-esum 34008

This theorem is referenced by: hasheuni 34065 esumcvg 34066 esumcvgre 34071 voliune 34209 volfiniune 34210

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