esumpinfval - Mathbox for Thierry Arnoux

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Theorem esumpinfval 30442

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 12447	. . 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 449	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞)))
6	3, 5	ralrimi 3093	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		nfcv 2900	. . . . 5 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 30399	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 696	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sseldi 3740	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		nfrab1 3259	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}
12		ssrab2 3826	. . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴)
14		0xr 10276	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		pnfxr 10282	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
16		0lepnf 12157	. . . . . . . 8 ⊢ 0 ≤ +∞
17		ubicc2 12480	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
18	14, 15, 16, 17	mp3an 1571	. . . . . . 7 ⊢ +∞ ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞))
20		0e0iccpnf 12474	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
21	20	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞))
22	19, 21	ifclda 4262	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞))
23		eldif 3723	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
24		rabid 3252	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = +∞))
25	24	simplbi2 656	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (𝐵 = +∞ → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
26	25	con3dimp 456	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞)
27	23, 26	sylbi 207	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞)
28	27	adantl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞)
29	28	iffalsed 4239	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0)
30	3, 11, 7, 13, 2, 22, 29	esumss 30441	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0))
31		eqidd 2759	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})
32	24	simprbi 483	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞)
33	32	iftrued 4236	. . . . . . 7 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → if(𝐵 = +∞, +∞, 0) = +∞)
34	33	adantl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → if(𝐵 = +∞, +∞, 0) = +∞)
35	3, 31, 34	esumeq12dvaf 30400	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞)
36	2, 13	ssexd 4955	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V)
37		nfcv 2900	. . . . . . 7 ⊢ Ⅎ𝑘+∞
38	11, 37	esumcst 30432	. . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞))
39	36, 18, 38	sylancl 697	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞))
40		hashxrcl 13338	. . . . . . 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 4099	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞)
44	42, 43	sylibr 224	. . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅)
45		hashgt0 13367	. . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
46	36, 44, 45	syl2anc 696	. . . . . 6 ⊢ (𝜑 → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
47		xmulpnf1 12295	. . . . . 6 ⊢ (((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ^* ∧ 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞) = +∞)
48	41, 46, 47	syl2anc 696	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞) = +∞)
49	35, 39, 48	3eqtrd 2796	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞)
50	30, 49	eqtr3d 2794	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞)
51		breq1 4805	. . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
52		breq1 4805	. . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
53		pnfge 12155	. . . . . . . 8 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
54	15, 53	ax-mp 5	. . . . . . 7 ⊢ +∞ ≤ +∞
55		breq2 4806	. . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞))
56	54, 55	mpbiri 248	. . . . . 6 ⊢ (𝐵 = +∞ → +∞ ≤ 𝐵)
57	56	adantl 473	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐵)
58	4	adantr 472	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ (0[,]+∞))
59		iccgelb 12421	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵)
60	14, 15, 59	mp3an12 1561	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵)
61	58, 60	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵)
62	51, 52, 57, 61	ifbothda 4265	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵)
63	3, 7, 2, 22, 4, 62	esumlef 30431	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ^𝑘 ∈ 𝐴𝐵)
64	50, 63	eqbrtrrd 4826	. 2 ⊢ (𝜑 → +∞ ≤ Σ^*𝑘 ∈ 𝐴𝐵)
65		xgepnf 12187	. . 3 ⊢ (Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^ → (+∞ ≤ Σ^𝑘 ∈ 𝐴𝐵 ↔ Σ^𝑘 ∈ 𝐴𝐵 = +∞))
66	65	biimpd 219	. 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 383 = wceq 1630 Ⅎwnf 1855 ∈ wcel 2137 ≠ wne 2930 ∀wral 3048 ∃wrex 3049 {crab 3052 Vcvv 3338 ∖ cdif 3710 ⊆ wss 3713 ∅c0 4056 ifcif 4228 class class class wbr 4802 ‘cfv 6047 (class class class)co 6811 0cc0 10126 +∞cpnf 10261 ℝ^*cxr 10263 < clt 10264 ≤ cle 10265 ·_e cxmu 12136 [,]cicc 12369 ♯chash 13309 Σ^*cesum 30396

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-rep 4921 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-inf2 8709 ax-cnex 10182 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 ax-pre-sup 10204 ax-addf 10205 ax-mulf 10206

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-int 4626 df-iun 4672 df-iin 4673 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-se 5224 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-isom 6056 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-of 7060 df-om 7229 df-1st 7331 df-2nd 7332 df-supp 7462 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-2o 7728 df-oadd 7731 df-er 7909 df-map 8023 df-pm 8024 df-ixp 8073 df-en 8120 df-dom 8121 df-sdom 8122 df-fin 8123 df-fsupp 8439 df-fi 8480 df-sup 8511 df-inf 8512 df-oi 8578 df-card 8953 df-cda 9180 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-div 10875 df-nn 11211 df-2 11269 df-3 11270 df-4 11271 df-5 11272 df-6 11273 df-7 11274 df-8 11275 df-9 11276 df-n0 11483 df-xnn0 11554 df-z 11568 df-dec 11684 df-uz 11878 df-q 11980 df-rp 12024 df-xneg 12137 df-xadd 12138 df-xmul 12139 df-ioo 12370 df-ioc 12371 df-ico 12372 df-icc 12373 df-fz 12518 df-fzo 12658 df-fl 12785 df-mod 12861 df-seq 12994 df-exp 13053 df-fac 13253 df-bc 13282 df-hash 13310 df-shft 14004 df-cj 14036 df-re 14037 df-im 14038 df-sqrt 14172 df-abs 14173 df-limsup 14399 df-clim 14416 df-rlim 14417 df-sum 14614 df-ef 14995 df-sin 14997 df-cos 14998 df-pi 15000 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16154 df-mulr 16155 df-starv 16156 df-sca 16157 df-vsca 16158 df-ip 16159 df-tset 16160 df-ple 16161 df-ds 16164 df-unif 16165 df-hom 16166 df-cco 16167 df-rest 16283 df-topn 16284 df-0g 16302 df-gsum 16303 df-topgen 16304 df-pt 16305 df-prds 16308 df-ordt 16361 df-xrs 16362 df-qtop 16367 df-imas 16368 df-xps 16370 df-mre 16446 df-mrc 16447 df-acs 16449 df-ps 17399 df-tsr 17400 df-plusf 17440 df-mgm 17441 df-sgrp 17483 df-mnd 17494 df-mhm 17534 df-submnd 17535 df-grp 17624 df-minusg 17625 df-sbg 17626 df-mulg 17740 df-subg 17790 df-cntz 17948 df-cmn 18393 df-abl 18394 df-mgp 18688 df-ur 18700 df-ring 18747 df-cring 18748 df-subrg 18978 df-abv 19017 df-lmod 19065 df-scaf 19066 df-sra 19372 df-rgmod 19373 df-psmet 19938 df-xmet 19939 df-met 19940 df-bl 19941 df-mopn 19942 df-fbas 19943 df-fg 19944 df-cnfld 19947 df-top 20899 df-topon 20916 df-topsp 20937 df-bases 20950 df-cld 21023 df-ntr 21024 df-cls 21025 df-nei 21102 df-lp 21140 df-perf 21141 df-cn 21231 df-cnp 21232 df-haus 21319 df-tx 21565 df-hmeo 21758 df-fil 21849 df-fm 21941 df-flim 21942 df-flf 21943 df-tmd 22075 df-tgp 22076 df-tsms 22129 df-trg 22162 df-xms 22324 df-ms 22325 df-tms 22326 df-nm 22586 df-ngp 22587 df-nrg 22589 df-nlm 22590 df-ii 22879 df-cncf 22880 df-limc 23827 df-dv 23828 df-log 24500 df-esum 30397

This theorem is referenced by: hasheuni 30454 esumcvg 30455 esumcvgre 30460 voliune 30599 volfiniune 30600

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