esumpinfval - Mathbox for Thierry Arnoux

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Theorem esumpinfval 34179

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 13344	. . 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 3232	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		nfcv 2896	. . . . 5 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 34136	. . . 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 3929	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		nfrab1 3417	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}
12		ssrab2 4030	. . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴)
14		0xr 11177	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		pnfxr 11184	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
16		0lepnf 13045	. . . . . . . 8 ⊢ 0 ≤ +∞
17		ubicc2 13379	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
18	14, 15, 16, 17	mp3an 1463	. . . . . . 7 ⊢ +∞ ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞))
20		0e0iccpnf 13373	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
21	20	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞))
22	19, 21	ifclda 4513	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞))
23		eldif 3909	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
24		rabid 3418	. . . . . . . . . 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 4488	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0)
30	3, 11, 7, 13, 2, 22, 29	esumss 34178	. . . 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 4485	. . . . . . 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 34137	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞)
36	2, 13	ssexd 5267	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V)
37		nfcv 2896	. . . . . . 7 ⊢ Ⅎ𝑘+∞
38	11, 37	esumcst 34169	. . . . . 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 14278	. . . . . . 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 4339	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞)
44	42, 43	sylibr 234	. . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅)
45		hashgt0 14309	. . . . . . 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 13187	. . . . . 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 2773	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞)
50	30, 49	eqtr3d 2771	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞)
51		breq1 5099	. . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
52		breq1 5099	. . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
53		pnfge 13042	. . . . . . . 8 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
54	15, 53	ax-mp 5	. . . . . . 7 ⊢ +∞ ≤ +∞
55		breq2 5100	. . . . . . 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 13316	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵)
60	14, 15, 59	mp3an12 1453	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵)
61	58, 60	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵)
62	51, 52, 57, 61	ifbothda 4516	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵)
63	3, 7, 2, 22, 4, 62	esumlef 34168	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ^𝑘 ∈ 𝐴𝐵)
64	50, 63	eqbrtrrd 5120	. 2 ⊢ (𝜑 → +∞ ≤ Σ^*𝑘 ∈ 𝐴𝐵)
65		xgepnf 13078	. . 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 1541 Ⅎwnf 1784 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∃wrex 3058 {crab 3397 Vcvv 3438 ∖ cdif 3896 ⊆ wss 3899 ∅c0 4283 ifcif 4477 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 0cc0 11024 +∞cpnf 11161 ℝ^*cxr 11163 < clt 11164 ≤ cle 11165 ·_e cxmu 13023 [,]cicc 13262 ♯chash 14251 Σ^*cesum 34133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-ordt 17420 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-ps 18487 df-tsr 18488 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20477 df-subrg 20501 df-abv 20740 df-lmod 20811 df-scaf 20812 df-sra 21123 df-rgmod 21124 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-tmd 24014 df-tgp 24015 df-tsms 24069 df-trg 24102 df-xms 24262 df-ms 24263 df-tms 24264 df-nm 24524 df-ngp 24525 df-nrg 24527 df-nlm 24528 df-ii 24824 df-cncf 24825 df-limc 25821 df-dv 25822 df-log 26519 df-esum 34134

This theorem is referenced by: hasheuni 34191 esumcvg 34192 esumcvgre 34197 voliune 34335 volfiniune 34336

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