esumpinfval - Mathbox for Thierry Arnoux

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Theorem esumpinfval 31941

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 13091	. . 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 3139	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞))
7		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑘𝐴
8	7	esumcl 31898	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
9	2, 6, 8	syl2anc 583	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
10	1, 9	sselid 3915	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^)
11		nfrab1 3310	. . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}
12		ssrab2 4009	. . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴)
14		0xr 10953	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		pnfxr 10960	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
16		0lepnf 12797	. . . . . . . 8 ⊢ 0 ≤ +∞
17		ubicc2 13126	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
18	14, 15, 16, 17	mp3an 1459	. . . . . . 7 ⊢ +∞ ∈ (0[,]+∞)
19	18	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞))
20		0e0iccpnf 13120	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
21	20	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞))
22	19, 21	ifclda 4491	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞))
23		eldif 3893	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
24		rabid 3304	. . . . . . . . . 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 216	. . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞)
28	27	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞)
29	28	iffalsed 4467	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0)
30	3, 11, 7, 13, 2, 22, 29	esumss 31940	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0))
31		eqidd 2739	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})
32	24	simprbi 496	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞)
33	32	iftrued 4464	. . . . . . 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 31899	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ^𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞)
36	2, 13	ssexd 5243	. . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V)
37		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑘+∞
38	11, 37	esumcst 31931	. . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞))
39	36, 18, 38	sylancl 585	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞))
40		hashxrcl 14000	. . . . . . 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 4316	. . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞)
44	42, 43	sylibr 233	. . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅)
45		hashgt0 14031	. . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
46	36, 44, 45	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}))
47		xmulpnf1 12937	. . . . . 6 ⊢ (((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ^* ∧ 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞) = +∞)
48	41, 46, 47	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·_e +∞) = +∞)
49	35, 39, 48	3eqtrd 2782	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞)
50	30, 49	eqtr3d 2780	. . 3 ⊢ (𝜑 → Σ^*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞)
51		breq1 5073	. . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
52		breq1 5073	. . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵))
53		pnfge 12795	. . . . . . . 8 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
54	15, 53	ax-mp 5	. . . . . . 7 ⊢ +∞ ≤ +∞
55		breq2 5074	. . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞))
56	54, 55	mpbiri 257	. . . . . 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 13064	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵)
60	14, 15, 59	mp3an12 1449	. . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵)
61	58, 60	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵)
62	51, 52, 57, 61	ifbothda 4494	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵)
63	3, 7, 2, 22, 4, 62	esumlef 31930	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ^𝑘 ∈ 𝐴𝐵)
64	50, 63	eqbrtrrd 5094	. 2 ⊢ (𝜑 → +∞ ≤ Σ^*𝑘 ∈ 𝐴𝐵)
65		xgepnf 12828	. . 3 ⊢ (Σ^𝑘 ∈ 𝐴𝐵 ∈ ℝ^ → (+∞ ≤ Σ^𝑘 ∈ 𝐴𝐵 ↔ Σ^𝑘 ∈ 𝐴𝐵 = +∞))
66	65	biimpd 228	. 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 1539 Ⅎwnf 1787 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4253 ifcif 4456 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 0cc0 10802 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 ·_e cxmu 12776 [,]cicc 13011 ♯chash 13972 Σ^*cesum 31895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-ordt 17129 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-ps 18199 df-tsr 18200 df-plusf 18240 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-abv 19992 df-lmod 20040 df-scaf 20041 df-sra 20349 df-rgmod 20350 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-tmd 23131 df-tgp 23132 df-tsms 23186 df-trg 23219 df-xms 23381 df-ms 23382 df-tms 23383 df-nm 23644 df-ngp 23645 df-nrg 23647 df-nlm 23648 df-ii 23946 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-esum 31896

This theorem is referenced by: hasheuni 31953 esumcvg 31954 esumcvgre 31959 voliune 32097 volfiniune 32098

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