esumcvgsum - Mathbox for Thierry Arnoux

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Theorem esumcvgsum 32056

Description: The value of the extended sum when the corresponding sum is convergent. (Contributed by Thierry Arnoux, 29-Oct-2019.)

**Hypotheses**
Ref	Expression
esumcvgsum.1	⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵)
esumcvgsum.2	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
esumcvgsum.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴)
esumcvgsum.4	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝐿)
esumcvgsum.5	⊢ (𝜑 → 𝐿 ∈ ℝ)

**Assertion**
Ref	Expression
esumcvgsum	⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Distinct variable groups: 𝑖,𝑘 𝐴,𝑖 𝐵,𝑘 𝑘,𝐹 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑖) 𝐴(𝑘) 𝐵(𝑖) 𝐹(𝑖) 𝐿(𝑖,𝑘)

Proof of Theorem esumcvgsum Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		esumcvgsum.2	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
2		esumcvgsum.1	. 2 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵)
3		simpll 764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑)
4		elfznn 13285	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
5	4	adantl 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
6		esumcvgsum.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴)
7	3, 5, 6	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴)
8		nnuz 12621	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
9	8	eleq2i 2830	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
10	9	biimpi 215	. . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ_≥‘1))
11	10	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
12		mnfxr 11032	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
13		pnfxr 11029	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
14		0re 10977	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15		mnflt 12859	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 0
17		pnfge 12866	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
18	13, 17	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
19		icossioo 13172	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞))
20	12, 13, 16, 18, 19	mp4an 690	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞)
21		ioomax 13154	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
22	20, 21	sseqtri 3957	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
23	3, 5, 1	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞))
24	22, 23	sselid 3919	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ)
25	24	recnd 11003	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ)
26	7, 11, 25	fsumser 15442	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗))
27	26	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
28		1z 12350	. . . . . . 7 ⊢ 1 ∈ ℤ
29		seqfn 13733	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ_≥‘1))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ_≥‘1)
31		fneq2 6525	. . . . . . 7 ⊢ (ℕ = (ℤ_≥‘1) → (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) Fn (ℤ_≥‘1)))
32	8, 31	ax-mp 5	. . . . . 6 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) Fn (ℤ_≥‘1))
33	30, 32	mpbir 230	. . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ
34		dffn5 6828	. . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
35	33, 34	mpbi 229	. . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))
36		seqex 13723	. . . . . 6 ⊢ seq1( + , 𝐹) ∈ V
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → seq1( + , 𝐹) ∈ V)
38		esumcvgsum.5	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ)
39		esumcvgsum.4	. . . . 5 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝐿)
40		breldmg 5818	. . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ )
41	37, 38, 39, 40	syl3anc 1370	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
42	35, 41	eqeltrrid 2844	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ )
43	27, 42	eqeltrd 2839	. 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ )
44	1, 2, 43	esumpcvgval 32046	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 ℕcn 11973 ℤcz 12319 ℤ_≥cuz 12582 (,)cioo 13079 [,)cico 13081 ...cfz 13239 seqcseq 13721 ⇝ cli 15193 Σcsu 15397 Σ^*cesum 31995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xadd 12849 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-ordt 17212 df-xrs 17213 df-mre 17295 df-mrc 17296 df-acs 17298 df-ps 18284 df-tsr 18285 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-ntr 22171 df-nei 22249 df-cn 22378 df-haus 22466 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tsms 23278 df-esum 31996

This theorem is referenced by: omssubadd 32267

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