esumcvgsum - Mathbox for Thierry Arnoux

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Theorem esumcvgsum 31768

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 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑)
4		elfznn 13141	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
5	4	adantl 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
6		esumcvgsum.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴)
7	3, 5, 6	syl2anc 587	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴)
8		nnuz 12477	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
9	8	eleq2i 2829	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
10	9	biimpi 219	. . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ_≥‘1))
11	10	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
12		mnfxr 10890	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
13		pnfxr 10887	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
14		0re 10835	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15		mnflt 12715	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 0
17		pnfge 12722	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
18	13, 17	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
19		icossioo 13028	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞))
20	12, 13, 16, 18, 19	mp4an 693	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞)
21		ioomax 13010	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
22	20, 21	sseqtri 3937	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
23	3, 5, 1	syl2anc 587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞))
24	22, 23	sseldi 3899	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ)
25	24	recnd 10861	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ)
26	7, 11, 25	fsumser 15294	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗))
27	26	mpteq2dva 5150	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
28		1z 12207	. . . . . . 7 ⊢ 1 ∈ ℤ
29		seqfn 13586	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ_≥‘1))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ_≥‘1)
31		fneq2 6471	. . . . . . 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 234	. . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ
34		dffn5 6771	. . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
35	33, 34	mpbi 233	. . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))
36		seqex 13576	. . . . . 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 5778	. . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ )
41	37, 38, 39, 40	syl3anc 1373	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
42	35, 41	eqeltrrid 2843	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ )
43	27, 42	eqeltrd 2838	. 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ )
44	1, 2, 43	esumpcvgval 31758	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 Fn wfn 6375 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 +∞cpnf 10864 -∞cmnf 10865 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 ℕcn 11830 ℤcz 12176 ℤ_≥cuz 12438 (,)cioo 12935 [,)cico 12937 ...cfz 13095 seqcseq 13574 ⇝ cli 15045 Σcsu 15249 Σ^*cesum 31707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xadd 12705 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-ordt 17006 df-xrs 17007 df-mre 17089 df-mrc 17090 df-acs 17092 df-ps 18072 df-tsr 18073 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-ntr 21917 df-nei 21995 df-cn 22124 df-haus 22212 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-tsms 23024 df-esum 31708

This theorem is referenced by: omssubadd 31979

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