esumcvgsum - Mathbox for Thierry Arnoux

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Theorem esumcvgsum 34119

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 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑)
4		elfznn 13570	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
5	4	adantl 481	. . . . . 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 12895	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
9	8	eleq2i 2826	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
10	9	biimpi 216	. . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ_≥‘1))
11	10	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
12		mnfxr 11292	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
13		pnfxr 11289	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
14		0re 11237	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15		mnflt 13139	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 0
17		pnfge 13146	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
18	13, 17	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
19		icossioo 13457	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞))
20	12, 13, 16, 18, 19	mp4an 693	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞)
21		ioomax 13439	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
22	20, 21	sseqtri 4007	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
23	3, 5, 1	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞))
24	22, 23	sselid 3956	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ)
25	24	recnd 11263	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ)
26	7, 11, 25	fsumser 15746	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗))
27	26	mpteq2dva 5214	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
28		1z 12622	. . . . . . 7 ⊢ 1 ∈ ℤ
29		seqfn 14031	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ_≥‘1))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ_≥‘1)
31		fneq2 6630	. . . . . . 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 231	. . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ
34		dffn5 6937	. . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
35	33, 34	mpbi 230	. . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))
36		seqex 14021	. . . . . 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 5889	. . . . 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 2839	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ )
43	27, 42	eqeltrd 2834	. 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ )
44	1, 2, 43	esumpcvgval 34109	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ⊆ wss 3926 class class class wbr 5119 ↦ cmpt 5201 dom cdm 5654 Fn wfn 6526 ‘cfv 6531 (class class class)co 7405 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 +∞cpnf 11266 -∞cmnf 11267 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 ℕcn 12240 ℤcz 12588 ℤ_≥cuz 12852 (,)cioo 13362 [,)cico 13364 ...cfz 13524 seqcseq 14019 ⇝ cli 15500 Σcsu 15702 Σ^*cesum 34058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xadd 13129 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-rlim 15505 df-sum 15703 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-ordt 17515 df-xrs 17516 df-mre 17598 df-mrc 17599 df-acs 17601 df-ps 18576 df-tsr 18577 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-cntz 19300 df-cmn 19763 df-abl 19764 df-mgp 20101 df-ur 20142 df-ring 20195 df-cring 20196 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-ntr 22958 df-nei 23036 df-cn 23165 df-haus 23253 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-tsms 24065 df-esum 34059

This theorem is referenced by: omssubadd 34332

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