esumcvgsum - Mathbox for Thierry Arnoux

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Theorem esumcvgsum 34089

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 13593	. . . . . . 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 12921	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
9	8	eleq2i 2833	. . . . . . 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 11318	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
13		pnfxr 11315	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
14		0re 11263	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
15		mnflt 13165	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 0
17		pnfge 13172	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
18	13, 17	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
19		icossioo 13480	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞))
20	12, 13, 16, 18, 19	mp4an 693	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞)
21		ioomax 13462	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
22	20, 21	sseqtri 4032	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
23	3, 5, 1	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞))
24	22, 23	sselid 3981	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ)
25	24	recnd 11289	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ)
26	7, 11, 25	fsumser 15766	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗))
27	26	mpteq2dva 5242	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
28		1z 12647	. . . . . . 7 ⊢ 1 ∈ ℤ
29		seqfn 14054	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ_≥‘1))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ_≥‘1)
31		fneq2 6660	. . . . . . 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 6967	. . . . 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 14044	. . . . . 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 5920	. . . . 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 2846	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ )
43	27, 42	eqeltrd 2841	. 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ )
44	1, 2, 43	esumpcvgval 34079	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 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 ℤcz 12613 ℤ_≥cuz 12878 (,)cioo 13387 [,)cico 13389 ...cfz 13547 seqcseq 14042 ⇝ cli 15520 Σcsu 15722 Σ^*cesum 34028

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-ordt 17546 df-xrs 17547 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-cring 20233 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-ntr 23028 df-nei 23106 df-cn 23235 df-haus 23323 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tsms 24135 df-esum 34029

This theorem is referenced by: omssubadd 34302

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