esumcvgsum - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcvgsum		Structured version Visualization version GIF version

Theorem esumcvgsum 34282

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 773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑)
4		elfznn 13502	. . . . . . 7 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ)
5	4	adantl 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ)
6		esumcvgsum.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴)
7	3, 5, 6	syl2anc 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴)
8		nnuz 12822	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
9	8	eleq2i 2833	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ_≥‘1))
10	9	bilani 506	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ_≥‘1))
11		mnfxr 11198	. . . . . . . . 9 ⊢ -∞ ∈ ℝ^*
12		pnfxr 11195	. . . . . . . . 9 ⊢ +∞ ∈ ℝ^*
13		0re 11142	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
14		mnflt 13069	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0)
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 0
16		pnfge 13076	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
17	12, 16	ax-mp 5	. . . . . . . . 9 ⊢ +∞ ≤ +∞
18		icossioo 13388	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞))
19	11, 12, 15, 17, 18	mp4an 700	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞)
20		ioomax 13370	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
21	19, 20	sseqtri 3964	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
22	3, 5, 1	syl2anc 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞))
23	21, 22	sselid 3914	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ)
24	23	recnd 11169	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ)
25	7, 10, 24	fsumser 15687	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗))
26	25	mpteq2dva 5167	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
27		1z 12552	. . . . . . 7 ⊢ 1 ∈ ℤ
28		seqfn 13970	. . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ_≥‘1))
29	27, 28	ax-mp 5	. . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ_≥‘1)
30		fneq2 6580	. . . . . . 7 ⊢ (ℕ = (ℤ_≥‘1) → (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) Fn (ℤ_≥‘1)))
31	8, 30	ax-mp 5	. . . . . 6 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) Fn (ℤ_≥‘1))
32	29, 31	mpbir 233	. . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ
33		dffn5 6888	. . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)))
34	32, 33	mpbi 232	. . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))
35		seqex 13960	. . . . . 6 ⊢ seq1( + , 𝐹) ∈ V
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → seq1( + , 𝐹) ∈ V)
37		esumcvgsum.5	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ)
38		esumcvgsum.4	. . . . 5 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝐿)
39		breldmg 5857	. . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ )
40	36, 37, 38, 39	syl3anc 1380	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
41	34, 40	eqeltrrid 2846	. . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ )
42	26, 41	eqeltrd 2841	. 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ )
43	1, 2, 42	esumpcvgval 34272	1 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 Vcvv 3433 ⊆ wss 3884 class class class wbr 5074 ↦ cmpt 5155 dom cdm 5620 Fn wfn 6483 ‘cfv 6488 (class class class)co 7359 ℝcr 11033 0cc0 11034 1c1 11035 + caddc 11037 +∞cpnf 11172 -∞cmnf 11173 ℝ^*cxr 11174 < clt 11175 ≤ cle 11176 ℕcn 12169 ℤcz 12519 ℤ_≥cuz 12783 (,)cioo 13293 [,)cico 13295 ...cfz 13456 seqcseq 13958 ⇝ cli 15441 Σcsu 15643 Σ^*cesum 34221

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 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-inf2 9557 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112 ax-addf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xadd 13059 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sum 15644 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-ordt 17460 df-xrs 17461 df-mre 17543 df-mrc 17544 df-acs 17546 df-ps 18527 df-tsr 18528 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18907 df-minusg 18908 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-ur 20157 df-ring 20210 df-cring 20211 df-fbas 21347 df-fg 21348 df-cnfld 21351 df-top 22880 df-topon 22897 df-topsp 22919 df-bases 22932 df-ntr 23006 df-nei 23084 df-cn 23213 df-haus 23301 df-fil 23832 df-fm 23924 df-flim 23925 df-flf 23926 df-tsms 24113 df-esum 34222

This theorem is referenced by: omssubadd 34494

W3C validator