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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcvgsum | Structured version Visualization version GIF version |
Description: The value of the extended sum when the corresponding sum is convergent. (Contributed by Thierry Arnoux, 29-Oct-2019.) |
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) |
Ref | Expression |
---|---|
esumcvgsum | ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumcvgsum.2 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) | |
2 | esumcvgsum.1 | . 2 ⊢ (𝑘 = 𝑖 → 𝐴 = 𝐵) | |
3 | simpll 763 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑) | |
4 | elfznn 12786 | . . . . . . 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 12130 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eleq2i 2873 | . . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1)) |
10 | 9 | biimpi 217 | . . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ≥‘1)) |
11 | 10 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ≥‘1)) |
12 | mnfxr 10547 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
13 | pnfxr 10544 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
14 | 0re 10492 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
15 | mnflt 12368 | . . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ -∞ < 0 |
17 | pnfge 12375 | . . . . . . . . . 10 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
18 | 13, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ +∞ ≤ +∞ |
19 | icossioo 12678 | . . . . . . . . 9 ⊢ (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞)) | |
20 | 12, 13, 16, 18, 19 | mp4an 689 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞) |
21 | ioomax 12661 | . . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ | |
22 | 20, 21 | sseqtri 3926 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ |
23 | 3, 5, 1 | syl2anc 584 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞)) |
24 | 22, 23 | sseldi 3889 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ) |
25 | 24 | recnd 10518 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ) |
26 | 7, 11, 25 | fsumser 14920 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗)) |
27 | 26 | mpteq2dva 5058 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) |
28 | 1z 11862 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
29 | seqfn 13231 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ≥‘1)) | |
30 | 28, 29 | ax-mp 5 | . . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ≥‘1) |
31 | fneq2 6318 | . . . . . . 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 232 | . . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ |
34 | dffn5 6595 | . . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) | |
35 | 33, 34 | mpbi 231 | . . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) |
36 | seqex 13221 | . . . . . 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 5667 | . . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ ) | |
41 | 37, 38, 39, 40 | syl3anc 1364 | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) |
42 | 35, 41 | syl5eqelr 2887 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ ) |
43 | 27, 42 | eqeltrd 2882 | . 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ ) |
44 | 1, 2, 43 | esumpcvgval 30946 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 Vcvv 3436 ⊆ wss 3861 class class class wbr 4964 ↦ cmpt 5043 dom cdm 5446 Fn wfn 6223 ‘cfv 6228 (class class class)co 7019 ℝcr 10385 0cc0 10386 1c1 10387 + caddc 10389 +∞cpnf 10521 -∞cmnf 10522 ℝ*cxr 10523 < clt 10524 ≤ cle 10525 ℕcn 11488 ℤcz 11831 ℤ≥cuz 12093 (,)cioo 12588 [,)cico 12590 ...cfz 12742 seqcseq 13219 ⇝ cli 14675 Σcsu 14876 Σ*cesum 30895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-of 7270 df-om 7440 df-1st 7548 df-2nd 7549 df-supp 7685 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-map 8261 df-pm 8262 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-fsupp 8683 df-fi 8724 df-sup 8755 df-inf 8756 df-oi 8823 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-z 11832 df-dec 11949 df-uz 12094 df-q 12198 df-rp 12240 df-xadd 12358 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-rlim 14680 df-sum 14877 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-ordt 16603 df-xrs 16604 df-mre 16686 df-mrc 16687 df-acs 16689 df-ps 17639 df-tsr 17640 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-cntz 18188 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-ntr 21312 df-nei 21390 df-cn 21519 df-haus 21607 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-tsms 22418 df-esum 30896 |
This theorem is referenced by: omssubadd 31167 |
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