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 790 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑) | |
4 | elfznn 12370 | . . . . . . 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 693 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴) |
8 | nnuz 11723 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eleq2i 2693 | . . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1)) |
10 | 9 | biimpi 206 | . . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ≥‘1)) |
11 | 10 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ≥‘1)) |
12 | mnfxr 10096 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
13 | pnfxr 10092 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
14 | 0re 10040 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
15 | mnflt 11957 | . . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ -∞ < 0 |
17 | pnfge 11964 | . . . . . . . . . 10 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
18 | 13, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ +∞ ≤ +∞ |
19 | icossioo 12264 | . . . . . . . . 9 ⊢ (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞)) | |
20 | 12, 13, 16, 18, 19 | mp4an 709 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞) |
21 | ioomax 12248 | . . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ | |
22 | 20, 21 | sseqtri 3637 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ |
23 | 3, 5, 1 | syl2anc 693 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞)) |
24 | 22, 23 | sseldi 3601 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ) |
25 | 24 | recnd 10068 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ) |
26 | 7, 11, 25 | fsumser 14461 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗)) |
27 | 26 | mpteq2dva 4744 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) |
28 | 1z 11407 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
29 | seqfn 12813 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ≥‘1)) | |
30 | 28, 29 | ax-mp 5 | . . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ≥‘1) |
31 | fneq2 5980 | . . . . . . 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 221 | . . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ |
34 | dffn5 6241 | . . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) | |
35 | 33, 34 | mpbi 220 | . . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) |
36 | seqex 12803 | . . . . . 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 5330 | . . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ ) | |
41 | 37, 38, 39, 40 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) |
42 | 35, 41 | syl5eqelr 2706 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ ) |
43 | 27, 42 | eqeltrd 2701 | . 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ ) |
44 | 1, 2, 43 | esumpcvgval 30140 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 ℤcz 11377 ℤ≥cuz 11687 (,)cioo 12175 [,)cico 12177 ...cfz 12326 seqcseq 12801 ⇝ cli 14215 Σcsu 14416 Σ*cesum 30089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-nei 20902 df-cn 21031 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-esum 30090 |
This theorem is referenced by: omssubadd 30362 |
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