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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 766 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑) | |
4 | elfznn 13477 | . . . . . . 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 585 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴) |
8 | nnuz 12813 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eleq2i 2830 | . . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1)) |
10 | 9 | biimpi 215 | . . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ≥‘1)) |
11 | 10 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ≥‘1)) |
12 | mnfxr 11219 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
13 | pnfxr 11216 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
14 | 0re 11164 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
15 | mnflt 13051 | . . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ -∞ < 0 |
17 | pnfge 13058 | . . . . . . . . . 10 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
18 | 13, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ +∞ ≤ +∞ |
19 | icossioo 13364 | . . . . . . . . 9 ⊢ (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞)) | |
20 | 12, 13, 16, 18, 19 | mp4an 692 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞) |
21 | ioomax 13346 | . . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ | |
22 | 20, 21 | sseqtri 3985 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ |
23 | 3, 5, 1 | syl2anc 585 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞)) |
24 | 22, 23 | sselid 3947 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ) |
25 | 24 | recnd 11190 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ) |
26 | 7, 11, 25 | fsumser 15622 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗)) |
27 | 26 | mpteq2dva 5210 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) |
28 | 1z 12540 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
29 | seqfn 13925 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ≥‘1)) | |
30 | 28, 29 | ax-mp 5 | . . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ≥‘1) |
31 | fneq2 6599 | . . . . . . 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 230 | . . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ |
34 | dffn5 6906 | . . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) | |
35 | 33, 34 | mpbi 229 | . . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) |
36 | seqex 13915 | . . . . . 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 5870 | . . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ ) | |
41 | 37, 38, 39, 40 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) |
42 | 35, 41 | eqeltrrid 2843 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ ) |
43 | 27, 42 | eqeltrd 2838 | . 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ ) |
44 | 1, 2, 43 | esumpcvgval 32717 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ⊆ wss 3915 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5638 Fn wfn 6496 ‘cfv 6501 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 +∞cpnf 11193 -∞cmnf 11194 ℝ*cxr 11195 < clt 11196 ≤ cle 11197 ℕcn 12160 ℤcz 12506 ℤ≥cuz 12770 (,)cioo 13271 [,)cico 13273 ...cfz 13431 seqcseq 13913 ⇝ cli 15373 Σcsu 15577 Σ*cesum 32666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-rlim 15378 df-sum 15578 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-ordt 17390 df-xrs 17391 df-mre 17473 df-mrc 17474 df-acs 17476 df-ps 18462 df-tsr 18463 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-ntr 22387 df-nei 22465 df-cn 22594 df-haus 22682 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tsms 23494 df-esum 32667 |
This theorem is referenced by: omssubadd 32940 |
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