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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 765 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝜑) | |
4 | elfznn 13584 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ) | |
5 | 4 | adantl 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ) |
6 | esumcvgsum.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴) | |
7 | 3, 5, 6 | syl2anc 582 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴) |
8 | nnuz 12917 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eleq2i 2818 | . . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1)) |
10 | 9 | biimpi 215 | . . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ≥‘1)) |
11 | 10 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ≥‘1)) |
12 | mnfxr 11321 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
13 | pnfxr 11318 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
14 | 0re 11266 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
15 | mnflt 13157 | . . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ -∞ < 0 |
17 | pnfge 13164 | . . . . . . . . . 10 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
18 | 13, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ +∞ ≤ +∞ |
19 | icossioo 13471 | . . . . . . . . 9 ⊢ (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞)) | |
20 | 12, 13, 16, 18, 19 | mp4an 691 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞) |
21 | ioomax 13453 | . . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ | |
22 | 20, 21 | sseqtri 4016 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ |
23 | 3, 5, 1 | syl2anc 582 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞)) |
24 | 22, 23 | sselid 3977 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ) |
25 | 24 | recnd 11292 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ) |
26 | 7, 11, 25 | fsumser 15734 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗)) |
27 | 26 | mpteq2dva 5253 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) |
28 | 1z 12644 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
29 | seqfn 14033 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ≥‘1)) | |
30 | 28, 29 | ax-mp 5 | . . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ≥‘1) |
31 | fneq2 6652 | . . . . . . 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 6961 | . . . . 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 14023 | . . . . . 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 5916 | . . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ ) | |
41 | 37, 38, 39, 40 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) |
42 | 35, 41 | eqeltrrid 2831 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ ) |
43 | 27, 42 | eqeltrd 2826 | . 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ ) |
44 | 1, 2, 43 | esumpcvgval 33911 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 class class class wbr 5153 ↦ cmpt 5236 dom cdm 5682 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 +∞cpnf 11295 -∞cmnf 11296 ℝ*cxr 11297 < clt 11298 ≤ cle 11299 ℕcn 12264 ℤcz 12610 ℤ≥cuz 12874 (,)cioo 13378 [,)cico 13380 ...cfz 13538 seqcseq 14021 ⇝ cli 15486 Σcsu 15690 Σ*cesum 33860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xadd 13147 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-sum 15691 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-ordt 17516 df-xrs 17517 df-mre 17599 df-mrc 17600 df-acs 17602 df-ps 18591 df-tsr 18592 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-ur 20165 df-ring 20218 df-cring 20219 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-ntr 23015 df-nei 23093 df-cn 23222 df-haus 23310 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-tsms 24122 df-esum 33861 |
This theorem is referenced by: omssubadd 34134 |
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