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 12937 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝑗) → 𝑘 ∈ ℕ) | |
5 | 4 | adantl 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝑘 ∈ ℕ) |
6 | esumcvgsum.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴) | |
7 | 3, 5, 6 | syl2anc 586 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → (𝐹‘𝑘) = 𝐴) |
8 | nnuz 12282 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
9 | 8 | eleq2i 2904 | . . . . . . 7 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1)) |
10 | 9 | biimpi 218 | . . . . . 6 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ (ℤ≥‘1)) |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ (ℤ≥‘1)) |
12 | mnfxr 10698 | . . . . . . . . 9 ⊢ -∞ ∈ ℝ* | |
13 | pnfxr 10695 | . . . . . . . . 9 ⊢ +∞ ∈ ℝ* | |
14 | 0re 10643 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
15 | mnflt 12519 | . . . . . . . . . 10 ⊢ (0 ∈ ℝ → -∞ < 0) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . . 9 ⊢ -∞ < 0 |
17 | pnfge 12526 | . . . . . . . . . 10 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
18 | 13, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ +∞ ≤ +∞ |
19 | icossioo 12829 | . . . . . . . . 9 ⊢ (((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ (-∞ < 0 ∧ +∞ ≤ +∞)) → (0[,)+∞) ⊆ (-∞(,)+∞)) | |
20 | 12, 13, 16, 18, 19 | mp4an 691 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (-∞(,)+∞) |
21 | ioomax 12812 | . . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ | |
22 | 20, 21 | sseqtri 4003 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ |
23 | 3, 5, 1 | syl2anc 586 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ (0[,)+∞)) |
24 | 22, 23 | sseldi 3965 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℝ) |
25 | 24 | recnd 10669 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑗)) → 𝐴 ∈ ℂ) |
26 | 7, 11, 25 | fsumser 15087 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → Σ𝑘 ∈ (1...𝑗)𝐴 = (seq1( + , 𝐹)‘𝑗)) |
27 | 26 | mpteq2dva 5161 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) |
28 | 1z 12013 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
29 | seqfn 13382 | . . . . . . 7 ⊢ (1 ∈ ℤ → seq1( + , 𝐹) Fn (ℤ≥‘1)) | |
30 | 28, 29 | ax-mp 5 | . . . . . 6 ⊢ seq1( + , 𝐹) Fn (ℤ≥‘1) |
31 | fneq2 6445 | . . . . . . 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 233 | . . . . 5 ⊢ seq1( + , 𝐹) Fn ℕ |
34 | dffn5 6724 | . . . . 5 ⊢ (seq1( + , 𝐹) Fn ℕ ↔ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗))) | |
35 | 33, 34 | mpbi 232 | . . . 4 ⊢ seq1( + , 𝐹) = (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) |
36 | seqex 13372 | . . . . . 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 5778 | . . . . 5 ⊢ ((seq1( + , 𝐹) ∈ V ∧ 𝐿 ∈ ℝ ∧ seq1( + , 𝐹) ⇝ 𝐿) → seq1( + , 𝐹) ∈ dom ⇝ ) | |
41 | 37, 38, 39, 40 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ ) |
42 | 35, 41 | eqeltrrid 2918 | . . 3 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑗)) ∈ dom ⇝ ) |
43 | 27, 42 | eqeltrd 2913 | . 2 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑗)𝐴) ∈ dom ⇝ ) |
44 | 1, 2, 43 | esumpcvgval 31337 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 +∞cpnf 10672 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 (,)cioo 12739 [,)cico 12741 ...cfz 12893 seqcseq 13370 ⇝ cli 14841 Σcsu 15042 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xadd 12509 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-ordt 16774 df-xrs 16775 df-mre 16857 df-mrc 16858 df-acs 16860 df-ps 17810 df-tsr 17811 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-ntr 21628 df-nei 21706 df-cn 21835 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-esum 31287 |
This theorem is referenced by: omssubadd 31558 |
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