Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isumneg | Structured version Visualization version GIF version |
Description: Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
isumneg.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumneg.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumneg.3 | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ) |
isumneg.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumneg.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
isumneg.6 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumneg | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = -Σ𝑘 ∈ 𝑍 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumneg.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
2 | 1 | mulm1d 10482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1 · 𝐴) = -𝐴) |
3 | 2 | eqcomd 2628 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -𝐴 = (-1 · 𝐴)) |
4 | 3 | sumeq2dv 14433 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = Σ𝑘 ∈ 𝑍 (-1 · 𝐴)) |
5 | isumneg.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | isumneg.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | isumneg.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
8 | isumneg.6 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
9 | 1cnd 10056 | . . . 4 ⊢ (𝜑 → 1 ∈ ℂ) | |
10 | 9 | negcld 10379 | . . 3 ⊢ (𝜑 → -1 ∈ ℂ) |
11 | 5, 6, 7, 1, 8, 10 | isummulc2 14493 | . 2 ⊢ (𝜑 → (-1 · Σ𝑘 ∈ 𝑍 𝐴) = Σ𝑘 ∈ 𝑍 (-1 · 𝐴)) |
12 | isumneg.3 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ) | |
13 | 12 | mulm1d 10482 | . 2 ⊢ (𝜑 → (-1 · Σ𝑘 ∈ 𝑍 𝐴) = -Σ𝑘 ∈ 𝑍 𝐴) |
14 | 4, 11, 13 | 3eqtr2d 2662 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = -Σ𝑘 ∈ 𝑍 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 -cneg 10267 ℤcz 11377 ℤ≥cuz 11687 seqcseq 12801 ⇝ cli 14215 Σcsu 14416 |
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 |
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-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-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 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-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 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-sum 14417 |
This theorem is referenced by: (None) |
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