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Mirrors > Home > MPE Home > Th. List > isumnn0nn | Structured version Visualization version GIF version |
Description: Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumnn0nn.1 | ⊢ (𝑘 = 0 → 𝐴 = 𝐵) |
isumnn0nn.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴) |
isumnn0nn.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ) |
isumnn0nn.4 | ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumnn0nn | ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 11722 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11389 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | isumnn0nn.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = 𝐴) | |
4 | isumnn0nn.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ) | |
5 | isumnn0nn.4 | . . 3 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ ) | |
6 | 1, 2, 3, 4, 5 | isum1p 14573 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴)) |
7 | 0nn0 11307 | . . . 4 ⊢ 0 ∈ ℕ0 | |
8 | 3 | ralrimiva 2966 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) = 𝐴) |
9 | fveq2 6191 | . . . . . 6 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0)) | |
10 | isumnn0nn.1 | . . . . . 6 ⊢ (𝑘 = 0 → 𝐴 = 𝐵) | |
11 | 9, 10 | eqeq12d 2637 | . . . . 5 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘0) = 𝐵)) |
12 | 11 | rspcv 3305 | . . . 4 ⊢ (0 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 (𝐹‘𝑘) = 𝐴 → (𝐹‘0) = 𝐵)) |
13 | 7, 8, 12 | mpsyl 68 | . . 3 ⊢ (𝜑 → (𝐹‘0) = 𝐵) |
14 | 0p1e1 11132 | . . . . . . 7 ⊢ (0 + 1) = 1 | |
15 | 14 | fveq2i 6194 | . . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
16 | nnuz 11723 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eqtr4i 2647 | . . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ |
18 | 17 | sumeq1i 14428 | . . . 4 ⊢ Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴 = Σ𝑘 ∈ ℕ 𝐴 |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴 = Σ𝑘 ∈ ℕ 𝐴) |
20 | 13, 19 | oveq12d 6668 | . 2 ⊢ (𝜑 → ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴) = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
21 | 6, 20 | eqtrd 2656 | 1 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 ℕcn 11020 ℕ0cn0 11292 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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