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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 12617 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12329 | . . 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 15549 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴)) |
7 | fveq2 6769 | . . . . 5 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0)) | |
8 | isumnn0nn.1 | . . . . 5 ⊢ (𝑘 = 0 → 𝐴 = 𝐵) | |
9 | 7, 8 | eqeq12d 2756 | . . . 4 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘0) = 𝐵)) |
10 | 3 | ralrimiva 3110 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘𝑘) = 𝐴) |
11 | 0nn0 12246 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℕ0) |
13 | 9, 10, 12 | rspcdva 3563 | . . 3 ⊢ (𝜑 → (𝐹‘0) = 𝐵) |
14 | 0p1e1 12093 | . . . . . . 7 ⊢ (0 + 1) = 1 | |
15 | 14 | fveq2i 6772 | . . . . . 6 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
16 | nnuz 12618 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eqtr4i 2771 | . . . . 5 ⊢ (ℤ≥‘(0 + 1)) = ℕ |
18 | 17 | sumeq1i 15406 | . . . 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 7287 | . 2 ⊢ (𝜑 → ((𝐹‘0) + Σ𝑘 ∈ (ℤ≥‘(0 + 1))𝐴) = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
21 | 6, 20 | eqtrd 2780 | 1 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐴 = (𝐵 + Σ𝑘 ∈ ℕ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 dom cdm 5589 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 0cc0 10870 1c1 10871 + caddc 10873 ℕcn 11971 ℕ0cn0 12231 ℤ≥cuz 12579 seqcseq 13717 ⇝ cli 15189 Σcsu 15393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-sup 9177 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12580 df-rp 12728 df-fz 13237 df-fzo 13380 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-clim 15193 df-sum 15394 |
This theorem is referenced by: (None) |
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