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Mirrors > Home > MPE Home > Th. List > Mathboxes > oddnumth | Structured version Visualization version GIF version |
Description: The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15892. (Contributed by SN, 21-Mar-2025.) |
Ref | Expression |
---|---|
oddnumth | ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 14010 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
2 | 2cnd 12341 | . . . . 5 ⊢ (𝑘 ∈ (1...𝑁) → 2 ∈ ℂ) | |
3 | elfznn 13589 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
4 | 3 | nncnd 12279 | . . . . 5 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℂ) |
5 | 2, 4 | mulcld 11278 | . . . 4 ⊢ (𝑘 ∈ (1...𝑁) → (2 · 𝑘) ∈ ℂ) |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → (2 · 𝑘) ∈ ℂ) |
7 | 1cnd 11253 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ ℂ) | |
8 | 1, 6, 7 | fsumsub 15820 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (Σ𝑘 ∈ (1...𝑁)(2 · 𝑘) − Σ𝑘 ∈ (1...𝑁)1)) |
9 | arisum 15892 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)𝑘 = (((𝑁↑2) + 𝑁) / 2)) | |
10 | 9 | oveq2d 7446 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑁)𝑘) = (2 · (((𝑁↑2) + 𝑁) / 2))) |
11 | 2cnd 12341 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) | |
12 | 4 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ) |
13 | 1, 11, 12 | fsummulc2 15816 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 · Σ𝑘 ∈ (1...𝑁)𝑘) = Σ𝑘 ∈ (1...𝑁)(2 · 𝑘)) |
14 | nn0cn 12533 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
15 | 14 | sqcld 14180 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁↑2) ∈ ℂ) |
16 | 15, 14 | addcld 11277 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁↑2) + 𝑁) ∈ ℂ) |
17 | 2ne0 12367 | . . . . . 6 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ≠ 0) |
19 | 16, 11, 18 | divcan2d 12042 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 · (((𝑁↑2) + 𝑁) / 2)) = ((𝑁↑2) + 𝑁)) |
20 | 10, 13, 19 | 3eqtr3d 2782 | . . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(2 · 𝑘) = ((𝑁↑2) + 𝑁)) |
21 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
22 | 1cnd 11253 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) | |
23 | 21, 22 | fz1sumconst 42321 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)1 = (𝑁 · 1)) |
24 | 14 | mulridd 11275 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 · 1) = 𝑁) |
25 | 23, 24 | eqtrd 2774 | . . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)1 = 𝑁) |
26 | 20, 25 | oveq12d 7448 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑁)(2 · 𝑘) − Σ𝑘 ∈ (1...𝑁)1) = (((𝑁↑2) + 𝑁) − 𝑁)) |
27 | 15, 14 | pncand 11618 | . 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁↑2) + 𝑁) − 𝑁) = (𝑁↑2)) |
28 | 8, 26, 27 | 3eqtrd 2778 | 1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 − cmin 11489 / cdiv 11917 2c2 12318 ℕ0cn0 12523 ...cfz 13543 ↑cexp 14098 Σcsu 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 |
This theorem is referenced by: nicomachus 42324 sumcubes 42325 |
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