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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nicomachus | Structured version Visualization version GIF version | ||
| Description: Nicomachus's Theorem. The sum of the odd numbers from 𝑁↑2 − 𝑁 + 1 to 𝑁↑2 + 𝑁 − 1 is 𝑁↑3. Proof 2 from https://proofwiki.org/wiki/Nicomachus%27s_Theorem. (Contributed by SN, 21-Mar-2025.) |
| Ref | Expression |
|---|---|
| nicomachus | ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(((𝑁↑2) − 𝑁) + ((2 · 𝑘) − 1)) = (𝑁↑3)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fzfid 13986 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin) | |
| 2 | nn0cn 12491 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
| 3 | 2 | adantr 484 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑁 ∈ ℂ) |
| 4 | 3 | sqcld 14157 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → (𝑁↑2) ∈ ℂ) |
| 5 | 4, 3 | subcld 11542 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((𝑁↑2) − 𝑁) ∈ ℂ) |
| 6 | 2cnd 12296 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 2 ∈ ℂ) | |
| 7 | elfznn 13558 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
| 8 | 7 | nncnd 12226 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℂ) |
| 9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ) |
| 10 | 6, 9 | mulcld 11202 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → (2 · 𝑘) ∈ ℂ) |
| 11 | 1cnd 11175 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → 1 ∈ ℂ) | |
| 12 | 10, 11 | subcld 11542 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (1...𝑁)) → ((2 · 𝑘) − 1) ∈ ℂ) |
| 13 | 1, 5, 12 | fsumadd 15767 | . 2 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(((𝑁↑2) − 𝑁) + ((2 · 𝑘) − 1)) = (Σ𝑘 ∈ (1...𝑁)((𝑁↑2) − 𝑁) + Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1))) |
| 14 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
| 15 | 2 | sqcld 14157 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁↑2) ∈ ℂ) |
| 16 | 15, 2 | subcld 11542 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁↑2) − 𝑁) ∈ ℂ) |
| 17 | 14, 16 | fz1sumconst 42915 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((𝑁↑2) − 𝑁) = (𝑁 · ((𝑁↑2) − 𝑁))) |
| 18 | 2, 15, 2 | subdid 11643 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 · ((𝑁↑2) − 𝑁)) = ((𝑁 · (𝑁↑2)) − (𝑁 · 𝑁))) |
| 19 | df-3 12281 | . . . . . . . 8 ⊢ 3 = (2 + 1) | |
| 20 | 19 | oveq2i 7407 | . . . . . . 7 ⊢ (𝑁↑3) = (𝑁↑(2 + 1)) |
| 21 | 2nn0 12498 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
| 22 | 21 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ0) |
| 23 | 2, 22 | expp1d 14160 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁↑(2 + 1)) = ((𝑁↑2) · 𝑁)) |
| 24 | 20, 23 | eqtrid 2809 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁↑3) = ((𝑁↑2) · 𝑁)) |
| 25 | 15, 2 | mulcomd 11203 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁↑2) · 𝑁) = (𝑁 · (𝑁↑2))) |
| 26 | 24, 25 | eqtr2d 2798 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 · (𝑁↑2)) = (𝑁↑3)) |
| 27 | 2 | sqvald 14156 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁↑2) = (𝑁 · 𝑁)) |
| 28 | 27 | eqcomd 2768 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 · 𝑁) = (𝑁↑2)) |
| 29 | 26, 28 | oveq12d 7414 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 · (𝑁↑2)) − (𝑁 · 𝑁)) = ((𝑁↑3) − (𝑁↑2))) |
| 30 | 17, 18, 29 | 3eqtrd 2801 | . . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((𝑁↑2) − 𝑁) = ((𝑁↑3) − (𝑁↑2))) |
| 31 | oddnumth 42917 | . . 3 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2)) | |
| 32 | 30, 31 | oveq12d 7414 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Σ𝑘 ∈ (1...𝑁)((𝑁↑2) − 𝑁) + Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1)) = (((𝑁↑3) − (𝑁↑2)) + (𝑁↑2))) |
| 33 | 3nn0 12499 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 34 | 33 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℕ0) |
| 35 | 2, 34 | expcld 14159 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁↑3) ∈ ℂ) |
| 36 | 35, 15 | npcand 11546 | . 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁↑3) − (𝑁↑2)) + (𝑁↑2)) = (𝑁↑3)) |
| 37 | 13, 32, 36 | 3eqtrd 2801 | 1 ⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(((𝑁↑2) − 𝑁) + ((2 · 𝑘) − 1)) = (𝑁↑3)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 (class class class)co 7396 ℂcc 11071 1c1 11074 + caddc 11076 · cmul 11078 − cmin 11414 2c2 12272 3c3 12273 ℕ0cn0 12481 ...cfz 13512 ↑cexp 14074 Σcsu 15713 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-n0 12482 df-z 12569 df-uz 12840 df-rp 12994 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-sum 15714 |
| This theorem is referenced by: sumcubes 42919 |
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