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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sum9cubes | Structured version Visualization version GIF version | ||
| Description: The sum of the first nine perfect cubes is 2025. (Contributed by SN, 30-Mar-2025.) |
| Ref | Expression |
|---|---|
| sum9cubes | ⊢ Σ𝑘 ∈ (1...9)(𝑘↑3) = ;;;2025 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 9nn0 12550 | . . 3 ⊢ 9 ∈ ℕ0 | |
| 2 | sumcubes 42347 | . . 3 ⊢ (9 ∈ ℕ0 → Σ𝑘 ∈ (1...9)(𝑘↑3) = (Σ𝑘 ∈ (1...9)𝑘↑2)) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ Σ𝑘 ∈ (1...9)(𝑘↑3) = (Σ𝑘 ∈ (1...9)𝑘↑2) |
| 4 | arisum 15896 | . . . . 5 ⊢ (9 ∈ ℕ0 → Σ𝑘 ∈ (1...9)𝑘 = (((9↑2) + 9) / 2)) | |
| 5 | 1, 4 | ax-mp 5 | . . . 4 ⊢ Σ𝑘 ∈ (1...9)𝑘 = (((9↑2) + 9) / 2) |
| 6 | 8nn0 12549 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
| 7 | 1nn0 12542 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 8 | sq9 42332 | . . . . . . 7 ⊢ (9↑2) = ;81 | |
| 9 | 8p1e9 12416 | . . . . . . 7 ⊢ (8 + 1) = 9 | |
| 10 | 9cn 12366 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
| 11 | ax-1cn 11213 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
| 12 | 9p1e10 12735 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
| 13 | 10, 11, 12 | addcomli 11453 | . . . . . . 7 ⊢ (1 + 9) = ;10 |
| 14 | 6, 7, 1, 8, 9, 13 | decaddci2 12795 | . . . . . 6 ⊢ ((9↑2) + 9) = ;90 |
| 15 | 2nn0 12543 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
| 16 | 4nn0 12545 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
| 17 | 5nn0 12546 | . . . . . . 7 ⊢ 5 ∈ ℕ0 | |
| 18 | eqid 2737 | . . . . . . 7 ⊢ ;45 = ;45 | |
| 19 | 0nn0 12541 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
| 20 | 4t2e8 12434 | . . . . . . . . 9 ⊢ (4 · 2) = 8 | |
| 21 | 20 | oveq1i 7441 | . . . . . . . 8 ⊢ ((4 · 2) + 1) = (8 + 1) |
| 22 | 21, 9 | eqtri 2765 | . . . . . . 7 ⊢ ((4 · 2) + 1) = 9 |
| 23 | 5t2e10 12833 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
| 24 | 15, 16, 17, 18, 19, 7, 22, 23 | decmul1c 12798 | . . . . . 6 ⊢ (;45 · 2) = ;90 |
| 25 | 14, 24 | eqtr4i 2768 | . . . . 5 ⊢ ((9↑2) + 9) = (;45 · 2) |
| 26 | 25 | oveq1i 7441 | . . . 4 ⊢ (((9↑2) + 9) / 2) = ((;45 · 2) / 2) |
| 27 | 16, 17 | deccl 12748 | . . . . . 6 ⊢ ;45 ∈ ℕ0 |
| 28 | 27 | nn0cni 12538 | . . . . 5 ⊢ ;45 ∈ ℂ |
| 29 | 2cn 12341 | . . . . 5 ⊢ 2 ∈ ℂ | |
| 30 | 2ne0 12370 | . . . . 5 ⊢ 2 ≠ 0 | |
| 31 | 28, 29, 30 | divcan4i 12014 | . . . 4 ⊢ ((;45 · 2) / 2) = ;45 |
| 32 | 5, 26, 31 | 3eqtri 2769 | . . 3 ⊢ Σ𝑘 ∈ (1...9)𝑘 = ;45 |
| 33 | 32 | oveq1i 7441 | . 2 ⊢ (Σ𝑘 ∈ (1...9)𝑘↑2) = (;45↑2) |
| 34 | sq45 42681 | . 2 ⊢ (;45↑2) = ;;;2025 | |
| 35 | 3, 33, 34 | 3eqtri 2769 | 1 ⊢ Σ𝑘 ∈ (1...9)(𝑘↑3) = ;;;2025 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 / cdiv 11920 2c2 12321 3c3 12322 4c4 12323 5c5 12324 8c8 12327 9c9 12328 ℕ0cn0 12526 ;cdc 12733 ...cfz 13547 ↑cexp 14102 Σcsu 15722 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 |
| This theorem is referenced by: (None) |
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