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Mirrors > Home > MPE Home > Th. List > basel | Structured version Visualization version GIF version |
Description: The sum of the inverse squares is π↑2 / 6. This is commonly known as the Basel problem, with the first known proof attributed to Euler. See http://en.wikipedia.org/wiki/Basel_problem. This particular proof approach is due to Cauchy (1821). This is Metamath 100 proof #14. (Contributed by Mario Carneiro, 30-Jul-2014.) |
Ref | Expression |
---|---|
basel | ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6986 | . . . . 5 ⊢ (𝑚 = 𝑛 → (2 · 𝑚) = (2 · 𝑛)) | |
2 | 1 | oveq1d 6993 | . . . 4 ⊢ (𝑚 = 𝑛 → ((2 · 𝑚) + 1) = ((2 · 𝑛) + 1)) |
3 | 2 | oveq2d 6994 | . . 3 ⊢ (𝑚 = 𝑛 → (1 / ((2 · 𝑚) + 1)) = (1 / ((2 · 𝑛) + 1))) |
4 | 3 | cbvmptv 5029 | . 2 ⊢ (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))) = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1))) |
5 | oveq1 6985 | . . . 4 ⊢ (𝑚 = 𝑛 → (𝑚↑-2) = (𝑛↑-2)) | |
6 | 5 | cbvmptv 5029 | . . 3 ⊢ (𝑚 ∈ ℕ ↦ (𝑚↑-2)) = (𝑛 ∈ ℕ ↦ (𝑛↑-2)) |
7 | seqeq3 13192 | . . 3 ⊢ ((𝑚 ∈ ℕ ↦ (𝑚↑-2)) = (𝑛 ∈ ℕ ↦ (𝑛↑-2)) → seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑-2))) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (𝑚↑-2))) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2))) |
9 | eqid 2778 | . 2 ⊢ ((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓 − (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) = ((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓 − (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) | |
10 | eqid 2778 | . 2 ⊢ (((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓 − (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) ∘𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1)))))) = (((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓 − (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) ∘𝑓 · ((ℕ × {1}) ∘𝑓 + ((ℕ × {-2}) ∘𝑓 · (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1)))))) | |
11 | eqid 2778 | . 2 ⊢ (((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓 − (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) ∘𝑓 · ((ℕ × {1}) ∘𝑓 + (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) = (((ℕ × {((π↑2) / 6)}) ∘𝑓 · ((ℕ × {1}) ∘𝑓 − (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) ∘𝑓 · ((ℕ × {1}) ∘𝑓 + (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))))) | |
12 | 4, 8, 9, 10, 11 | basellem9 25371 | 1 ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 {csn 4442 ↦ cmpt 5009 × cxp 5406 (class class class)co 6978 ∘𝑓 cof 7227 1c1 10338 + caddc 10340 · cmul 10342 − cmin 10672 -cneg 10673 / cdiv 11100 ℕcn 11441 2c2 11498 6c6 11502 seqcseq 13187 ↑cexp 13247 Σcsu 14906 πcpi 15283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-dju 9126 df-card 9164 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-xnn0 11783 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ioc 12562 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-fac 13452 df-bc 13481 df-hash 13509 df-shft 14290 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-sum 14907 df-ef 15284 df-sin 15286 df-cos 15287 df-tan 15288 df-pi 15289 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-haus 21630 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cncf 23192 df-0p 23977 df-limc 24170 df-dv 24171 df-ply 24484 df-idp 24485 df-coe 24486 df-dgr 24487 df-quot 24586 |
This theorem is referenced by: (None) |
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