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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 6799 | . . . . 5 ⊢ (𝑚 = 𝑛 → (2 · 𝑚) = (2 · 𝑛)) | |
2 | 1 | oveq1d 6806 | . . . 4 ⊢ (𝑚 = 𝑛 → ((2 · 𝑚) + 1) = ((2 · 𝑛) + 1)) |
3 | 2 | oveq2d 6807 | . . 3 ⊢ (𝑚 = 𝑛 → (1 / ((2 · 𝑚) + 1)) = (1 / ((2 · 𝑛) + 1))) |
4 | 3 | cbvmptv 4884 | . 2 ⊢ (𝑚 ∈ ℕ ↦ (1 / ((2 · 𝑚) + 1))) = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1))) |
5 | oveq1 6798 | . . . 4 ⊢ (𝑚 = 𝑛 → (𝑚↑-2) = (𝑛↑-2)) | |
6 | 5 | cbvmptv 4884 | . . 3 ⊢ (𝑚 ∈ ℕ ↦ (𝑚↑-2)) = (𝑛 ∈ ℕ ↦ (𝑛↑-2)) |
7 | seqeq3 13006 | . . 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 2771 | . 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 2771 | . 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 2771 | . 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 25029 | 1 ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 {csn 4316 ↦ cmpt 4863 × cxp 5247 (class class class)co 6791 ∘𝑓 cof 7040 1c1 10137 + caddc 10139 · cmul 10141 − cmin 10466 -cneg 10467 / cdiv 10884 ℕcn 11220 2c2 11270 6c6 11274 seqcseq 13001 ↑cexp 13060 Σcsu 14617 πcpi 14996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 ax-addf 10215 ax-mulf 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-of 7042 df-om 7211 df-1st 7313 df-2nd 7314 df-supp 7445 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-er 7894 df-map 8009 df-pm 8010 df-ixp 8061 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-fsupp 8430 df-fi 8471 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-xnn0 11564 df-z 11578 df-dec 11694 df-uz 11887 df-q 11990 df-rp 12029 df-xneg 12144 df-xadd 12145 df-xmul 12146 df-ioo 12377 df-ioc 12378 df-ico 12379 df-icc 12380 df-fz 12527 df-fzo 12667 df-fl 12794 df-mod 12870 df-seq 13002 df-exp 13061 df-fac 13258 df-bc 13287 df-hash 13315 df-shft 14008 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-limsup 14403 df-clim 14420 df-rlim 14421 df-sum 14618 df-ef 14997 df-sin 14999 df-cos 15000 df-tan 15001 df-pi 15002 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-mulr 16156 df-starv 16157 df-sca 16158 df-vsca 16159 df-ip 16160 df-tset 16161 df-ple 16162 df-ds 16165 df-unif 16166 df-hom 16167 df-cco 16168 df-rest 16284 df-topn 16285 df-0g 16303 df-gsum 16304 df-topgen 16305 df-pt 16306 df-prds 16309 df-xrs 16363 df-qtop 16368 df-imas 16369 df-xps 16371 df-mre 16447 df-mrc 16448 df-acs 16450 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19946 df-xmet 19947 df-met 19948 df-bl 19949 df-mopn 19950 df-fbas 19951 df-fg 19952 df-cnfld 19955 df-top 20912 df-topon 20929 df-topsp 20951 df-bases 20964 df-cld 21037 df-ntr 21038 df-cls 21039 df-nei 21116 df-lp 21154 df-perf 21155 df-cn 21245 df-cnp 21246 df-haus 21333 df-tx 21579 df-hmeo 21772 df-fil 21863 df-fm 21955 df-flim 21956 df-flf 21957 df-xms 22338 df-ms 22339 df-tms 22340 df-cncf 22894 df-0p 23650 df-limc 23843 df-dv 23844 df-ply 24157 df-idp 24158 df-coe 24159 df-dgr 24160 df-quot 24259 |
This theorem is referenced by: (None) |
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