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Mirrors > Home > MPE Home > Th. List > bpoly0 | Structured version Visualization version GIF version |
Description: The value of the Bernoulli polynomials at zero. (Contributed by Scott Fenton, 16-May-2014.) |
Ref | Expression |
---|---|
bpoly0 | โข (๐ โ โ โ (0 BernPoly ๐) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 12523 | . . 3 โข 0 โ โ0 | |
2 | bpolyval 16031 | . . 3 โข ((0 โ โ0 โง ๐ โ โ) โ (0 BernPoly ๐) = ((๐โ0) โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))))) | |
3 | 1, 2 | mpan 688 | . 2 โข (๐ โ โ โ (0 BernPoly ๐) = ((๐โ0) โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))))) |
4 | exp0 14068 | . . . 4 โข (๐ โ โ โ (๐โ0) = 1) | |
5 | 4 | oveq1d 7439 | . . 3 โข (๐ โ โ โ ((๐โ0) โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1)))) = (1 โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))))) |
6 | risefall0lem 16008 | . . . . . . 7 โข (0...(0 โ 1)) = โ | |
7 | 6 | sumeq1i 15682 | . . . . . 6 โข ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))) = ฮฃ๐ โ โ ((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))) |
8 | sum0 15705 | . . . . . 6 โข ฮฃ๐ โ โ ((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))) = 0 | |
9 | 7, 8 | eqtri 2755 | . . . . 5 โข ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1))) = 0 |
10 | 9 | oveq2i 7435 | . . . 4 โข (1 โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1)))) = (1 โ 0) |
11 | 1m0e1 12369 | . . . 4 โข (1 โ 0) = 1 | |
12 | 10, 11 | eqtri 2755 | . . 3 โข (1 โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1)))) = 1 |
13 | 5, 12 | eqtrdi 2783 | . 2 โข (๐ โ โ โ ((๐โ0) โ ฮฃ๐ โ (0...(0 โ 1))((0C๐) ยท ((๐ BernPoly ๐) / ((0 โ ๐) + 1)))) = 1) |
14 | 3, 13 | eqtrd 2767 | 1 โข (๐ โ โ โ (0 BernPoly ๐) = 1) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โ c0 4324 (class class class)co 7424 โcc 11142 0cc0 11144 1c1 11145 + caddc 11147 ยท cmul 11149 โ cmin 11480 / cdiv 11907 โ0cn0 12508 ...cfz 13522 โcexp 14064 Ccbc 14299 ฮฃcsu 15670 BernPoly cbp 16028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671 df-bpoly 16029 |
This theorem is referenced by: bpoly1 16033 bpolydiflem 16036 bpoly2 16039 bpoly3 16040 bpoly4 16041 |
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