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Mathbox for Asger C. Ipsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppf | Structured version Visualization version GIF version |
Description: Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
Ref | Expression |
---|---|
knoppf.t | β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) |
knoppf.f | β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) |
knoppf.w | β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) |
knoppf.c | β’ (π β πΆ β (-1(,)1)) |
knoppf.n | β’ (π β π β β) |
Ref | Expression |
---|---|
knoppf | β’ (π β π:ββΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12871 | . . 3 β’ β0 = (β€β₯β0) | |
2 | 0zd 12577 | . . 3 β’ ((π β§ π€ β β) β 0 β β€) | |
3 | eqidd 2732 | . . 3 β’ (((π β§ π€ β β) β§ π β β0) β ((πΉβπ€)βπ) = ((πΉβπ€)βπ)) | |
4 | knoppf.t | . . . 4 β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) | |
5 | knoppf.f | . . . 4 β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) | |
6 | knoppf.n | . . . . . 6 β’ (π β π β β) | |
7 | 6 | adantr 480 | . . . . 5 β’ ((π β§ π€ β β) β π β β) |
8 | 7 | adantr 480 | . . . 4 β’ (((π β§ π€ β β) β§ π β β0) β π β β) |
9 | knoppf.c | . . . . . . . 8 β’ (π β πΆ β (-1(,)1)) | |
10 | 9 | knoppndvlem3 35854 | . . . . . . 7 β’ (π β (πΆ β β β§ (absβπΆ) < 1)) |
11 | 10 | simpld 494 | . . . . . 6 β’ (π β πΆ β β) |
12 | 11 | adantr 480 | . . . . 5 β’ ((π β§ π€ β β) β πΆ β β) |
13 | 12 | adantr 480 | . . . 4 β’ (((π β§ π€ β β) β§ π β β0) β πΆ β β) |
14 | simpr 484 | . . . . 5 β’ ((π β§ π€ β β) β π€ β β) | |
15 | 14 | adantr 480 | . . . 4 β’ (((π β§ π€ β β) β§ π β β0) β π€ β β) |
16 | simpr 484 | . . . 4 β’ (((π β§ π€ β β) β§ π β β0) β π β β0) | |
17 | 4, 5, 8, 13, 15, 16 | knoppcnlem3 35835 | . . 3 β’ (((π β§ π€ β β) β§ π β β0) β ((πΉβπ€)βπ) β β) |
18 | knoppf.w | . . . . . 6 β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) | |
19 | fveq2 6891 | . . . . . . . . 9 β’ (π€ = π§ β (πΉβπ€) = (πΉβπ§)) | |
20 | 19 | fveq1d 6893 | . . . . . . . 8 β’ (π€ = π§ β ((πΉβπ€)βπ) = ((πΉβπ§)βπ)) |
21 | 20 | sumeq2sdv 15657 | . . . . . . 7 β’ (π€ = π§ β Ξ£π β β0 ((πΉβπ€)βπ) = Ξ£π β β0 ((πΉβπ§)βπ)) |
22 | 21 | cbvmptv 5261 | . . . . . 6 β’ (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) = (π§ β β β¦ Ξ£π β β0 ((πΉβπ§)βπ)) |
23 | 18, 22 | eqtri 2759 | . . . . 5 β’ π = (π§ β β β¦ Ξ£π β β0 ((πΉβπ§)βπ)) |
24 | 9 | adantr 480 | . . . . 5 β’ ((π β§ π€ β β) β πΆ β (-1(,)1)) |
25 | 4, 5, 23, 14, 24, 7 | knoppndvlem4 35855 | . . . 4 β’ ((π β§ π€ β β) β seq0( + , (πΉβπ€)) β (πβπ€)) |
26 | seqex 13975 | . . . . 5 β’ seq0( + , (πΉβπ€)) β V | |
27 | fvex 6904 | . . . . 5 β’ (πβπ€) β V | |
28 | 26, 27 | breldm 5908 | . . . 4 β’ (seq0( + , (πΉβπ€)) β (πβπ€) β seq0( + , (πΉβπ€)) β dom β ) |
29 | 25, 28 | syl 17 | . . 3 β’ ((π β§ π€ β β) β seq0( + , (πΉβπ€)) β dom β ) |
30 | 1, 2, 3, 17, 29 | isumrecl 15718 | . 2 β’ ((π β§ π€ β β) β Ξ£π β β0 ((πΉβπ€)βπ) β β) |
31 | 30, 18 | fmptd 7115 | 1 β’ (π β π:ββΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 class class class wbr 5148 β¦ cmpt 5231 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcr 11115 0cc0 11116 1c1 11117 + caddc 11119 Β· cmul 11121 < clt 11255 β cmin 11451 -cneg 11452 / cdiv 11878 βcn 12219 2c2 12274 β0cn0 12479 (,)cioo 13331 βcfl 13762 seqcseq 13973 βcexp 14034 abscabs 15188 β cli 15435 Ξ£csu 15639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-ioo 13335 df-ico 13337 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ulm 26228 |
This theorem is referenced by: knoppcn2 35876 |
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