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Mirrors > Home > MPE Home > Th. List > pcmptcl | Structured version Visualization version GIF version |
Description: Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.) |
Ref | Expression |
---|---|
pcmpt.1 | β’ πΉ = (π β β β¦ if(π β β, (πβπ΄), 1)) |
pcmpt.2 | β’ (π β βπ β β π΄ β β0) |
Ref | Expression |
---|---|
pcmptcl | β’ (π β (πΉ:ββΆβ β§ seq1( Β· , πΉ):ββΆβ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcmpt.2 | . . . 4 β’ (π β βπ β β π΄ β β0) | |
2 | pm2.27 42 | . . . . . . . 8 β’ (π β β β ((π β β β π΄ β β0) β π΄ β β0)) | |
3 | iftrue 4533 | . . . . . . . . . . 11 β’ (π β β β if(π β β, (πβπ΄), 1) = (πβπ΄)) | |
4 | 3 | adantr 481 | . . . . . . . . . 10 β’ ((π β β β§ π΄ β β0) β if(π β β, (πβπ΄), 1) = (πβπ΄)) |
5 | prmnn 16607 | . . . . . . . . . . 11 β’ (π β β β π β β) | |
6 | nnexpcl 14036 | . . . . . . . . . . 11 β’ ((π β β β§ π΄ β β0) β (πβπ΄) β β) | |
7 | 5, 6 | sylan 580 | . . . . . . . . . 10 β’ ((π β β β§ π΄ β β0) β (πβπ΄) β β) |
8 | 4, 7 | eqeltrd 2833 | . . . . . . . . 9 β’ ((π β β β§ π΄ β β0) β if(π β β, (πβπ΄), 1) β β) |
9 | 8 | ex 413 | . . . . . . . 8 β’ (π β β β (π΄ β β0 β if(π β β, (πβπ΄), 1) β β)) |
10 | 2, 9 | syld 47 | . . . . . . 7 β’ (π β β β ((π β β β π΄ β β0) β if(π β β, (πβπ΄), 1) β β)) |
11 | iffalse 4536 | . . . . . . . . 9 β’ (Β¬ π β β β if(π β β, (πβπ΄), 1) = 1) | |
12 | 1nn 12219 | . . . . . . . . 9 β’ 1 β β | |
13 | 11, 12 | eqeltrdi 2841 | . . . . . . . 8 β’ (Β¬ π β β β if(π β β, (πβπ΄), 1) β β) |
14 | 13 | a1d 25 | . . . . . . 7 β’ (Β¬ π β β β ((π β β β π΄ β β0) β if(π β β, (πβπ΄), 1) β β)) |
15 | 10, 14 | pm2.61i 182 | . . . . . 6 β’ ((π β β β π΄ β β0) β if(π β β, (πβπ΄), 1) β β) |
16 | 15 | a1d 25 | . . . . 5 β’ ((π β β β π΄ β β0) β (π β β β if(π β β, (πβπ΄), 1) β β)) |
17 | 16 | ralimi2 3078 | . . . 4 β’ (βπ β β π΄ β β0 β βπ β β if(π β β, (πβπ΄), 1) β β) |
18 | 1, 17 | syl 17 | . . 3 β’ (π β βπ β β if(π β β, (πβπ΄), 1) β β) |
19 | pcmpt.1 | . . . 4 β’ πΉ = (π β β β¦ if(π β β, (πβπ΄), 1)) | |
20 | 19 | fmpt 7106 | . . 3 β’ (βπ β β if(π β β, (πβπ΄), 1) β β β πΉ:ββΆβ) |
21 | 18, 20 | sylib 217 | . 2 β’ (π β πΉ:ββΆβ) |
22 | nnuz 12861 | . . 3 β’ β = (β€β₯β1) | |
23 | 1zzd 12589 | . . 3 β’ (π β 1 β β€) | |
24 | 21 | ffvelcdmda 7083 | . . 3 β’ ((π β§ π β β) β (πΉβπ) β β) |
25 | nnmulcl 12232 | . . . 4 β’ ((π β β β§ π β β) β (π Β· π) β β) | |
26 | 25 | adantl 482 | . . 3 β’ ((π β§ (π β β β§ π β β)) β (π Β· π) β β) |
27 | 22, 23, 24, 26 | seqf 13985 | . 2 β’ (π β seq1( Β· , πΉ):ββΆβ) |
28 | 21, 27 | jca 512 | 1 β’ (π β (πΉ:ββΆβ β§ seq1( Β· , πΉ):ββΆβ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 ifcif 4527 β¦ cmpt 5230 βΆwf 6536 (class class class)co 7405 1c1 11107 Β· cmul 11111 βcn 12208 β0cn0 12468 seqcseq 13962 βcexp 14023 βcprime 16604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963 df-exp 14024 df-prm 16605 |
This theorem is referenced by: pcmpt 16821 pcmpt2 16822 pcmptdvds 16823 pcprod 16824 1arithlem4 16855 bposlem3 26778 bposlem5 26780 bposlem6 26781 |
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