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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnsrplycl | Structured version Visualization version GIF version |
Description: Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
Ref | Expression |
---|---|
cnsrplycl.s | β’ (π β π β (SubRingββfld)) |
cnsrplycl.p | β’ (π β π β (PolyβπΆ)) |
cnsrplycl.x | β’ (π β π β π) |
cnsrplycl.c | β’ (π β πΆ β π) |
Ref | Expression |
---|---|
cnsrplycl | β’ (π β (πβπ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsrplycl.c | . . . . 5 β’ (π β πΆ β π) | |
2 | cnsrplycl.s | . . . . . 6 β’ (π β π β (SubRingββfld)) | |
3 | cnfldbas 21287 | . . . . . . 7 β’ β = (Baseββfld) | |
4 | 3 | subrgss 20515 | . . . . . 6 β’ (π β (SubRingββfld) β π β β) |
5 | 2, 4 | syl 17 | . . . . 5 β’ (π β π β β) |
6 | plyss 26151 | . . . . 5 β’ ((πΆ β π β§ π β β) β (PolyβπΆ) β (Polyβπ)) | |
7 | 1, 5, 6 | syl2anc 582 | . . . 4 β’ (π β (PolyβπΆ) β (Polyβπ)) |
8 | cnsrplycl.p | . . . 4 β’ (π β π β (PolyβπΆ)) | |
9 | 7, 8 | sseldd 3973 | . . 3 β’ (π β π β (Polyβπ)) |
10 | cnsrplycl.x | . . . 4 β’ (π β π β π) | |
11 | 5, 10 | sseldd 3973 | . . 3 β’ (π β π β β) |
12 | eqid 2725 | . . . 4 β’ (coeffβπ) = (coeffβπ) | |
13 | eqid 2725 | . . . 4 β’ (degβπ) = (degβπ) | |
14 | 12, 13 | coeid2 26191 | . . 3 β’ ((π β (Polyβπ) β§ π β β) β (πβπ) = Ξ£π β (0...(degβπ))(((coeffβπ)βπ) Β· (πβπ))) |
15 | 9, 11, 14 | syl2anc 582 | . 2 β’ (π β (πβπ) = Ξ£π β (0...(degβπ))(((coeffβπ)βπ) Β· (πβπ))) |
16 | fzfid 13970 | . . 3 β’ (π β (0...(degβπ)) β Fin) | |
17 | 2 | adantr 479 | . . . 4 β’ ((π β§ π β (0...(degβπ))) β π β (SubRingββfld)) |
18 | subrgsubg 20520 | . . . . . . . 8 β’ (π β (SubRingββfld) β π β (SubGrpββfld)) | |
19 | cnfld0 21324 | . . . . . . . . 9 β’ 0 = (0gββfld) | |
20 | 19 | subg0cl 19093 | . . . . . . . 8 β’ (π β (SubGrpββfld) β 0 β π) |
21 | 2, 18, 20 | 3syl 18 | . . . . . . 7 β’ (π β 0 β π) |
22 | 12 | coef2 26183 | . . . . . . 7 β’ ((π β (Polyβπ) β§ 0 β π) β (coeffβπ):β0βΆπ) |
23 | 9, 21, 22 | syl2anc 582 | . . . . . 6 β’ (π β (coeffβπ):β0βΆπ) |
24 | 23 | adantr 479 | . . . . 5 β’ ((π β§ π β (0...(degβπ))) β (coeffβπ):β0βΆπ) |
25 | elfznn0 13626 | . . . . . 6 β’ (π β (0...(degβπ)) β π β β0) | |
26 | 25 | adantl 480 | . . . . 5 β’ ((π β§ π β (0...(degβπ))) β π β β0) |
27 | 24, 26 | ffvelcdmd 7090 | . . . 4 β’ ((π β§ π β (0...(degβπ))) β ((coeffβπ)βπ) β π) |
28 | 10 | adantr 479 | . . . . 5 β’ ((π β§ π β (0...(degβπ))) β π β π) |
29 | 17, 28, 26 | cnsrexpcl 42654 | . . . 4 β’ ((π β§ π β (0...(degβπ))) β (πβπ) β π) |
30 | cnfldmul 21291 | . . . . 5 β’ Β· = (.rββfld) | |
31 | 30 | subrgmcl 20527 | . . . 4 β’ ((π β (SubRingββfld) β§ ((coeffβπ)βπ) β π β§ (πβπ) β π) β (((coeffβπ)βπ) Β· (πβπ)) β π) |
32 | 17, 27, 29, 31 | syl3anc 1368 | . . 3 β’ ((π β§ π β (0...(degβπ))) β (((coeffβπ)βπ) Β· (πβπ)) β π) |
33 | 2, 16, 32 | fsumcnsrcl 42655 | . 2 β’ (π β Ξ£π β (0...(degβπ))(((coeffβπ)βπ) Β· (πβπ)) β π) |
34 | 15, 33 | eqeltrd 2825 | 1 β’ (π β (πβπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3939 βΆwf 6539 βcfv 6543 (class class class)co 7416 βcc 11136 0cc0 11138 Β· cmul 11143 β0cn0 12502 ...cfz 13516 βcexp 14058 Ξ£csu 15664 SubGrpcsubg 19079 SubRingcsubrg 20510 βfldccnfld 21283 Polycply 26136 coeffccoe 26138 degcdgr 26139 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-subg 19082 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-subrng 20487 df-subrg 20512 df-cnfld 21284 df-0p 25617 df-ply 26140 df-coe 26142 df-dgr 26143 |
This theorem is referenced by: rngunsnply 42662 |
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