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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 20948 | . . . . . . 7 β’ β = (Baseββfld) | |
4 | 3 | subrgss 20320 | . . . . . 6 β’ (π β (SubRingββfld) β π β β) |
5 | 2, 4 | syl 17 | . . . . 5 β’ (π β π β β) |
6 | plyss 25713 | . . . . 5 β’ ((πΆ β π β§ π β β) β (PolyβπΆ) β (Polyβπ)) | |
7 | 1, 5, 6 | syl2anc 585 | . . . 4 β’ (π β (PolyβπΆ) β (Polyβπ)) |
8 | cnsrplycl.p | . . . 4 β’ (π β π β (PolyβπΆ)) | |
9 | 7, 8 | sseldd 3984 | . . 3 β’ (π β π β (Polyβπ)) |
10 | cnsrplycl.x | . . . 4 β’ (π β π β π) | |
11 | 5, 10 | sseldd 3984 | . . 3 β’ (π β π β β) |
12 | eqid 2733 | . . . 4 β’ (coeffβπ) = (coeffβπ) | |
13 | eqid 2733 | . . . 4 β’ (degβπ) = (degβπ) | |
14 | 12, 13 | coeid2 25753 | . . 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 585 | . 2 β’ (π β (πβπ) = Ξ£π β (0...(degβπ))(((coeffβπ)βπ) Β· (πβπ))) |
16 | fzfid 13938 | . . 3 β’ (π β (0...(degβπ)) β Fin) | |
17 | 2 | adantr 482 | . . . 4 β’ ((π β§ π β (0...(degβπ))) β π β (SubRingββfld)) |
18 | subrgsubg 20325 | . . . . . . . 8 β’ (π β (SubRingββfld) β π β (SubGrpββfld)) | |
19 | cnfld0 20969 | . . . . . . . . 9 β’ 0 = (0gββfld) | |
20 | 19 | subg0cl 19014 | . . . . . . . 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 25745 | . . . . . . 7 β’ ((π β (Polyβπ) β§ 0 β π) β (coeffβπ):β0βΆπ) |
23 | 9, 21, 22 | syl2anc 585 | . . . . . 6 β’ (π β (coeffβπ):β0βΆπ) |
24 | 23 | adantr 482 | . . . . 5 β’ ((π β§ π β (0...(degβπ))) β (coeffβπ):β0βΆπ) |
25 | elfznn0 13594 | . . . . . 6 β’ (π β (0...(degβπ)) β π β β0) | |
26 | 25 | adantl 483 | . . . . 5 β’ ((π β§ π β (0...(degβπ))) β π β β0) |
27 | 24, 26 | ffvelcdmd 7088 | . . . 4 β’ ((π β§ π β (0...(degβπ))) β ((coeffβπ)βπ) β π) |
28 | 10 | adantr 482 | . . . . 5 β’ ((π β§ π β (0...(degβπ))) β π β π) |
29 | 17, 28, 26 | cnsrexpcl 41907 | . . . 4 β’ ((π β§ π β (0...(degβπ))) β (πβπ) β π) |
30 | cnfldmul 20950 | . . . . 5 β’ Β· = (.rββfld) | |
31 | 30 | subrgmcl 20331 | . . . 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 1372 | . . 3 β’ ((π β§ π β (0...(degβπ))) β (((coeffβπ)βπ) Β· (πβπ)) β π) |
33 | 2, 16, 32 | fsumcnsrcl 41908 | . 2 β’ (π β Ξ£π β (0...(degβπ))(((coeffβπ)βπ) Β· (πβπ)) β π) |
34 | 15, 33 | eqeltrd 2834 | 1 β’ (π β (πβπ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcc 11108 0cc0 11110 Β· cmul 11115 β0cn0 12472 ...cfz 13484 βcexp 14027 Ξ£csu 15632 SubGrpcsubg 19000 SubRingcsubrg 20315 βfldccnfld 20944 Polycply 25698 coeffccoe 25700 degcdgr 25701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-subg 19003 df-cmn 19650 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-subrg 20317 df-cnfld 20945 df-0p 25187 df-ply 25702 df-coe 25704 df-dgr 25705 |
This theorem is referenced by: rngunsnply 41915 |
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