Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnsrplycl Structured version   Unicode version

Theorem cnsrplycl 27340
Description: Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
Hypotheses
Ref Expression
cnsrplycl.s  |-  ( ph  ->  S  e.  (SubRing ` fld ) )
cnsrplycl.p  |-  ( ph  ->  P  e.  (Poly `  C ) )
cnsrplycl.x  |-  ( ph  ->  X  e.  S )
cnsrplycl.c  |-  ( ph  ->  C  C_  S )
Assertion
Ref Expression
cnsrplycl  |-  ( ph  ->  ( P `  X
)  e.  S )

Proof of Theorem cnsrplycl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cnsrplycl.c . . . . 5  |-  ( ph  ->  C  C_  S )
2 cnsrplycl.s . . . . . 6  |-  ( ph  ->  S  e.  (SubRing ` fld ) )
3 cnfldbas 16699 . . . . . . 7  |-  CC  =  ( Base ` fld )
43subrgss 15861 . . . . . 6  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
52, 4syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
6 plyss 20110 . . . . 5  |-  ( ( C  C_  S  /\  S  C_  CC )  -> 
(Poly `  C )  C_  (Poly `  S )
)
71, 5, 6syl2anc 643 . . . 4  |-  ( ph  ->  (Poly `  C )  C_  (Poly `  S )
)
8 cnsrplycl.p . . . 4  |-  ( ph  ->  P  e.  (Poly `  C ) )
97, 8sseldd 3341 . . 3  |-  ( ph  ->  P  e.  (Poly `  S ) )
10 cnsrplycl.x . . . 4  |-  ( ph  ->  X  e.  S )
115, 10sseldd 3341 . . 3  |-  ( ph  ->  X  e.  CC )
12 eqid 2435 . . . 4  |-  (coeff `  P )  =  (coeff `  P )
13 eqid 2435 . . . 4  |-  (deg `  P )  =  (deg
`  P )
1412, 13coeid2 20150 . . 3  |-  ( ( P  e.  (Poly `  S )  /\  X  e.  CC )  ->  ( P `  X )  =  sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) ) )
159, 11, 14syl2anc 643 . 2  |-  ( ph  ->  ( P `  X
)  =  sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) ) )
16 fzfid 11304 . . 3  |-  ( ph  ->  ( 0 ... (deg `  P ) )  e. 
Fin )
172adantr 452 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  S  e.  (SubRing ` fld ) )
18 subrgsubg 15866 . . . . . . . 8  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
19 cnfld0 16717 . . . . . . . . 9  |-  0  =  ( 0g ` fld )
2019subg0cl 14944 . . . . . . . 8  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
212, 18, 203syl 19 . . . . . . 7  |-  ( ph  ->  0  e.  S )
2212coef2 20142 . . . . . . 7  |-  ( ( P  e.  (Poly `  S )  /\  0  e.  S )  ->  (coeff `  P ) : NN0 --> S )
239, 21, 22syl2anc 643 . . . . . 6  |-  ( ph  ->  (coeff `  P ) : NN0 --> S )
2423adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  (coeff `  P
) : NN0 --> S )
25 elfznn0 11075 . . . . . 6  |-  ( k  e.  ( 0 ... (deg `  P )
)  ->  k  e.  NN0 )
2625adantl 453 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  k  e.  NN0 )
2724, 26ffvelrnd 5863 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( (coeff `  P ) `  k
)  e.  S )
2810adantr 452 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  X  e.  S
)
2917, 28, 26cnsrexpcl 27338 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( X ^
k )  e.  S
)
30 cnfldmul 16701 . . . . 5  |-  x.  =  ( .r ` fld )
3130subrgmcl 15872 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  (
(coeff `  P ) `  k )  e.  S  /\  ( X ^ k
)  e.  S )  ->  ( ( (coeff `  P ) `  k
)  x.  ( X ^ k ) )  e.  S )
3217, 27, 29, 31syl3anc 1184 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( ( (coeff `  P ) `  k
)  x.  ( X ^ k ) )  e.  S )
332, 16, 32fsumcnsrcl 27339 . 2  |-  ( ph  -> 
sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) )  e.  S
)
3415, 33eqeltrd 2509 1  |-  ( ph  ->  ( P `  X
)  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987   NN0cn0 10213   ...cfz 11035   ^cexp 11374   sum_csu 12471  SubGrpcsubg 14930  SubRingcsubrg 15856  ℂfldccnfld 16695  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  rngunsnply  27346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-grp 14804  df-subg 14933  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-subrg 15858  df-cnfld 16696  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
  Copyright terms: Public domain W3C validator