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Theorem cnsrplycl 35747
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 18913 . . . . . . 7  |-  CC  =  ( Base ` fld )
43subrgss 17948 . . . . . 6  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
52, 4syl 17 . . . . 5  |-  ( ph  ->  S  C_  CC )
6 plyss 23029 . . . . 5  |-  ( ( C  C_  S  /\  S  C_  CC )  -> 
(Poly `  C )  C_  (Poly `  S )
)
71, 5, 6syl2anc 665 . . . 4  |-  ( ph  ->  (Poly `  C )  C_  (Poly `  S )
)
8 cnsrplycl.p . . . 4  |-  ( ph  ->  P  e.  (Poly `  C ) )
97, 8sseldd 3471 . . 3  |-  ( ph  ->  P  e.  (Poly `  S ) )
10 cnsrplycl.x . . . 4  |-  ( ph  ->  X  e.  S )
115, 10sseldd 3471 . . 3  |-  ( ph  ->  X  e.  CC )
12 eqid 2429 . . . 4  |-  (coeff `  P )  =  (coeff `  P )
13 eqid 2429 . . . 4  |-  (deg `  P )  =  (deg
`  P )
1412, 13coeid2 23069 . . 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 665 . 2  |-  ( ph  ->  ( P `  X
)  =  sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) ) )
16 fzfid 12183 . . 3  |-  ( ph  ->  ( 0 ... (deg `  P ) )  e. 
Fin )
172adantr 466 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  S  e.  (SubRing ` fld ) )
18 subrgsubg 17953 . . . . . . . 8  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
19 cnfld0 18931 . . . . . . . . 9  |-  0  =  ( 0g ` fld )
2019subg0cl 16780 . . . . . . . 8  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
212, 18, 203syl 18 . . . . . . 7  |-  ( ph  ->  0  e.  S )
2212coef2 23061 . . . . . . 7  |-  ( ( P  e.  (Poly `  S )  /\  0  e.  S )  ->  (coeff `  P ) : NN0 --> S )
239, 21, 22syl2anc 665 . . . . . 6  |-  ( ph  ->  (coeff `  P ) : NN0 --> S )
2423adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  (coeff `  P
) : NN0 --> S )
25 elfznn0 11885 . . . . . 6  |-  ( k  e.  ( 0 ... (deg `  P )
)  ->  k  e.  NN0 )
2625adantl 467 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  k  e.  NN0 )
2724, 26ffvelrnd 6038 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( (coeff `  P ) `  k
)  e.  S )
2810adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  X  e.  S
)
2917, 28, 26cnsrexpcl 35745 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( X ^
k )  e.  S
)
30 cnfldmul 18915 . . . . 5  |-  x.  =  ( .r ` fld )
3130subrgmcl 17959 . . . 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 1264 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( ( (coeff `  P ) `  k
)  x.  ( X ^ k ) )  e.  S )
332, 16, 32fsumcnsrcl 35746 . 2  |-  ( ph  -> 
sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) )  e.  S
)
3415, 33eqeltrd 2517 1  |-  ( ph  ->  ( P `  X
)  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    C_ wss 3442   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538    x. cmul 9543   NN0cn0 10869   ...cfz 11782   ^cexp 12269   sum_csu 13730  SubGrpcsubg 16766  SubRingcsubrg 17943  ℂfldccnfld 18909  Polycply 23014  coeffccoe 23016  degcdgr 23017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15166  df-mulr 15167  df-starv 15168  df-tset 15172  df-ple 15173  df-ds 15175  df-unif 15176  df-0g 15303  df-mgm 16443  df-sgrp 16482  df-mnd 16492  df-grp 16628  df-subg 16769  df-cmn 17371  df-mgp 17663  df-ur 17675  df-ring 17721  df-cring 17722  df-subrg 17945  df-cnfld 18910  df-0p 22513  df-ply 23018  df-coe 23020  df-dgr 23021
This theorem is referenced by:  rngunsnply  35753
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