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Theorem cnsrplycl 30749
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 18223 . . . . . . 7  |-  CC  =  ( Base ` fld )
43subrgss 17230 . . . . . 6  |-  ( S  e.  (SubRing ` fld )  ->  S  C_  CC )
52, 4syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
6 plyss 22359 . . . . 5  |-  ( ( C  C_  S  /\  S  C_  CC )  -> 
(Poly `  C )  C_  (Poly `  S )
)
71, 5, 6syl2anc 661 . . . 4  |-  ( ph  ->  (Poly `  C )  C_  (Poly `  S )
)
8 cnsrplycl.p . . . 4  |-  ( ph  ->  P  e.  (Poly `  C ) )
97, 8sseldd 3505 . . 3  |-  ( ph  ->  P  e.  (Poly `  S ) )
10 cnsrplycl.x . . . 4  |-  ( ph  ->  X  e.  S )
115, 10sseldd 3505 . . 3  |-  ( ph  ->  X  e.  CC )
12 eqid 2467 . . . 4  |-  (coeff `  P )  =  (coeff `  P )
13 eqid 2467 . . . 4  |-  (deg `  P )  =  (deg
`  P )
1412, 13coeid2 22399 . . 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 661 . 2  |-  ( ph  ->  ( P `  X
)  =  sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) ) )
16 fzfid 12051 . . 3  |-  ( ph  ->  ( 0 ... (deg `  P ) )  e. 
Fin )
172adantr 465 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  S  e.  (SubRing ` fld ) )
18 subrgsubg 17235 . . . . . . . 8  |-  ( S  e.  (SubRing ` fld )  ->  S  e.  (SubGrp ` fld ) )
19 cnfld0 18241 . . . . . . . . 9  |-  0  =  ( 0g ` fld )
2019subg0cl 16014 . . . . . . . 8  |-  ( S  e.  (SubGrp ` fld )  ->  0  e.  S )
212, 18, 203syl 20 . . . . . . 7  |-  ( ph  ->  0  e.  S )
2212coef2 22391 . . . . . . 7  |-  ( ( P  e.  (Poly `  S )  /\  0  e.  S )  ->  (coeff `  P ) : NN0 --> S )
239, 21, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  (coeff `  P ) : NN0 --> S )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  (coeff `  P
) : NN0 --> S )
25 elfznn0 11770 . . . . . 6  |-  ( k  e.  ( 0 ... (deg `  P )
)  ->  k  e.  NN0 )
2625adantl 466 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  k  e.  NN0 )
2724, 26ffvelrnd 6022 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( (coeff `  P ) `  k
)  e.  S )
2810adantr 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  X  e.  S
)
2917, 28, 26cnsrexpcl 30747 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( X ^
k )  e.  S
)
30 cnfldmul 18225 . . . . 5  |-  x.  =  ( .r ` fld )
3130subrgmcl 17241 . . . 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 1228 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... (deg `  P ) ) )  ->  ( ( (coeff `  P ) `  k
)  x.  ( X ^ k ) )  e.  S )
332, 16, 32fsumcnsrcl 30748 . 2  |-  ( ph  -> 
sum_ k  e.  ( 0 ... (deg `  P ) ) ( ( (coeff `  P
) `  k )  x.  ( X ^ k
) )  e.  S
)
3415, 33eqeltrd 2555 1  |-  ( ph  ->  ( P `  X
)  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492    x. cmul 9497   NN0cn0 10795   ...cfz 11672   ^cexp 12134   sum_csu 13471  SubGrpcsubg 16000  SubRingcsubrg 17225  ℂfldccnfld 18219  Polycply 22344  coeffccoe 22346  degcdgr 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-0g 14697  df-mnd 15732  df-grp 15867  df-subg 16003  df-cmn 16606  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-subrg 17227  df-cnfld 18220  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351
This theorem is referenced by:  rngunsnply  30755
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