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Theorem plyssc 19987
Description: Every polynomial ring is contained in the ring of polynomials over  CC. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plyssc  |-  (Poly `  S )  C_  (Poly `  CC )

Proof of Theorem plyssc
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 0ss 3600 . . 3  |-  (/)  C_  (Poly `  CC )
2 sseq1 3313 . . 3  |-  ( (Poly `  S )  =  (/)  ->  ( (Poly `  S
)  C_  (Poly `  CC ) 
<->  (/)  C_  (Poly `  CC ) ) )
31, 2mpbiri 225 . 2  |-  ( (Poly `  S )  =  (/)  ->  (Poly `  S )  C_  (Poly `  CC )
)
4 n0 3581 . . 3  |-  ( (Poly `  S )  =/=  (/)  <->  E. f 
f  e.  (Poly `  S ) )
5 plybss 19981 . . . . 5  |-  ( f  e.  (Poly `  S
)  ->  S  C_  CC )
6 ssid 3311 . . . . 5  |-  CC  C_  CC
7 plyss 19986 . . . . 5  |-  ( ( S  C_  CC  /\  CC  C_  CC )  ->  (Poly `  S )  C_  (Poly `  CC ) )
85, 6, 7sylancl 644 . . . 4  |-  ( f  e.  (Poly `  S
)  ->  (Poly `  S
)  C_  (Poly `  CC ) )
98exlimiv 1641 . . 3  |-  ( E. f  f  e.  (Poly `  S )  ->  (Poly `  S )  C_  (Poly `  CC ) )
104, 9sylbi 188 . 2  |-  ( (Poly `  S )  =/=  (/)  ->  (Poly `  S )  C_  (Poly `  CC ) )
113, 10pm2.61ine 2627 1  |-  (Poly `  S )  C_  (Poly `  CC )
Colors of variables: wff set class
Syntax hints:   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551    C_ wss 3264   (/)c0 3572   ` cfv 5395   CCcc 8922  Polycply 19971
This theorem is referenced by:  plyaddcl  20007  plymulcl  20008  plysubcl  20009  coeval  20010  coeeu  20012  dgrval  20015  coef3  20019  coeidlem  20024  coemulc  20041  coesub  20043  dgrmulc  20057  dgrsub  20058  dgrcolem1  20059  dgrcolem2  20060  dgrco  20061  coecj  20064  dvply2  20071  dvnply  20073  quotval  20077  quotlem  20085  quotcl2  20087  quotdgr  20088  plyrem  20090  facth  20091  fta1  20093  quotcan  20094  vieta1lem1  20095  vieta1  20097  plyexmo  20098  ftalem7  20729  dgrsub2  27009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-i2m1 8992  ax-1ne0 8993  ax-rrecex 8996  ax-cnre 8997
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-map 6957  df-nn 9934  df-n0 10155  df-ply 19975
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