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Theorem plyssc 22325
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 3807 . . 3  |-  (/)  C_  (Poly `  CC )
2 sseq1 3518 . . 3  |-  ( (Poly `  S )  =  (/)  ->  ( (Poly `  S
)  C_  (Poly `  CC ) 
<->  (/)  C_  (Poly `  CC ) ) )
31, 2mpbiri 233 . 2  |-  ( (Poly `  S )  =  (/)  ->  (Poly `  S )  C_  (Poly `  CC )
)
4 n0 3787 . . 3  |-  ( (Poly `  S )  =/=  (/)  <->  E. f 
f  e.  (Poly `  S ) )
5 plybss 22319 . . . . 5  |-  ( f  e.  (Poly `  S
)  ->  S  C_  CC )
6 ssid 3516 . . . . 5  |-  CC  C_  CC
7 plyss 22324 . . . . 5  |-  ( ( S  C_  CC  /\  CC  C_  CC )  ->  (Poly `  S )  C_  (Poly `  CC ) )
85, 6, 7sylancl 662 . . . 4  |-  ( f  e.  (Poly `  S
)  ->  (Poly `  S
)  C_  (Poly `  CC ) )
98exlimiv 1693 . . 3  |-  ( E. f  f  e.  (Poly `  S )  ->  (Poly `  S )  C_  (Poly `  CC ) )
104, 9sylbi 195 . 2  |-  ( (Poly `  S )  =/=  (/)  ->  (Poly `  S )  C_  (Poly `  CC ) )
113, 10pm2.61ine 2773 1  |-  (Poly `  S )  C_  (Poly `  CC )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2655    C_ wss 3469   (/)c0 3778   ` cfv 5579   CCcc 9479  Polycply 22309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-map 7412  df-nn 10526  df-n0 10785  df-ply 22313
This theorem is referenced by:  plyaddcl  22345  plymulcl  22346  plysubcl  22347  coeval  22348  coeeu  22350  dgrval  22353  coef3  22357  coeidlem  22362  coemulc  22379  coesub  22381  dgrmulc  22395  dgrsub  22396  dgrcolem1  22397  dgrcolem2  22398  dgrco  22399  coecj  22402  dvply2  22409  dvnply  22411  quotval  22415  quotlem  22423  quotcl2  22425  quotdgr  22426  plyrem  22428  facth  22429  fta1  22431  quotcan  22432  vieta1lem1  22433  vieta1  22435  plyexmo  22436  ftalem7  23073  dgrsub2  30541
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