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Theorem plyssc 22723
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 3823 . . 3  |-  (/)  C_  (Poly `  CC )
2 sseq1 3520 . . 3  |-  ( (Poly `  S )  =  (/)  ->  ( (Poly `  S
)  C_  (Poly `  CC ) 
<->  (/)  C_  (Poly `  CC ) ) )
31, 2mpbiri 233 . 2  |-  ( (Poly `  S )  =  (/)  ->  (Poly `  S )  C_  (Poly `  CC )
)
4 n0 3803 . . 3  |-  ( (Poly `  S )  =/=  (/)  <->  E. f 
f  e.  (Poly `  S ) )
5 plybss 22717 . . . . 5  |-  ( f  e.  (Poly `  S
)  ->  S  C_  CC )
6 ssid 3518 . . . . 5  |-  CC  C_  CC
7 plyss 22722 . . . . 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 1723 . . 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 2770 1  |-  (Poly `  S )  C_  (Poly `  CC )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652    C_ wss 3471   (/)c0 3793   ` cfv 5594   CCcc 9507  Polycply 22707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-map 7440  df-nn 10557  df-n0 10817  df-ply 22711
This theorem is referenced by:  plyaddcl  22743  plymulcl  22744  plysubcl  22745  coeval  22746  coeeu  22748  dgrval  22751  coef3  22755  coeidlem  22760  coemulc  22778  coesub  22780  dgrmulc  22794  dgrsub  22795  dgrcolem1  22796  dgrcolem2  22797  dgrco  22798  coecj  22801  dvply2  22808  dvnply  22810  quotval  22814  quotlem  22822  quotcl2  22824  quotdgr  22825  plyrem  22827  facth  22828  fta1  22830  quotcan  22831  vieta1lem1  22832  vieta1  22834  plyexmo  22835  ftalem7  23478  dgrsub2  31288
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