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Theorem plyssc 24705
Description: Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plyssc (Poly‘𝑆) ⊆ (Poly‘ℂ)

Proof of Theorem plyssc
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 0ss 4354 . . 3 ∅ ⊆ (Poly‘ℂ)
2 sseq1 3996 . . 3 ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ)))
31, 2mpbiri 259 . 2 ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ))
4 n0 4314 . . 3 ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆))
5 plybss 24699 . . . . 5 (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
6 ssid 3993 . . . . 5 ℂ ⊆ ℂ
7 plyss 24704 . . . . 5 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘ℂ))
85, 6, 7sylancl 586 . . . 4 (𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ))
98exlimiv 1924 . . 3 (∃𝑓 𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ))
104, 9sylbi 218 . 2 ((Poly‘𝑆) ≠ ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ))
113, 10pm2.61ine 3105 1 (Poly‘𝑆) ⊆ (Poly‘ℂ)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wex 1773  wcel 2107  wne 3021  wss 3940  c0 4295  cfv 6352  cc 10524  Polycply 24689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-1cn 10584  ax-addcl 10586
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-map 8398  df-nn 11628  df-n0 11887  df-ply 24693
This theorem is referenced by:  plyaddcl  24725  plymulcl  24726  plysubcl  24727  coeval  24728  coeeu  24730  dgrval  24733  coef3  24737  coeidlem  24742  coemulc  24760  coesub  24762  dgrmulc  24776  dgrsub  24777  dgrcolem1  24778  dgrcolem2  24779  dgrco  24780  coecj  24783  dvply2  24790  dvnply  24792  quotval  24796  quotlem  24804  quotcl2  24806  quotdgr  24807  plyrem  24809  facth  24810  fta1  24812  quotcan  24813  vieta1lem1  24814  vieta1  24816  plyexmo  24817  ftalem7  25570  dgrsub2  39600
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