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Theorem plyssc 26239
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 4400 . . 3 ∅ ⊆ (Poly‘ℂ)
2 sseq1 4009 . . 3 ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ)))
31, 2mpbiri 258 . 2 ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ))
4 n0 4353 . . 3 ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆))
5 plybss 26233 . . . . 5 (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
6 ssid 4006 . . . . 5 ℂ ⊆ ℂ
7 plyss 26238 . . . . 5 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘ℂ))
85, 6, 7sylancl 586 . . . 4 (𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ))
98exlimiv 1930 . . 3 (∃𝑓 𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ))
104, 9sylbi 217 . 2 ((Poly‘𝑆) ≠ ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ))
113, 10pm2.61ine 3025 1 (Poly‘𝑆) ⊆ (Poly‘ℂ)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  wne 2940  wss 3951  c0 4333  cfv 6561  cc 11153  Polycply 26223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-nn 12267  df-n0 12527  df-ply 26227
This theorem is referenced by:  plyaddcl  26259  plymulcl  26260  plysubcl  26261  coeval  26262  coeeu  26264  dgrval  26267  coef3  26271  coeidlem  26276  coemulc  26294  coesub  26296  dgrmulc  26311  dgrsub  26312  dgrcolem1  26313  dgrcolem2  26314  dgrco  26315  coecj  26318  coecjOLD  26320  dvply2  26328  dvnply  26330  quotval  26334  quotlem  26342  quotcl2  26344  quotdgr  26345  plyrem  26347  facth  26348  fta1  26350  quotcan  26351  vieta1lem1  26352  vieta1  26354  plyexmo  26355  ftalem7  27122  dgrsub2  43147
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