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Mirrors > Home > MPE Home > Th. List > plyssc | Structured version Visualization version GIF version |
Description: Every polynomial ring is contained in the ring of polynomials over ℂ. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
plyssc | ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4005 | . . 3 ⊢ ∅ ⊆ (Poly‘ℂ) | |
2 | sseq1 3659 | . . 3 ⊢ ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ))) | |
3 | 1, 2 | mpbiri 248 | . 2 ⊢ ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
4 | n0 3964 | . . 3 ⊢ ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆)) | |
5 | plybss 23995 | . . . . 5 ⊢ (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
6 | ssid 3657 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
7 | plyss 24000 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) | |
8 | 5, 6, 7 | sylancl 695 | . . . 4 ⊢ (𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
9 | 8 | exlimiv 1898 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
10 | 4, 9 | sylbi 207 | . 2 ⊢ ((Poly‘𝑆) ≠ ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
11 | 3, 10 | pm2.61ine 2906 | 1 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∃wex 1744 ∈ wcel 2030 ≠ wne 2823 ⊆ wss 3607 ∅c0 3948 ‘cfv 5926 ℂcc 9972 Polycply 23985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-map 7901 df-nn 11059 df-n0 11331 df-ply 23989 |
This theorem is referenced by: plyaddcl 24021 plymulcl 24022 plysubcl 24023 coeval 24024 coeeu 24026 dgrval 24029 coef3 24033 coeidlem 24038 coemulc 24056 coesub 24058 dgrmulc 24072 dgrsub 24073 dgrcolem1 24074 dgrcolem2 24075 dgrco 24076 coecj 24079 dvply2 24086 dvnply 24088 quotval 24092 quotlem 24100 quotcl2 24102 quotdgr 24103 plyrem 24105 facth 24106 fta1 24108 quotcan 24109 vieta1lem1 24110 vieta1 24112 plyexmo 24113 ftalem7 24850 dgrsub2 38022 |
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