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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 4304 | . . 3 ⊢ ∅ ⊆ (Poly‘ℂ) | |
2 | sseq1 3940 | . . 3 ⊢ ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ))) | |
3 | 1, 2 | mpbiri 261 | . 2 ⊢ ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
4 | n0 4260 | . . 3 ⊢ ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆)) | |
5 | plybss 24791 | . . . . 5 ⊢ (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
6 | ssid 3937 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
7 | plyss 24796 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) | |
8 | 5, 6, 7 | sylancl 589 | . . . 4 ⊢ (𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
9 | 8 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
10 | 4, 9 | sylbi 220 | . 2 ⊢ ((Poly‘𝑆) ≠ ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
11 | 3, 10 | pm2.61ine 3070 | 1 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ‘cfv 6324 ℂcc 10524 Polycply 24781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-nn 11626 df-n0 11886 df-ply 24785 |
This theorem is referenced by: plyaddcl 24817 plymulcl 24818 plysubcl 24819 coeval 24820 coeeu 24822 dgrval 24825 coef3 24829 coeidlem 24834 coemulc 24852 coesub 24854 dgrmulc 24868 dgrsub 24869 dgrcolem1 24870 dgrcolem2 24871 dgrco 24872 coecj 24875 dvply2 24882 dvnply 24884 quotval 24888 quotlem 24896 quotcl2 24898 quotdgr 24899 plyrem 24901 facth 24902 fta1 24904 quotcan 24905 vieta1lem1 24906 vieta1 24908 plyexmo 24909 ftalem7 25664 dgrsub2 40079 |
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