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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 4400 | . . 3 ⊢ ∅ ⊆ (Poly‘ℂ) | |
| 2 | sseq1 4009 | . . 3 ⊢ ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ))) | |
| 3 | 1, 2 | mpbiri 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)) | |
| 8 | 5, 6, 7 | sylancl 586 | . . . 4 ⊢ (𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
| 9 | 8 | exlimiv 1930 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (Poly‘𝑆) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
| 10 | 4, 9 | sylbi 217 | . 2 ⊢ ((Poly‘𝑆) ≠ ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
| 11 | 3, 10 | pm2.61ine 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 ℂcc 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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