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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 4406 | . . 3 ⊢ ∅ ⊆ (Poly‘ℂ) | |
2 | sseq1 4021 | . . 3 ⊢ ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ))) | |
3 | 1, 2 | mpbiri 258 | . 2 ⊢ ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
4 | n0 4359 | . . 3 ⊢ ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆)) | |
5 | plybss 26248 | . . . . 5 ⊢ (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
6 | ssid 4018 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
7 | plyss 26253 | . . . . 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 1928 | . . 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 3023 | 1 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 ℂcc 11151 Polycply 26238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-map 8867 df-nn 12265 df-n0 12525 df-ply 26242 |
This theorem is referenced by: plyaddcl 26274 plymulcl 26275 plysubcl 26276 coeval 26277 coeeu 26279 dgrval 26282 coef3 26286 coeidlem 26291 coemulc 26309 coesub 26311 dgrmulc 26326 dgrsub 26327 dgrcolem1 26328 dgrcolem2 26329 dgrco 26330 coecj 26333 coecjOLD 26335 dvply2 26343 dvnply 26345 quotval 26349 quotlem 26357 quotcl2 26359 quotdgr 26360 plyrem 26362 facth 26363 fta1 26365 quotcan 26366 vieta1lem1 26367 vieta1 26369 plyexmo 26370 ftalem7 27137 dgrsub2 43124 |
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