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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 4353 | . . 3 ⊢ ∅ ⊆ (Poly‘ℂ) | |
2 | sseq1 3995 | . . 3 ⊢ ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ))) | |
3 | 1, 2 | mpbiri 260 | . 2 ⊢ ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
4 | n0 4313 | . . 3 ⊢ ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆)) | |
5 | plybss 24787 | . . . . 5 ⊢ (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
6 | ssid 3992 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
7 | plyss 24792 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘ℂ)) | |
8 | 5, 6, 7 | sylancl 588 | . . . 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 219 | . 2 ⊢ ((Poly‘𝑆) ≠ ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
11 | 3, 10 | pm2.61ine 3103 | 1 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 ∅c0 4294 ‘cfv 6358 ℂcc 10538 Polycply 24777 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-1cn 10598 ax-addcl 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-map 8411 df-nn 11642 df-n0 11901 df-ply 24781 |
This theorem is referenced by: plyaddcl 24813 plymulcl 24814 plysubcl 24815 coeval 24816 coeeu 24818 dgrval 24821 coef3 24825 coeidlem 24830 coemulc 24848 coesub 24850 dgrmulc 24864 dgrsub 24865 dgrcolem1 24866 dgrcolem2 24867 dgrco 24868 coecj 24871 dvply2 24878 dvnply 24880 quotval 24884 quotlem 24892 quotcl2 24894 quotdgr 24895 plyrem 24897 facth 24898 fta1 24900 quotcan 24901 vieta1lem1 24902 vieta1 24904 plyexmo 24905 ftalem7 25659 dgrsub2 39741 |
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