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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 4350 | . . 3 ⊢ ∅ ⊆ (Poly‘ℂ) | |
2 | sseq1 3992 | . . 3 ⊢ ((Poly‘𝑆) = ∅ → ((Poly‘𝑆) ⊆ (Poly‘ℂ) ↔ ∅ ⊆ (Poly‘ℂ))) | |
3 | 1, 2 | mpbiri 260 | . 2 ⊢ ((Poly‘𝑆) = ∅ → (Poly‘𝑆) ⊆ (Poly‘ℂ)) |
4 | n0 4310 | . . 3 ⊢ ((Poly‘𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (Poly‘𝑆)) | |
5 | plybss 24784 | . . . . 5 ⊢ (𝑓 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
6 | ssid 3989 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
7 | plyss 24789 | . . . . 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 1931 | . . 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 3100 | 1 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 ℂcc 10535 Polycply 24774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-map 8408 df-nn 11639 df-n0 11899 df-ply 24778 |
This theorem is referenced by: plyaddcl 24810 plymulcl 24811 plysubcl 24812 coeval 24813 coeeu 24815 dgrval 24818 coef3 24822 coeidlem 24827 coemulc 24845 coesub 24847 dgrmulc 24861 dgrsub 24862 dgrcolem1 24863 dgrcolem2 24864 dgrco 24865 coecj 24868 dvply2 24875 dvnply 24877 quotval 24881 quotlem 24889 quotcl2 24891 quotdgr 24892 plyrem 24894 facth 24895 fta1 24897 quotcan 24898 vieta1lem1 24899 vieta1 24901 plyexmo 24902 ftalem7 25656 dgrsub2 39755 |
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