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Mirrors > Home > MPE Home > Th. List > plybss | Structured version Visualization version GIF version |
Description: Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
plybss | ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ply 24778 | . . 3 ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))}) | |
2 | 1 | mptrcl 6777 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4550 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 ∪ cun 3934 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℂcc 10535 0cc0 10537 · cmul 10542 ℕ0cn0 11898 ...cfz 12893 ↑cexp 13430 Σcsu 15042 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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 df-ply 24778 |
This theorem is referenced by: elply 24785 plyf 24788 plyssc 24790 plyaddlem 24805 plymullem 24806 plysub 24809 dgrlem 24819 coeidlem 24827 plyco 24831 plycj 24867 plyreres 24872 plydivlem3 24884 plydivlem4 24885 elmnc 39785 |
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