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| 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 26228 | . . 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 7024 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ) | 
| 3 | 2 | elpwid 4608 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 ∪ cun 3948 ⊆ wss 3950 𝒫 cpw 4599 {csn 4625 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 ℂcc 11154 0cc0 11156 · cmul 11161 ℕ0cn0 12528 ...cfz 13548 ↑cexp 14103 Σcsu 15723 Polycply 26224 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fv 6568 df-ply 26228 | 
| This theorem is referenced by: elply 26235 plyf 26238 plyssc 26240 plyaddlem 26255 plymullem 26256 plysub 26259 dgrlem 26269 coeidlem 26277 plyco 26281 plycj 26318 plycjOLD 26320 plyreres 26325 plydivlem3 26338 plydivlem4 26339 elmnc 43153 | 
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