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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 26242 | . . 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 7025 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4614 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 ∪ cun 3961 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 0cc0 11153 · cmul 11158 ℕ0cn0 12524 ...cfz 13544 ↑cexp 14099 Σcsu 15719 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-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 df-ply 26242 |
This theorem is referenced by: elply 26249 plyf 26252 plyssc 26254 plyaddlem 26269 plymullem 26270 plysub 26273 dgrlem 26283 coeidlem 26291 plyco 26295 plycj 26332 plycjOLD 26334 plyreres 26339 plydivlem3 26352 plydivlem4 26353 elmnc 43125 |
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