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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 25082 | . . 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 6827 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ) |
3 | 2 | elpwid 4524 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 {cab 2714 ∃wrex 3062 ∪ cun 3864 ⊆ wss 3866 𝒫 cpw 4513 {csn 4541 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 ℂcc 10727 0cc0 10729 · cmul 10734 ℕ0cn0 12090 ...cfz 13095 ↑cexp 13635 Σcsu 15249 Polycply 25078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fv 6388 df-ply 25082 |
This theorem is referenced by: elply 25089 plyf 25092 plyssc 25094 plyaddlem 25109 plymullem 25110 plysub 25113 dgrlem 25123 coeidlem 25131 plyco 25135 plycj 25171 plyreres 25176 plydivlem3 25188 plydivlem4 25189 elmnc 40664 |
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