Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23748 | . . . 4 ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))}) | |
2 | 1 | dmmptss 5548 | . . 3 ⊢ dom Poly ⊆ 𝒫 ℂ |
3 | elfvdm 6130 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ dom Poly) | |
4 | 2, 3 | sseldi 3566 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ) |
5 | 4 | elpwid 4118 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℂcc 9813 0cc0 9815 · cmul 9820 ℕ0cn0 11169 ...cfz 12197 ↑cexp 12722 Σcsu 14264 Polycply 23744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 df-ply 23748 |
This theorem is referenced by: elply 23755 plyf 23758 plyssc 23760 plyaddlem 23775 plymullem 23776 plysub 23779 dgrlem 23789 coeidlem 23797 plyco 23801 plycj 23837 plyreres 23842 plydivlem3 23854 plydivlem4 23855 elmnc 36725 |
Copyright terms: Public domain | W3C validator |