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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 23944 | . . . 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 5631 | . . 3 ⊢ dom Poly ⊆ 𝒫 ℂ |
3 | elfvdm 6220 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ dom Poly) | |
4 | 2, 3 | sseldi 3601 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ) |
5 | 4 | elpwid 4170 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℂcc 9934 0cc0 9936 · cmul 9941 ℕ0cn0 11292 ...cfz 12326 ↑cexp 12860 Σcsu 14416 Polycply 23940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ply 23944 |
This theorem is referenced by: elply 23951 plyf 23954 plyssc 23956 plyaddlem 23971 plymullem 23972 plysub 23975 dgrlem 23985 coeidlem 23993 plyco 23997 plycj 24033 plyreres 24038 plydivlem3 24050 plydivlem4 24051 elmnc 37706 |
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