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Mirrors > Home > MPE Home > Th. List > plyun0 | Structured version Visualization version GIF version |
Description: The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
plyun0 | ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 9911 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
2 | snssi 4280 | . . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ {0} ⊆ ℂ |
4 | 3 | biantru 525 | . . . . 5 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ)) |
5 | unss 3749 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ) | |
6 | 4, 5 | bitr2i 264 | . . . 4 ⊢ ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ) |
7 | unass 3732 | . . . . . . . 8 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0})) | |
8 | unidm 3718 | . . . . . . . . 9 ⊢ ({0} ∪ {0}) = {0} | |
9 | 8 | uneq2i 3726 | . . . . . . . 8 ⊢ (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0}) |
10 | 7, 9 | eqtri 2632 | . . . . . . 7 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0}) |
11 | 10 | oveq1i 6559 | . . . . . 6 ⊢ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚 ℕ0) = ((𝑆 ∪ {0}) ↑𝑚 ℕ0) |
12 | 11 | rexeqi 3120 | . . . . 5 ⊢ (∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) |
13 | 12 | rexbii 3023 | . . . 4 ⊢ (∃𝑛 ∈ ℕ0 ∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))) ↔ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) |
14 | 6, 13 | anbi12i 729 | . . 3 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) |
15 | elply 23755 | . . 3 ⊢ (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ ((𝑆 ∪ {0}) ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ (((𝑆 ∪ {0}) ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) | |
16 | elply 23755 | . . 3 ⊢ (𝑓 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) | |
17 | 14, 15, 16 | 3bitr4i 291 | . 2 ⊢ (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆)) |
18 | 17 | eqriv 2607 | 1 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∪ cun 3538 ⊆ wss 3540 {csn 4125 ↦ cmpt 4643 ‘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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-n0 11170 df-ply 23748 |
This theorem is referenced by: elplyd 23762 ply1term 23764 ply0 23768 plyaddlem 23775 plymullem 23776 plyco 23801 plycj 23837 |
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