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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 11251 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
2 | snssi 4813 | . . . . . . 7 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ) | |
3 | 1, 2 | ax-mp 5 | . . . . . 6 ⊢ {0} ⊆ ℂ |
4 | 3 | biantru 529 | . . . . 5 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ)) |
5 | unss 4200 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ {0} ⊆ ℂ) ↔ (𝑆 ∪ {0}) ⊆ ℂ) | |
6 | 4, 5 | bitr2i 276 | . . . 4 ⊢ ((𝑆 ∪ {0}) ⊆ ℂ ↔ 𝑆 ⊆ ℂ) |
7 | unass 4182 | . . . . . . . 8 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ ({0} ∪ {0})) | |
8 | unidm 4167 | . . . . . . . . 9 ⊢ ({0} ∪ {0}) = {0} | |
9 | 8 | uneq2i 4175 | . . . . . . . 8 ⊢ (𝑆 ∪ ({0} ∪ {0})) = (𝑆 ∪ {0}) |
10 | 7, 9 | eqtri 2763 | . . . . . . 7 ⊢ ((𝑆 ∪ {0}) ∪ {0}) = (𝑆 ∪ {0}) |
11 | 10 | oveq1i 7441 | . . . . . 6 ⊢ (((𝑆 ∪ {0}) ∪ {0}) ↑m ℕ0) = ((𝑆 ∪ {0}) ↑m ℕ0) |
12 | 11 | rexeqi 3323 | . . . . 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 3092 | . . . 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 628 | . . 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 26249 | . . 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 26249 | . . 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 303 | . 2 ⊢ (𝑓 ∈ (Poly‘(𝑆 ∪ {0})) ↔ 𝑓 ∈ (Poly‘𝑆)) |
18 | 17 | eqriv 2732 | 1 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ∪ cun 3961 ⊆ wss 3963 {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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-n0 12525 df-ply 26242 |
This theorem is referenced by: elplyd 26256 ply1term 26258 ply0 26262 plyaddlem 26269 plymullem 26270 plyco 26295 plycj 26332 plycjOLD 26334 |
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