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Mirrors > Home > MPE Home > Th. List > 0dgr | Structured version Visualization version GIF version |
Description: A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
0dgr | ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3988 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | plyconst 24795 | . . . 4 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ)) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ)) |
4 | 0nn0 11911 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℕ0) |
6 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ (0...0)) → 𝐴 ∈ ℂ) | |
7 | 0z 11991 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | exp0 13432 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (𝑧↑0) = 1) | |
9 | 8 | oveq2d 7171 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (𝐴 · (𝑧↑0)) = (𝐴 · 1)) |
10 | mulid1 10638 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
11 | 9, 10 | sylan9eqr 2878 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) = 𝐴) |
12 | simpl 485 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝐴 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2913 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) ∈ ℂ) |
14 | oveq2 7163 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑧↑𝑘) = (𝑧↑0)) | |
15 | 14 | oveq2d 7171 | . . . . . . . 8 ⊢ (𝑘 = 0 → (𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
16 | 15 | fsum1 15101 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ (𝐴 · (𝑧↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
17 | 7, 13, 16 | sylancr 589 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
18 | 17, 11 | eqtrd 2856 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = 𝐴) |
19 | 18 | mpteq2dva 5160 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ 𝐴)) |
20 | fconstmpt 5613 | . . . 4 ⊢ (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ 𝐴) | |
21 | 19, 20 | syl6reqr 2875 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)))) |
22 | 3, 5, 6, 21 | dgrle 24832 | . 2 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) ≤ 0) |
23 | dgrcl 24822 | . . 3 ⊢ ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → (deg‘(ℂ × {𝐴})) ∈ ℕ0) | |
24 | nn0le0eq0 11924 | . . 3 ⊢ ((deg‘(ℂ × {𝐴})) ∈ ℕ0 → ((deg‘(ℂ × {𝐴})) ≤ 0 ↔ (deg‘(ℂ × {𝐴})) = 0)) | |
25 | 3, 23, 24 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ℂ → ((deg‘(ℂ × {𝐴})) ≤ 0 ↔ (deg‘(ℂ × {𝐴})) = 0)) |
26 | 22, 25 | mpbid 234 | 1 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4566 class class class wbr 5065 ↦ cmpt 5145 × cxp 5552 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 · cmul 10541 ≤ cle 10675 ℕ0cn0 11896 ℤcz 11980 ...cfz 12891 ↑cexp 13428 Σcsu 15041 Polycply 24773 degcdgr 24776 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845 df-sum 15042 df-0p 24270 df-ply 24777 df-coe 24779 df-dgr 24780 |
This theorem is referenced by: 0dgrb 24835 coemulc 24844 dgr0 24851 dgrmulc 24860 dgrcolem2 24863 plyremlem 24892 vieta1lem2 24899 |
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