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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 3624 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | plyconst 23962 | . . . 4 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ)) | |
3 | 1, 2 | mpan 706 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ)) |
4 | 0nn0 11307 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℕ0) |
6 | simpl 473 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ (0...0)) → 𝐴 ∈ ℂ) | |
7 | 0z 11388 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | exp0 12864 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (𝑧↑0) = 1) | |
9 | 8 | oveq2d 6666 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (𝐴 · (𝑧↑0)) = (𝐴 · 1)) |
10 | mulid1 10037 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
11 | 9, 10 | sylan9eqr 2678 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) = 𝐴) |
12 | simpl 473 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝐴 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2701 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) ∈ ℂ) |
14 | oveq2 6658 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑧↑𝑘) = (𝑧↑0)) | |
15 | 14 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝑘 = 0 → (𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
16 | 15 | fsum1 14476 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ (𝐴 · (𝑧↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
17 | 7, 13, 16 | sylancr 695 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
18 | 17, 11 | eqtrd 2656 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = 𝐴) |
19 | 18 | mpteq2dva 4744 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ 𝐴)) |
20 | fconstmpt 5163 | . . . 4 ⊢ (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ 𝐴) | |
21 | 19, 20 | syl6reqr 2675 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)))) |
22 | 3, 5, 6, 21 | dgrle 23999 | . 2 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) ≤ 0) |
23 | dgrcl 23989 | . . 3 ⊢ ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → (deg‘(ℂ × {𝐴})) ∈ ℕ0) | |
24 | nn0le0eq0 11321 | . . 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 222 | 1 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 ≤ cle 10075 ℕ0cn0 11292 ℤcz 11377 ...cfz 12326 ↑cexp 12860 Σcsu 14416 Polycply 23940 degcdgr 23943 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-0p 23437 df-ply 23944 df-coe 23946 df-dgr 23947 |
This theorem is referenced by: 0dgrb 24002 coemulc 24011 dgr0 24018 dgrmulc 24027 dgrcolem2 24030 plyremlem 24059 vieta1lem2 24066 |
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