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Mirrors > Home > MPE Home > Th. List > coe0 | Structured version Visualization version GIF version |
Description: The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
coe0 | ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cnd 10826 | . . . 4 ⊢ (⊤ → 0 ∈ ℂ) | |
2 | ssid 3923 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
3 | ply0 25102 | . . . . 5 ⊢ (ℂ ⊆ ℂ → 0𝑝 ∈ (Poly‘ℂ)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 0𝑝 ∈ (Poly‘ℂ) |
5 | coemulc 25149 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0𝑝 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = ((ℕ0 × {0}) ∘f · (coeff‘0𝑝))) | |
6 | 1, 4, 5 | sylancl 589 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = ((ℕ0 × {0}) ∘f · (coeff‘0𝑝))) |
7 | cnex 10810 | . . . . . . 7 ⊢ ℂ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℂ ∈ V) |
9 | plyf 25092 | . . . . . . 7 ⊢ (0𝑝 ∈ (Poly‘ℂ) → 0𝑝:ℂ⟶ℂ) | |
10 | 4, 9 | mp1i 13 | . . . . . 6 ⊢ (⊤ → 0𝑝:ℂ⟶ℂ) |
11 | mul02 11010 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0) | |
12 | 11 | adantl 485 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
13 | 8, 10, 1, 1, 12 | caofid2 7502 | . . . . 5 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = (ℂ × {0})) |
14 | df-0p 24567 | . . . . 5 ⊢ 0𝑝 = (ℂ × {0}) | |
15 | 13, 14 | eqtr4di 2796 | . . . 4 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = 0𝑝) |
16 | 15 | fveq2d 6721 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = (coeff‘0𝑝)) |
17 | nn0ex 12096 | . . . . 5 ⊢ ℕ0 ∈ V | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
19 | eqid 2737 | . . . . . 6 ⊢ (coeff‘0𝑝) = (coeff‘0𝑝) | |
20 | 19 | coef3 25126 | . . . . 5 ⊢ (0𝑝 ∈ (Poly‘ℂ) → (coeff‘0𝑝):ℕ0⟶ℂ) |
21 | 4, 20 | mp1i 13 | . . . 4 ⊢ (⊤ → (coeff‘0𝑝):ℕ0⟶ℂ) |
22 | 18, 21, 1, 1, 12 | caofid2 7502 | . . 3 ⊢ (⊤ → ((ℕ0 × {0}) ∘f · (coeff‘0𝑝)) = (ℕ0 × {0})) |
23 | 6, 16, 22 | 3eqtr3d 2785 | . 2 ⊢ (⊤ → (coeff‘0𝑝) = (ℕ0 × {0})) |
24 | 23 | mptru 1550 | 1 ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 {csn 4541 × cxp 5549 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ∘f cof 7467 ℂcc 10727 0cc0 10729 · cmul 10734 ℕ0cn0 12090 0𝑝c0p 24566 Polycply 25078 coeffccoe 25080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-0p 24567 df-ply 25082 df-coe 25084 df-dgr 25085 |
This theorem is referenced by: dgreq0 25159 dgrlt 25160 plymul0or 25174 plydivlem4 25189 plymulx 32239 mncn0 40670 aaitgo 40693 n0p 42267 elaa2 43453 aacllem 46179 |
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