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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 10622 | . . . 4 ⊢ (⊤ → 0 ∈ ℂ) | |
2 | ssid 3986 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
3 | ply0 24725 | . . . . 5 ⊢ (ℂ ⊆ ℂ → 0𝑝 ∈ (Poly‘ℂ)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 0𝑝 ∈ (Poly‘ℂ) |
5 | coemulc 24772 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0𝑝 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = ((ℕ0 × {0}) ∘f · (coeff‘0𝑝))) | |
6 | 1, 4, 5 | sylancl 586 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = ((ℕ0 × {0}) ∘f · (coeff‘0𝑝))) |
7 | cnex 10606 | . . . . . . 7 ⊢ ℂ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℂ ∈ V) |
9 | plyf 24715 | . . . . . . 7 ⊢ (0𝑝 ∈ (Poly‘ℂ) → 0𝑝:ℂ⟶ℂ) | |
10 | 4, 9 | mp1i 13 | . . . . . 6 ⊢ (⊤ → 0𝑝:ℂ⟶ℂ) |
11 | mul02 10806 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0) | |
12 | 11 | adantl 482 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
13 | 8, 10, 1, 1, 12 | caofid2 7429 | . . . . 5 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = (ℂ × {0})) |
14 | df-0p 24198 | . . . . 5 ⊢ 0𝑝 = (ℂ × {0}) | |
15 | 13, 14 | syl6eqr 2871 | . . . 4 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = 0𝑝) |
16 | 15 | fveq2d 6667 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = (coeff‘0𝑝)) |
17 | nn0ex 11891 | . . . . 5 ⊢ ℕ0 ∈ V | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
19 | eqid 2818 | . . . . . 6 ⊢ (coeff‘0𝑝) = (coeff‘0𝑝) | |
20 | 19 | coef3 24749 | . . . . 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 7429 | . . 3 ⊢ (⊤ → ((ℕ0 × {0}) ∘f · (coeff‘0𝑝)) = (ℕ0 × {0})) |
23 | 6, 16, 22 | 3eqtr3d 2861 | . 2 ⊢ (⊤ → (coeff‘0𝑝) = (ℕ0 × {0})) |
24 | 23 | mptru 1535 | 1 ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 ℂcc 10523 0cc0 10525 · cmul 10530 ℕ0cn0 11885 0𝑝c0p 24197 Polycply 24701 coeffccoe 24703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-0p 24198 df-ply 24705 df-coe 24707 df-dgr 24708 |
This theorem is referenced by: dgreq0 24782 dgrlt 24783 plymul0or 24797 plydivlem4 24812 plymulx 31717 mncn0 39617 aaitgo 39640 n0p 41182 elaa2 42396 aacllem 44830 |
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