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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 10432 | . . . 4 ⊢ (⊤ → 0 ∈ ℂ) | |
2 | ssid 3880 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
3 | ply0 24501 | . . . . 5 ⊢ (ℂ ⊆ ℂ → 0𝑝 ∈ (Poly‘ℂ)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 0𝑝 ∈ (Poly‘ℂ) |
5 | coemulc 24548 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0𝑝 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {0}) ∘𝑓 · 0𝑝)) = ((ℕ0 × {0}) ∘𝑓 · (coeff‘0𝑝))) | |
6 | 1, 4, 5 | sylancl 577 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘𝑓 · 0𝑝)) = ((ℕ0 × {0}) ∘𝑓 · (coeff‘0𝑝))) |
7 | cnex 10416 | . . . . . . 7 ⊢ ℂ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℂ ∈ V) |
9 | plyf 24491 | . . . . . . 7 ⊢ (0𝑝 ∈ (Poly‘ℂ) → 0𝑝:ℂ⟶ℂ) | |
10 | 4, 9 | mp1i 13 | . . . . . 6 ⊢ (⊤ → 0𝑝:ℂ⟶ℂ) |
11 | mul02 10618 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0) | |
12 | 11 | adantl 474 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
13 | 8, 10, 1, 1, 12 | caofid2 7258 | . . . . 5 ⊢ (⊤ → ((ℂ × {0}) ∘𝑓 · 0𝑝) = (ℂ × {0})) |
14 | df-0p 23974 | . . . . 5 ⊢ 0𝑝 = (ℂ × {0}) | |
15 | 13, 14 | syl6eqr 2833 | . . . 4 ⊢ (⊤ → ((ℂ × {0}) ∘𝑓 · 0𝑝) = 0𝑝) |
16 | 15 | fveq2d 6503 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘𝑓 · 0𝑝)) = (coeff‘0𝑝)) |
17 | nn0ex 11714 | . . . . 5 ⊢ ℕ0 ∈ V | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
19 | eqid 2779 | . . . . . 6 ⊢ (coeff‘0𝑝) = (coeff‘0𝑝) | |
20 | 19 | coef3 24525 | . . . . 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 7258 | . . 3 ⊢ (⊤ → ((ℕ0 × {0}) ∘𝑓 · (coeff‘0𝑝)) = (ℕ0 × {0})) |
23 | 6, 16, 22 | 3eqtr3d 2823 | . 2 ⊢ (⊤ → (coeff‘0𝑝) = (ℕ0 × {0})) |
24 | 23 | mptru 1514 | 1 ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 Vcvv 3416 ⊆ wss 3830 {csn 4441 × cxp 5405 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∘𝑓 cof 7225 ℂcc 10333 0cc0 10335 · cmul 10340 ℕ0cn0 11707 0𝑝c0p 23973 Polycply 24477 coeffccoe 24479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fz 12709 df-fzo 12850 df-fl 12977 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-rlim 14707 df-sum 14904 df-0p 23974 df-ply 24481 df-coe 24483 df-dgr 24484 |
This theorem is referenced by: dgreq0 24558 dgrlt 24559 plymul0or 24573 plydivlem4 24588 plymulx 31461 mncn0 39132 aaitgo 39155 n0p 40721 elaa2 41948 aacllem 44267 |
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