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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 10899 | . . . 4 ⊢ (⊤ → 0 ∈ ℂ) | |
2 | ssid 3939 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
3 | ply0 25274 | . . . . 5 ⊢ (ℂ ⊆ ℂ → 0𝑝 ∈ (Poly‘ℂ)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 0𝑝 ∈ (Poly‘ℂ) |
5 | coemulc 25321 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0𝑝 ∈ (Poly‘ℂ)) → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = ((ℕ0 × {0}) ∘f · (coeff‘0𝑝))) | |
6 | 1, 4, 5 | sylancl 585 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = ((ℕ0 × {0}) ∘f · (coeff‘0𝑝))) |
7 | cnex 10883 | . . . . . . 7 ⊢ ℂ ∈ V | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℂ ∈ V) |
9 | plyf 25264 | . . . . . . 7 ⊢ (0𝑝 ∈ (Poly‘ℂ) → 0𝑝:ℂ⟶ℂ) | |
10 | 4, 9 | mp1i 13 | . . . . . 6 ⊢ (⊤ → 0𝑝:ℂ⟶ℂ) |
11 | mul02 11083 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0) | |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
13 | 8, 10, 1, 1, 12 | caofid2 7545 | . . . . 5 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = (ℂ × {0})) |
14 | df-0p 24739 | . . . . 5 ⊢ 0𝑝 = (ℂ × {0}) | |
15 | 13, 14 | eqtr4di 2797 | . . . 4 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = 0𝑝) |
16 | 15 | fveq2d 6760 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = (coeff‘0𝑝)) |
17 | nn0ex 12169 | . . . . 5 ⊢ ℕ0 ∈ V | |
18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
19 | eqid 2738 | . . . . . 6 ⊢ (coeff‘0𝑝) = (coeff‘0𝑝) | |
20 | 19 | coef3 25298 | . . . . 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 7545 | . . 3 ⊢ (⊤ → ((ℕ0 × {0}) ∘f · (coeff‘0𝑝)) = (ℕ0 × {0})) |
23 | 6, 16, 22 | 3eqtr3d 2786 | . 2 ⊢ (⊤ → (coeff‘0𝑝) = (ℕ0 × {0})) |
24 | 23 | mptru 1546 | 1 ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 Vcvv 3422 ⊆ wss 3883 {csn 4558 × cxp 5578 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘f cof 7509 ℂcc 10800 0cc0 10802 · cmul 10807 ℕ0cn0 12163 0𝑝c0p 24738 Polycply 25250 coeffccoe 25252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-0p 24739 df-ply 25254 df-coe 25256 df-dgr 25257 |
This theorem is referenced by: dgreq0 25331 dgrlt 25332 plymul0or 25346 plydivlem4 25361 plymulx 32427 mncn0 40880 aaitgo 40903 n0p 42480 elaa2 43665 aacllem 46391 |
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