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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 11254 | . . . 4 ⊢ (⊤ → 0 ∈ ℂ) | |
| 2 | ssid 4006 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
| 3 | ply0 26247 | . . . . 5 ⊢ (ℂ ⊆ ℂ → 0𝑝 ∈ (Poly‘ℂ)) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 0𝑝 ∈ (Poly‘ℂ) |
| 5 | coemulc 26294 | . . . 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 11236 | . . . . . . 7 ⊢ ℂ ∈ V | |
| 8 | 7 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℂ ∈ V) |
| 9 | plyf 26237 | . . . . . . 7 ⊢ (0𝑝 ∈ (Poly‘ℂ) → 0𝑝:ℂ⟶ℂ) | |
| 10 | 4, 9 | mp1i 13 | . . . . . 6 ⊢ (⊤ → 0𝑝:ℂ⟶ℂ) |
| 11 | mul02 11439 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0) | |
| 12 | 11 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0) |
| 13 | 8, 10, 1, 1, 12 | caofid2 7733 | . . . . 5 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = (ℂ × {0})) |
| 14 | df-0p 25705 | . . . . 5 ⊢ 0𝑝 = (ℂ × {0}) | |
| 15 | 13, 14 | eqtr4di 2795 | . . . 4 ⊢ (⊤ → ((ℂ × {0}) ∘f · 0𝑝) = 0𝑝) |
| 16 | 15 | fveq2d 6910 | . . 3 ⊢ (⊤ → (coeff‘((ℂ × {0}) ∘f · 0𝑝)) = (coeff‘0𝑝)) |
| 17 | nn0ex 12532 | . . . . 5 ⊢ ℕ0 ∈ V | |
| 18 | 17 | a1i 11 | . . . 4 ⊢ (⊤ → ℕ0 ∈ V) |
| 19 | eqid 2737 | . . . . . 6 ⊢ (coeff‘0𝑝) = (coeff‘0𝑝) | |
| 20 | 19 | coef3 26271 | . . . . 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 7733 | . . 3 ⊢ (⊤ → ((ℕ0 × {0}) ∘f · (coeff‘0𝑝)) = (ℕ0 × {0})) |
| 23 | 6, 16, 22 | 3eqtr3d 2785 | . 2 ⊢ (⊤ → (coeff‘0𝑝) = (ℕ0 × {0})) |
| 24 | 23 | mptru 1547 | 1 ⊢ (coeff‘0𝑝) = (ℕ0 × {0}) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 {csn 4626 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ℂcc 11153 0cc0 11155 · cmul 11160 ℕ0cn0 12526 0𝑝c0p 25704 Polycply 26223 coeffccoe 26225 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-0p 25705 df-ply 26227 df-coe 26229 df-dgr 26230 |
| This theorem is referenced by: dgreq0 26305 dgrlt 26306 plymul0or 26322 plydivlem4 26338 plymulx 34563 mncn0 43151 aaitgo 43174 n0p 45050 elaa2 46249 aacllem 49320 |
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