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Mirrors > Home > MPE Home > Th. List > dgrub2 | Structured version Visualization version GIF version |
Description: All the coefficients above the degree of 𝐹 are zero. (Contributed by Mario Carneiro, 23-Jul-2014.) |
Ref | Expression |
---|---|
dgrub.1 | ⊢ 𝐴 = (coeff‘𝐹) |
dgrub.2 | ⊢ 𝑁 = (deg‘𝐹) |
Ref | Expression |
---|---|
dgrub2 | ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dgrub.1 | . . . . 5 ⊢ 𝐴 = (coeff‘𝐹) | |
2 | dgrub.2 | . . . . 5 ⊢ 𝑁 = (deg‘𝐹) | |
3 | 1, 2 | dgrub 25391 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑁) |
4 | 3 | 3expia 1120 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
5 | 4 | ralrimiva 3110 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
6 | dgrcl 25390 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
7 | 2, 6 | eqeltrid 2845 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0) |
8 | 1 | coef3 25389 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
9 | plyco0 25349 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
10 | 7, 8, 9 | syl2anc 584 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
11 | 5, 10 | mpbird 256 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 {csn 4567 class class class wbr 5079 “ cima 5592 ⟶wf 6427 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 0cc0 10870 1c1 10871 + caddc 10873 ≤ cle 11009 ℕ0cn0 12231 ℤ≥cuz 12579 Polycply 25341 coeffccoe 25343 degcdgr 25344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-pm 8599 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12580 df-rp 12728 df-fz 13237 df-fzo 13380 df-fl 13508 df-seq 13718 df-exp 13779 df-hash 14041 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-clim 15193 df-rlim 15194 df-sum 15394 df-0p 24830 df-ply 25345 df-coe 25347 df-dgr 25348 |
This theorem is referenced by: coeaddlem 25406 coemullem 25407 coecj 25435 |
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