![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > deg1lt | Structured version Visualization version GIF version |
Description: If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
deg1leb.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1leb.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1leb.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1leb.y | ⊢ 0 = (0g‘𝑅) |
deg1leb.a | ⊢ 𝐴 = (coe1‘𝐹) |
Ref | Expression |
---|---|
deg1lt | ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1129 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐷‘𝐹) < 𝐺) | |
2 | breq2 4890 | . . . 4 ⊢ (𝑥 = 𝐺 → ((𝐷‘𝐹) < 𝑥 ↔ (𝐷‘𝐹) < 𝐺)) | |
3 | fveqeq2 6455 | . . . 4 ⊢ (𝑥 = 𝐺 → ((𝐴‘𝑥) = 0 ↔ (𝐴‘𝐺) = 0 )) | |
4 | 2, 3 | imbi12d 336 | . . 3 ⊢ (𝑥 = 𝐺 → (((𝐷‘𝐹) < 𝑥 → (𝐴‘𝑥) = 0 ) ↔ ((𝐷‘𝐹) < 𝐺 → (𝐴‘𝐺) = 0 ))) |
5 | deg1leb.d | . . . . . . 7 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
6 | deg1leb.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | deg1leb.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑃) | |
8 | 5, 6, 7 | deg1xrcl 24279 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
9 | 8 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐷‘𝐹) ∈ ℝ*) |
10 | 9 | xrleidd 12295 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
11 | simp1 1127 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → 𝐹 ∈ 𝐵) | |
12 | deg1leb.y | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
13 | deg1leb.a | . . . . . 6 ⊢ 𝐴 = (coe1‘𝐹) | |
14 | 5, 6, 7, 12, 13 | deg1leb 24292 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ∈ ℝ*) → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ ℕ0 ((𝐷‘𝐹) < 𝑥 → (𝐴‘𝑥) = 0 ))) |
15 | 11, 9, 14 | syl2anc 579 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ ℕ0 ((𝐷‘𝐹) < 𝑥 → (𝐴‘𝑥) = 0 ))) |
16 | 10, 15 | mpbid 224 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → ∀𝑥 ∈ ℕ0 ((𝐷‘𝐹) < 𝑥 → (𝐴‘𝑥) = 0 )) |
17 | simp2 1128 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → 𝐺 ∈ ℕ0) | |
18 | 4, 16, 17 | rspcdva 3517 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → ((𝐷‘𝐹) < 𝐺 → (𝐴‘𝐺) = 0 )) |
19 | 1, 18 | mpd 15 | 1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 class class class wbr 4886 ‘cfv 6135 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 ℕ0cn0 11642 Basecbs 16255 0gc0g 16486 Poly1cpl1 19943 coe1cco1 19944 deg1 cdg1 24251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-gsum 16489 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-psr 19753 df-mpl 19755 df-opsr 19757 df-psr1 19946 df-ply1 19948 df-coe1 19949 df-cnfld 20143 df-mdeg 24252 df-deg1 24253 |
This theorem is referenced by: deg1ge 24295 coe1mul3 24296 deg1add 24300 |
Copyright terms: Public domain | W3C validator |