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Mirrors > Home > MPE Home > Th. List > deg1ge | Structured version Visualization version GIF version |
Description: Conversely, a nonzero coefficient sets a lower bound on the degree. (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 |
---|---|
deg1ge | ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1leb.d | . . . . . 6 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | deg1leb.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | deg1leb.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
4 | 1, 2, 3 | deg1xrcl 25942 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) ∈ ℝ*) |
5 | nn0re 12479 | . . . . . 6 ⊢ (𝐺 ∈ ℕ0 → 𝐺 ∈ ℝ) | |
6 | 5 | rexrd 11262 | . . . . 5 ⊢ (𝐺 ∈ ℕ0 → 𝐺 ∈ ℝ*) |
7 | xrltnle 11279 | . . . . 5 ⊢ (((𝐷‘𝐹) ∈ ℝ* ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) < 𝐺 ↔ ¬ 𝐺 ≤ (𝐷‘𝐹))) | |
8 | 4, 6, 7 | syl2an 595 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0) → ((𝐷‘𝐹) < 𝐺 ↔ ¬ 𝐺 ≤ (𝐷‘𝐹))) |
9 | deg1leb.y | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
10 | deg1leb.a | . . . . . 6 ⊢ 𝐴 = (coe1‘𝐹) | |
11 | 1, 2, 3, 9, 10 | deg1lt 25957 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 ) |
12 | 11 | 3expia 1118 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0) → ((𝐷‘𝐹) < 𝐺 → (𝐴‘𝐺) = 0 )) |
13 | 8, 12 | sylbird 260 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0) → (¬ 𝐺 ≤ (𝐷‘𝐹) → (𝐴‘𝐺) = 0 )) |
14 | 13 | necon1ad 2949 | . 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0) → ((𝐴‘𝐺) ≠ 0 → 𝐺 ≤ (𝐷‘𝐹))) |
15 | 14 | 3impia 1114 | 1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5139 ‘cfv 6534 ℝ*cxr 11245 < clt 11246 ≤ cle 11247 ℕ0cn0 12470 Basecbs 17145 0gc0g 17386 Poly1cpl1 22021 coe1cco1 22022 deg1 cdg1 25911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-fzo 13626 df-seq 13965 df-hash 14289 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-sca 17214 df-vsca 17215 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-0g 17388 df-gsum 17389 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-mulg 18988 df-cntz 19225 df-cmn 19694 df-abl 19695 df-mgp 20032 df-ur 20079 df-ring 20132 df-cring 20133 df-cnfld 21231 df-psr 21773 df-mpl 21775 df-opsr 21777 df-psr1 22024 df-ply1 22026 df-coe1 22027 df-mdeg 25912 df-deg1 25913 |
This theorem is referenced by: deg1add 25963 deg1mul2 25974 deg1tm 25978 plypf1 26068 |
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