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Mirrors > Home > MPE Home > Th. List > deg1le0 | Structured version Visualization version GIF version |
Description: A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
deg1le0.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1le0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1le0.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1le0.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
deg1le0 | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2777 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | deg1le0.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | 2 | deg1fval 24277 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
4 | 1on 7850 | . . . 4 ⊢ 1o ∈ On | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 1o ∈ On) |
6 | simpl 476 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝑅 ∈ Ring) | |
7 | deg1le0.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
8 | eqid 2777 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
9 | deg1le0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
10 | 7, 8, 9 | ply1bas 19961 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
11 | deg1le0.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
12 | 7, 11 | ply1ascl 20024 | . . 3 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
13 | simpr 479 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
14 | 1, 3, 5, 6, 10, 12, 13 | mdegle0 24274 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(1o × {0}))))) |
15 | 0nn0 11659 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2777 | . . . . . 6 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
17 | 16 | coe1fv 19972 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝐹)‘0) = (𝐹‘(1o × {0}))) |
18 | 13, 15, 17 | sylancl 580 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((coe1‘𝐹)‘0) = (𝐹‘(1o × {0}))) |
19 | 18 | fveq2d 6450 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐴‘((coe1‘𝐹)‘0)) = (𝐴‘(𝐹‘(1o × {0})))) |
20 | 19 | eqeq2d 2787 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 = (𝐴‘((coe1‘𝐹)‘0)) ↔ 𝐹 = (𝐴‘(𝐹‘(1o × {0}))))) |
21 | 14, 20 | bitr4d 274 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {csn 4397 class class class wbr 4886 × cxp 5353 Oncon0 5976 ‘cfv 6135 (class class class)co 6922 1oc1o 7836 0cc0 10272 ≤ cle 10412 ℕ0cn0 11642 Basecbs 16255 Ringcrg 18934 algSccascl 19708 mPoly cmpl 19750 PwSer1cps1 19941 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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 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-ofr 7175 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-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 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-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-subrg 19170 df-ascl 19711 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: deg1sclle 24309 ply1rem 24360 fta1g 24364 |
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