![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > deg1nn0cl | Structured version Visualization version GIF version |
Description: Degree of a nonzero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
deg1z.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1z.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1z.z | ⊢ 0 = (0g‘𝑃) |
deg1nn0cl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
deg1nn0cl | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1z.d | . . 3 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | 1 | deg1fval 24178 | . 2 ⊢ 𝐷 = (1𝑜 mDeg 𝑅) |
3 | eqid 2797 | . 2 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
4 | deg1z.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | deg1z.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
6 | 3, 4, 5 | ply1mpl0 19944 | . 2 ⊢ 0 = (0g‘(1𝑜 mPoly 𝑅)) |
7 | eqid 2797 | . . 3 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
8 | deg1nn0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
9 | 4, 7, 8 | ply1bas 19884 | . 2 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅)) |
10 | 2, 3, 6, 9 | mdegnn0cl 24169 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ‘cfv 6099 (class class class)co 6876 1𝑜c1o 7790 ℕ0cn0 11576 Basecbs 16181 0gc0g 16412 Ringcrg 18860 mPoly cmpl 19673 PwSer1cps1 19864 Poly1cpl1 19866 deg1 cdg1 24152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-sup 8588 df-oi 8655 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-fzo 12717 df-seq 13052 df-hash 13367 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-0g 16414 df-gsum 16415 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-grp 17738 df-minusg 17739 df-subg 17901 df-cntz 18059 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-cring 18863 df-psr 19676 df-mpl 19678 df-opsr 19680 df-psr1 19869 df-ply1 19871 df-cnfld 20066 df-mdeg 24153 df-deg1 24154 |
This theorem is referenced by: deg1n0ima 24187 deg1nn0clb 24188 deg1lt0 24189 deg1ldg 24190 deg1ldgdomn 24192 coe1mul4 24198 deg1add 24201 deg1scl 24211 deg1mul2 24212 ply1domn 24221 ply1divmo 24233 ply1divex 24234 uc1pdeg 24245 deg1submon1p 24250 fta1glem1 24263 fta1g 24265 drnguc1p 24268 mon1psubm 38557 deg1mhm 38558 |
Copyright terms: Public domain | W3C validator |