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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 25373 | . 2 ⊢ 𝐷 = (1o mDeg 𝑅) |
3 | eqid 2738 | . 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 21554 | . 2 ⊢ 0 = (0g‘(1o mPoly 𝑅)) |
7 | eqid 2738 | . . 3 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
8 | deg1nn0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
9 | 4, 7, 8 | ply1bas 21494 | . 2 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
10 | 2, 3, 6, 9 | mdegnn0cl 25364 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ‘cfv 6492 (class class class)co 7350 1oc1o 8373 ℕ0cn0 12347 Basecbs 17019 0gc0g 17257 Ringcrg 19894 mPoly cmpl 21237 PwSer1cps1 21474 Poly1cpl1 21476 deg1 cdg1 25344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-sup 9312 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-fz 13355 df-fzo 13498 df-seq 13837 df-hash 14160 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-starv 17084 df-sca 17085 df-vsca 17086 df-tset 17088 df-ple 17089 df-ds 17091 df-unif 17092 df-0g 17259 df-gsum 17260 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-grp 18687 df-minusg 18688 df-subg 18860 df-cntz 19032 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-ring 19896 df-cring 19897 df-cnfld 20726 df-psr 21240 df-mpl 21242 df-opsr 21244 df-psr1 21479 df-ply1 21481 df-mdeg 25345 df-deg1 25346 |
This theorem is referenced by: deg1n0ima 25382 deg1nn0clb 25383 deg1lt0 25384 deg1ldg 25385 deg1ldgdomn 25387 coe1mul4 25393 deg1add 25396 deg1scl 25406 deg1mul2 25407 ply1domn 25416 ply1divmo 25428 ply1divex 25429 uc1pdeg 25440 deg1submon1p 25445 fta1glem1 25458 fta1g 25460 drnguc1p 25463 minplyeulem 32153 mon1psubm 41435 deg1mhm 41436 |
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