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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 2733 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | deg1le0.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | 2 | deg1fval 25273 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
4 | 1on 8329 | . . . 4 ⊢ 1o ∈ On | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 1o ∈ On) |
6 | simpl 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝑅 ∈ Ring) | |
7 | deg1le0.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
8 | eqid 2733 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
9 | deg1le0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
10 | 7, 8, 9 | ply1bas 21394 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
11 | deg1le0.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
12 | 7, 11 | ply1ascl 21457 | . . 3 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
13 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
14 | 1, 3, 5, 6, 10, 12, 13 | mdegle0 25270 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(1o × {0}))))) |
15 | 0nn0 12276 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2733 | . . . . . 6 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
17 | 16 | coe1fv 21405 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝐹)‘0) = (𝐹‘(1o × {0}))) |
18 | 13, 15, 17 | sylancl 585 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((coe1‘𝐹)‘0) = (𝐹‘(1o × {0}))) |
19 | 18 | fveq2d 6796 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐴‘((coe1‘𝐹)‘0)) = (𝐴‘(𝐹‘(1o × {0})))) |
20 | 19 | eqeq2d 2744 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 = (𝐴‘((coe1‘𝐹)‘0)) ↔ 𝐹 = (𝐴‘(𝐹‘(1o × {0}))))) |
21 | 14, 20 | bitr4d 281 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1537 ∈ wcel 2101 {csn 4564 class class class wbr 5077 × cxp 5589 Oncon0 6270 ‘cfv 6447 (class class class)co 7295 1oc1o 8310 0cc0 10899 ≤ cle 11038 ℕ0cn0 12261 Basecbs 16940 Ringcrg 19811 algSccascl 21087 mPoly cmpl 21137 PwSer1cps1 21374 Poly1cpl1 21376 coe1cco1 21377 deg1 cdg1 25244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 ax-addf 10978 ax-mulf 10979 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-ofr 7554 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-pm 8638 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-sup 9229 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-fzo 13411 df-seq 13750 df-hash 14073 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-sca 17006 df-vsca 17007 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-0g 17180 df-gsum 17181 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-submnd 18459 df-grp 18608 df-minusg 18609 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cntz 18951 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-ring 19813 df-cring 19814 df-subrg 20050 df-cnfld 20626 df-ascl 21090 df-psr 21140 df-mpl 21142 df-opsr 21144 df-psr1 21379 df-ply1 21381 df-coe1 21382 df-mdeg 25245 df-deg1 25246 |
This theorem is referenced by: deg1sclle 25305 ply1rem 25356 fta1g 25360 |
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