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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 2737 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | deg1le0.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
3 | 2 | deg1fval 24983 | . . 3 ⊢ 𝐷 = (1o mDeg 𝑅) |
4 | 1on 8214 | . . . 4 ⊢ 1o ∈ On | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 1o ∈ On) |
6 | simpl 486 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝑅 ∈ Ring) | |
7 | deg1le0.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
8 | eqid 2737 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
9 | deg1le0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
10 | 7, 8, 9 | ply1bas 21121 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
11 | deg1le0.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
12 | 7, 11 | ply1ascl 21184 | . . 3 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
13 | simpr 488 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ 𝐵) | |
14 | 1, 3, 5, 6, 10, 12, 13 | mdegle0 24980 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(1o × {0}))))) |
15 | 0nn0 12110 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2737 | . . . . . 6 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
17 | 16 | coe1fv 21132 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝐹)‘0) = (𝐹‘(1o × {0}))) |
18 | 13, 15, 17 | sylancl 589 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((coe1‘𝐹)‘0) = (𝐹‘(1o × {0}))) |
19 | 18 | fveq2d 6726 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐴‘((coe1‘𝐹)‘0)) = (𝐴‘(𝐹‘(1o × {0})))) |
20 | 19 | eqeq2d 2748 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 = (𝐴‘((coe1‘𝐹)‘0)) ↔ 𝐹 = (𝐴‘(𝐹‘(1o × {0}))))) |
21 | 14, 20 | bitr4d 285 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {csn 4546 class class class wbr 5058 × cxp 5554 Oncon0 6218 ‘cfv 6385 (class class class)co 7218 1oc1o 8200 0cc0 10734 ≤ cle 10873 ℕ0cn0 12095 Basecbs 16765 Ringcrg 19567 algSccascl 20819 mPoly cmpl 20870 PwSer1cps1 21101 Poly1cpl1 21103 coe1cco1 21104 deg1 cdg1 24954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 ax-pre-sup 10812 ax-addf 10813 ax-mulf 10814 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-int 4865 df-iun 4911 df-iin 4912 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-se 5515 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-isom 6394 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-of 7474 df-ofr 7475 df-om 7650 df-1st 7766 df-2nd 7767 df-supp 7909 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-er 8396 df-map 8515 df-pm 8516 df-ixp 8584 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-fsupp 8991 df-sup 9063 df-oi 9131 df-card 9560 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-7 11903 df-8 11904 df-9 11905 df-n0 12096 df-z 12182 df-dec 12299 df-uz 12444 df-fz 13101 df-fzo 13244 df-seq 13580 df-hash 13902 df-struct 16705 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-starv 16822 df-sca 16823 df-vsca 16824 df-tset 16826 df-ple 16827 df-ds 16829 df-unif 16830 df-0g 16951 df-gsum 16952 df-mre 17094 df-mrc 17095 df-acs 17097 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-mhm 18223 df-submnd 18224 df-grp 18373 df-minusg 18374 df-mulg 18494 df-subg 18545 df-ghm 18625 df-cntz 18716 df-cmn 19177 df-abl 19178 df-mgp 19510 df-ur 19522 df-ring 19569 df-cring 19570 df-subrg 19803 df-cnfld 20369 df-ascl 20822 df-psr 20873 df-mpl 20875 df-opsr 20877 df-psr1 21106 df-ply1 21108 df-coe1 21109 df-mdeg 24955 df-deg1 24956 |
This theorem is referenced by: deg1sclle 25015 ply1rem 25066 fta1g 25070 |
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