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Mirrors > Home > MPE Home > Th. List > deg1ldgdomn | Structured version Visualization version GIF version |
Description: A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
deg1z.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1z.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1z.z | ⊢ 0 = (0g‘𝑃) |
deg1nn0cl.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1ldgdomn.e | ⊢ 𝐸 = (RLReg‘𝑅) |
deg1ldgdomn.a | ⊢ 𝐴 = (coe1‘𝐹) |
Ref | Expression |
---|---|
deg1ldgdomn | ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1138 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝑅 ∈ Domn) | |
2 | deg1ldgdomn.a | . . . . 5 ⊢ 𝐴 = (coe1‘𝐹) | |
3 | deg1nn0cl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
4 | deg1z.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | coe1f 21132 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑅)) |
7 | 6 | 3ad2ant2 1136 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐴:ℕ0⟶(Base‘𝑅)) |
8 | domnring 20334 | . . . 4 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
9 | deg1z.d | . . . . 5 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
10 | deg1z.z | . . . . 5 ⊢ 0 = (0g‘𝑃) | |
11 | 9, 4, 10, 3 | deg1nn0cl 24986 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
12 | 8, 11 | syl3an1 1165 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
13 | 7, 12 | ffvelrnd 6905 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ (Base‘𝑅)) |
14 | eqid 2737 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 9, 4, 10, 3, 14, 2 | deg1ldg 24990 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ (0g‘𝑅)) |
16 | 8, 15 | syl3an1 1165 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ (0g‘𝑅)) |
17 | deg1ldgdomn.e | . . 3 ⊢ 𝐸 = (RLReg‘𝑅) | |
18 | 5, 17, 14 | domnrrg 20338 | . 2 ⊢ ((𝑅 ∈ Domn ∧ (𝐴‘(𝐷‘𝐹)) ∈ (Base‘𝑅) ∧ (𝐴‘(𝐷‘𝐹)) ≠ (0g‘𝑅)) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) |
19 | 1, 13, 16, 18 | syl3anc 1373 | 1 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ⟶wf 6376 ‘cfv 6380 ℕ0cn0 12090 Basecbs 16760 0gc0g 16944 Ringcrg 19562 RLRegcrlreg 20317 Domncdomn 20318 Poly1cpl1 21098 coe1cco1 21099 deg1 cdg1 24949 |
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 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 ax-mulf 10809 |
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 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-gsum 16947 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-mulg 18489 df-subg 18540 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-nzr 20296 df-rlreg 20321 df-domn 20322 df-cnfld 20364 df-psr 20868 df-mpl 20870 df-opsr 20872 df-psr1 21101 df-ply1 21103 df-coe1 21104 df-mdeg 24950 df-deg1 24951 |
This theorem is referenced by: ply1domn 25021 deg1mhm 40735 |
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