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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 1129 | . 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 2795 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | 2, 3, 4, 5 | coe1f 20062 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴:ℕ0⟶(Base‘𝑅)) |
7 | 6 | 3ad2ant2 1127 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐴:ℕ0⟶(Base‘𝑅)) |
8 | domnring 19758 | . . . 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 24365 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
12 | 8, 11 | syl3an1 1156 | . . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
13 | 7, 12 | ffvelrnd 6717 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ (Base‘𝑅)) |
14 | eqid 2795 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | 9, 4, 10, 3, 14, 2 | deg1ldg 24369 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ (0g‘𝑅)) |
16 | 8, 15 | syl3an1 1156 | . 2 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ (0g‘𝑅)) |
17 | deg1ldgdomn.e | . . 3 ⊢ 𝐸 = (RLReg‘𝑅) | |
18 | 5, 17, 14 | domnrrg 19762 | . 2 ⊢ ((𝑅 ∈ Domn ∧ (𝐴‘(𝐷‘𝐹)) ∈ (Base‘𝑅) ∧ (𝐴‘(𝐷‘𝐹)) ≠ (0g‘𝑅)) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) |
19 | 1, 13, 16, 18 | syl3anc 1364 | 1 ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ⟶wf 6221 ‘cfv 6225 ℕ0cn0 11745 Basecbs 16312 0gc0g 16542 Ringcrg 18987 RLRegcrlreg 19741 Domncdomn 19742 Poly1cpl1 20028 coe1cco1 20029 deg1 cdg1 24331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-sup 8752 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-fz 12743 df-fzo 12884 df-seq 13220 df-hash 13541 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-gsum 16545 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-mulg 17982 df-subg 18030 df-cntz 18188 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-nzr 19720 df-rlreg 19745 df-domn 19746 df-psr 19824 df-mpl 19826 df-opsr 19828 df-psr1 20031 df-ply1 20033 df-coe1 20034 df-cnfld 20228 df-mdeg 24332 df-deg1 24333 |
This theorem is referenced by: ply1domn 24400 deg1mhm 39292 |
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