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Mirrors > Home > MPE Home > Th. List > deg1scl | Structured version Visualization version GIF version |
Description: A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
deg1sclle.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1sclle.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1sclle.k | ⊢ 𝐾 = (Base‘𝑅) |
deg1sclle.a | ⊢ 𝐴 = (algSc‘𝑃) |
deg1scl.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
deg1scl | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1sclle.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
2 | deg1sclle.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | deg1sclle.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | deg1sclle.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑃) | |
5 | 1, 2, 3, 4 | deg1sclle 25612 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐷‘(𝐴‘𝐹)) ≤ 0) |
6 | 5 | 3adant3 1133 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) ≤ 0) |
7 | simp1 1137 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → 𝑅 ∈ Ring) | |
8 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
9 | 2, 4, 3, 8 | ply1sclcl 21790 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐴‘𝐹) ∈ (Base‘𝑃)) |
10 | 9 | 3adant3 1133 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐴‘𝐹) ∈ (Base‘𝑃)) |
11 | deg1scl.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
12 | eqid 2733 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
13 | 2, 4, 11, 12, 3 | ply1scln0 21796 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐴‘𝐹) ≠ (0g‘𝑃)) |
14 | 1, 2, 12, 8 | deg1nn0cl 25588 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴‘𝐹) ∈ (Base‘𝑃) ∧ (𝐴‘𝐹) ≠ (0g‘𝑃)) → (𝐷‘(𝐴‘𝐹)) ∈ ℕ0) |
15 | 7, 10, 13, 14 | syl3anc 1372 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) ∈ ℕ0) |
16 | nn0le0eq0 12496 | . . 3 ⊢ ((𝐷‘(𝐴‘𝐹)) ∈ ℕ0 → ((𝐷‘(𝐴‘𝐹)) ≤ 0 ↔ (𝐷‘(𝐴‘𝐹)) = 0)) | |
17 | 15, 16 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → ((𝐷‘(𝐴‘𝐹)) ≤ 0 ↔ (𝐷‘(𝐴‘𝐹)) = 0)) |
18 | 6, 17 | mpbid 231 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5147 ‘cfv 6540 0cc0 11106 ≤ cle 11245 ℕ0cn0 12468 Basecbs 17140 0gc0g 17381 Ringcrg 20047 algSccascl 21391 Poly1cpl1 21683 deg1 cdg1 25551 |
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 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-ofr 7666 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-subrg 20349 df-lmod 20461 df-lss 20531 df-cnfld 20930 df-ascl 21394 df-psr 21444 df-mvr 21445 df-mpl 21446 df-opsr 21448 df-psr1 21686 df-vr1 21687 df-ply1 21688 df-coe1 21689 df-mdeg 25552 df-deg1 25553 |
This theorem is referenced by: lgsqrlem4 26832 mon1pid 41880 |
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