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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 24633 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐷‘(𝐴‘𝐹)) ≤ 0) |
6 | 5 | 3adant3 1124 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) ≤ 0) |
7 | simp1 1128 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → 𝑅 ∈ Ring) | |
8 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
9 | 2, 4, 3, 8 | ply1sclcl 20382 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐴‘𝐹) ∈ (Base‘𝑃)) |
10 | 9 | 3adant3 1124 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐴‘𝐹) ∈ (Base‘𝑃)) |
11 | deg1scl.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
12 | eqid 2818 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
13 | 2, 4, 11, 12, 3 | ply1scln0 20387 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐴‘𝐹) ≠ (0g‘𝑃)) |
14 | 1, 2, 12, 8 | deg1nn0cl 24609 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴‘𝐹) ∈ (Base‘𝑃) ∧ (𝐴‘𝐹) ≠ (0g‘𝑃)) → (𝐷‘(𝐴‘𝐹)) ∈ ℕ0) |
15 | 7, 10, 13, 14 | syl3anc 1363 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) ∈ ℕ0) |
16 | nn0le0eq0 11913 | . . 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 233 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 0cc0 10525 ≤ cle 10664 ℕ0cn0 11885 Basecbs 16471 0gc0g 16701 Ringcrg 19226 algSccascl 20012 Poly1cpl1 20273 deg1 cdg1 24575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-lmod 19565 df-lss 19633 df-ascl 20015 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-vr1 20277 df-ply1 20278 df-coe1 20279 df-cnfld 20474 df-mdeg 24576 df-deg1 24577 |
This theorem is referenced by: lgsqrlem4 25852 mon1pid 39683 |
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