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| Mirrors > Home > MPE Home > Th. List > mon1pid | Structured version Visualization version GIF version | ||
| Description: Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
| Ref | Expression |
|---|---|
| mon1pid.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| mon1pid.o | ⊢ 1 = (1r‘𝑃) |
| mon1pid.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
| mon1pid.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
| Ref | Expression |
|---|---|
| mon1pid | ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mon1pid.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 2 | 1 | ply1nz 26050 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
| 3 | nzrring 20448 | . . . 4 ⊢ (𝑃 ∈ NzRing → 𝑃 ∈ Ring) | |
| 4 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 5 | mon1pid.o | . . . . 5 ⊢ 1 = (1r‘𝑃) | |
| 6 | 4, 5 | ringidcl 20195 | . . . 4 ⊢ (𝑃 ∈ Ring → 1 ∈ (Base‘𝑃)) |
| 7 | 2, 3, 6 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ∈ (Base‘𝑃)) |
| 8 | eqid 2728 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 9 | 5, 8 | nzrnz 20447 | . . . 4 ⊢ (𝑃 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
| 10 | 2, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
| 11 | nzrring 20448 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
| 12 | eqid 2728 | . . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
| 13 | eqid 2728 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 14 | 1, 12, 13, 5 | ply1scl1 22205 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
| 15 | 11, 14 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
| 16 | 15 | fveq2d 6895 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (coe1‘ 1 )) |
| 17 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 18 | 17, 13 | ringidcl 20195 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 19 | eqid 2728 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 20 | 1, 12, 17, 19 | coe1scl 22199 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
| 21 | 11, 18, 20 | syl2anc2 584 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
| 22 | 16, 21 | eqtr3d 2770 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (coe1‘ 1 ) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
| 23 | 15 | fveq2d 6895 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝐷‘ 1 )) |
| 24 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 25 | 13, 19 | nzrnz 20447 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
| 26 | mon1pid.d | . . . . . . . 8 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
| 27 | 26, 1, 17, 12, 19 | deg1scl 26042 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
| 28 | 11, 24, 25, 27 | syl3anc 1369 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
| 29 | 23, 28 | eqtr3d 2770 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (𝐷‘ 1 ) = 0) |
| 30 | 22, 29 | fveq12d 6898 | . . . 4 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0)) |
| 31 | 0nn0 12511 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 32 | iftrue 4530 | . . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)) | |
| 33 | eqid 2728 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) | |
| 34 | fvex 6904 | . . . . . 6 ⊢ (1r‘𝑅) ∈ V | |
| 35 | 32, 33, 34 | fvmpt 6999 | . . . . 5 ⊢ (0 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0) = (1r‘𝑅)) |
| 36 | 31, 35 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0) = (1r‘𝑅) |
| 37 | 30, 36 | eqtrdi 2784 | . . 3 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅)) |
| 38 | mon1pid.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
| 39 | 1, 4, 8, 26, 38, 13 | ismon1p 26071 | . . 3 ⊢ ( 1 ∈ 𝑀 ↔ ( 1 ∈ (Base‘𝑃) ∧ 1 ≠ (0g‘𝑃) ∧ ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅))) |
| 40 | 7, 10, 37, 39 | syl3anbrc 1341 | . 2 ⊢ (𝑅 ∈ NzRing → 1 ∈ 𝑀) |
| 41 | 40, 29 | jca 511 | 1 ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ifcif 4524 ↦ cmpt 5225 ‘cfv 6542 0cc0 11132 ℕ0cn0 12496 Basecbs 17173 0gc0g 17414 1rcur 20114 Ringcrg 20166 NzRingcnzr 20444 algSccascl 21779 Poly1cpl1 22089 coe1cco1 22090 deg1 cdg1 25980 Monic1pcmn1 26054 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-nzr 20445 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-cnfld 21273 df-ascl 21782 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mdeg 25981 df-deg1 25982 df-mon1 26059 |
| This theorem is referenced by: ply1unit 33249 mon1psubm 42621 deg1mhm 42622 |
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