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Mirrors > Home > MPE Home > Th. List > ply1pid | Structured version Visualization version GIF version |
Description: The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
ply1lpir.p | ⊢ 𝑃 = (Poly1‘𝑅) |
Ref | Expression |
---|---|
ply1pid | ⊢ (𝑅 ∈ Field → 𝑃 ∈ PID) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldidom 20574 | . . 3 ⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn) | |
2 | ply1lpir.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1idom 25287 | . . 3 ⊢ (𝑅 ∈ IDomn → 𝑃 ∈ IDomn) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ Field → 𝑃 ∈ IDomn) |
5 | isfld 19998 | . . . 4 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) | |
6 | 5 | simplbi 498 | . . 3 ⊢ (𝑅 ∈ Field → 𝑅 ∈ DivRing) |
7 | 2 | ply1lpir 25341 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR) |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝑅 ∈ Field → 𝑃 ∈ LPIR) |
9 | df-pid 20555 | . . 3 ⊢ PID = (IDomn ∩ LPIR) | |
10 | 9 | elin2 4136 | . 2 ⊢ (𝑃 ∈ PID ↔ (𝑃 ∈ IDomn ∧ 𝑃 ∈ LPIR)) |
11 | 4, 8, 10 | sylanbrc 583 | 1 ⊢ (𝑅 ∈ Field → 𝑃 ∈ PID) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 CRingccrg 19782 DivRingcdr 19989 Fieldcfield 19990 LPIRclpir 20511 IDomncidom 20550 PIDcpid 20551 Poly1cpl1 21346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-ofr 7528 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-tpos 8033 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-0g 17150 df-gsum 17151 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-drng 19991 df-field 19992 df-subrg 20020 df-lmod 20123 df-lss 20192 df-lsp 20232 df-sra 20432 df-rgmod 20433 df-lidl 20434 df-rsp 20435 df-lpidl 20512 df-lpir 20513 df-nzr 20527 df-rlreg 20552 df-domn 20553 df-idom 20554 df-pid 20555 df-cnfld 20596 df-ascl 21060 df-psr 21110 df-mvr 21111 df-mpl 21112 df-opsr 21114 df-psr1 21349 df-vr1 21350 df-ply1 21351 df-coe1 21352 df-mdeg 25215 df-deg1 25216 df-mon1 25293 df-uc1p 25294 df-q1p 25295 df-r1p 25296 df-ig1p 25297 |
This theorem is referenced by: (None) |
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