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Mirrors > Home > MPE Home > Th. List > ply1lpir | Structured version Visualization version GIF version |
Description: The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
ply1lpir.p | ⊢ 𝑃 = (Poly1‘𝑅) |
Ref | Expression |
---|---|
ply1lpir | ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 19511 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | ply1lpir.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1ring 20418 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring) |
5 | eqid 2823 | . . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
6 | eqid 2823 | . . . . . . . . 9 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃) | |
7 | 5, 6 | lidlss 19985 | . . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ⊆ (Base‘𝑃)) |
8 | 7 | adantl 484 | . . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ⊆ (Base‘𝑃)) |
9 | eqid 2823 | . . . . . . . 8 ⊢ (idlGen1p‘𝑅) = (idlGen1p‘𝑅) | |
10 | 2, 9, 6 | ig1pcl 24771 | . . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen1p‘𝑅)‘𝑖) ∈ 𝑖) |
11 | 8, 10 | sseldd 3970 | . . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen1p‘𝑅)‘𝑖) ∈ (Base‘𝑃)) |
12 | eqid 2823 | . . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
13 | 2, 9, 6, 12 | ig1prsp 24773 | . . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 = ((RSpan‘𝑃)‘{((idlGen1p‘𝑅)‘𝑖)})) |
14 | sneq 4579 | . . . . . . . 8 ⊢ (𝑗 = ((idlGen1p‘𝑅)‘𝑖) → {𝑗} = {((idlGen1p‘𝑅)‘𝑖)}) | |
15 | 14 | fveq2d 6676 | . . . . . . 7 ⊢ (𝑗 = ((idlGen1p‘𝑅)‘𝑖) → ((RSpan‘𝑃)‘{𝑗}) = ((RSpan‘𝑃)‘{((idlGen1p‘𝑅)‘𝑖)})) |
16 | 15 | rspceeqv 3640 | . . . . . 6 ⊢ ((((idlGen1p‘𝑅)‘𝑖) ∈ (Base‘𝑃) ∧ 𝑖 = ((RSpan‘𝑃)‘{((idlGen1p‘𝑅)‘𝑖)})) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗})) |
17 | 11, 13, 16 | syl2anc 586 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗})) |
18 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑃 ∈ Ring) |
19 | eqid 2823 | . . . . . . 7 ⊢ (LPIdeal‘𝑃) = (LPIdeal‘𝑃) | |
20 | 19, 12, 5 | islpidl 20021 | . . . . . 6 ⊢ (𝑃 ∈ Ring → (𝑖 ∈ (LPIdeal‘𝑃) ↔ ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))) |
21 | 18, 20 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → (𝑖 ∈ (LPIdeal‘𝑃) ↔ ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))) |
22 | 17, 21 | mpbird 259 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ∈ (LPIdeal‘𝑃)) |
23 | 22 | ex 415 | . . 3 ⊢ (𝑅 ∈ DivRing → (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ∈ (LPIdeal‘𝑃))) |
24 | 23 | ssrdv 3975 | . 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃)) |
25 | 19, 6 | islpir2 20026 | . 2 ⊢ (𝑃 ∈ LPIR ↔ (𝑃 ∈ Ring ∧ (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃))) |
26 | 4, 24, 25 | sylanbrc 585 | 1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ⊆ wss 3938 {csn 4569 ‘cfv 6357 Basecbs 16485 Ringcrg 19299 DivRingcdr 19504 LIdealclidl 19944 RSpancrsp 19945 LPIdealclpidl 20016 LPIRclpir 20017 Poly1cpl1 20347 idlGen1pcig1p 24725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-lpidl 20018 df-lpir 20019 df-rlreg 20058 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-vr1 20351 df-ply1 20352 df-coe1 20353 df-cnfld 20548 df-mdeg 24651 df-deg1 24652 df-mon1 24726 df-uc1p 24727 df-q1p 24728 df-r1p 24729 df-ig1p 24730 |
This theorem is referenced by: ply1pid 24775 |
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