ply1lpir - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ply1lpir		Structured version Visualization version GIF version

Theorem ply1lpir 24344

Description: The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)

**Hypothesis**
Ref	Expression
ply1lpir.p	⊢ 𝑃 = (Poly₁‘𝑅)

**Assertion**
Ref	Expression
ply1lpir	⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)

Proof of Theorem ply1lpir Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		drngring 19117	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		ply1lpir.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 19985	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
5		eqid 2825	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
6		eqid 2825	. . . . . . . . 9 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
7	5, 6	lidlss 19578	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ⊆ (Base‘𝑃))
8	7	adantl 475	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ⊆ (Base‘𝑃))
9		eqid 2825	. . . . . . . 8 ⊢ (idlGen_1p‘𝑅) = (idlGen_1p‘𝑅)
10	2, 9, 6	ig1pcl 24341	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen_1p‘𝑅)‘𝑖) ∈ 𝑖)
11	8, 10	sseldd 3828	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen_1p‘𝑅)‘𝑖) ∈ (Base‘𝑃))
12		eqid 2825	. . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
13	2, 9, 6, 12	ig1prsp 24343	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
14		sneq 4409	. . . . . . . 8 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → {𝑗} = {((idlGen_1p‘𝑅)‘𝑖)})
15	14	fveq2d 6441	. . . . . . 7 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → ((RSpan‘𝑃)‘{𝑗}) = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
16	15	rspceeqv 3544	. . . . . 6 ⊢ ((((idlGen_1p‘𝑅)‘𝑖) ∈ (Base‘𝑃) ∧ 𝑖 = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)})) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))
17	11, 13, 16	syl2anc 579	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))
18	4	adantr 474	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑃 ∈ Ring)
19		eqid 2825	. . . . . . 7 ⊢ (LPIdeal‘𝑃) = (LPIdeal‘𝑃)
20	19, 12, 5	islpidl 19614	. . . . . 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 249	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ∈ (LPIdeal‘𝑃))
23	22	ex 403	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ∈ (LPIdeal‘𝑃)))
24	23	ssrdv 3833	. 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃))
25	19, 6	islpir2 19619	. 2 ⊢ (𝑃 ∈ LPIR ↔ (𝑃 ∈ Ring ∧ (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃)))
26	4, 24, 25	sylanbrc 578	1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ⊆ wss 3798 {csn 4399 ‘cfv 6127 Basecbs 16229 Ringcrg 18908 DivRingcdr 19110 LIdealclidl 19538 RSpancrsp 19539 LPIdealclpidl 19609 LPIRclpir 19610 Poly₁cpl1 19914 idlGen_1pcig1p 24295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-ofr 7163 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-gsum 16463 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-drng 19112 df-subrg 19141 df-lmod 19228 df-lss 19296 df-lsp 19338 df-sra 19540 df-rgmod 19541 df-lidl 19542 df-rsp 19543 df-lpidl 19611 df-lpir 19612 df-rlreg 19651 df-ascl 19682 df-psr 19724 df-mvr 19725 df-mpl 19726 df-opsr 19728 df-psr1 19917 df-vr1 19918 df-ply1 19919 df-coe1 19920 df-cnfld 20114 df-mdeg 24221 df-deg1 24222 df-mon1 24296 df-uc1p 24297 df-q1p 24298 df-r1p 24299 df-ig1p 24300

This theorem is referenced by: ply1pid 24345

W3C validator