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Theorem ply1lpir 25687

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 20314	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		ply1lpir.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 21761	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
5		eqid 2732	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
6		eqid 2732	. . . . . . . . 9 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
7	5, 6	lidlss 20825	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ⊆ (Base‘𝑃))
8	7	adantl 482	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ⊆ (Base‘𝑃))
9		eqid 2732	. . . . . . . 8 ⊢ (idlGen_1p‘𝑅) = (idlGen_1p‘𝑅)
10	2, 9, 6	ig1pcl 25684	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen_1p‘𝑅)‘𝑖) ∈ 𝑖)
11	8, 10	sseldd 3982	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen_1p‘𝑅)‘𝑖) ∈ (Base‘𝑃))
12		eqid 2732	. . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
13	2, 9, 6, 12	ig1prsp 25686	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
14		sneq 4637	. . . . . . . 8 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → {𝑗} = {((idlGen_1p‘𝑅)‘𝑖)})
15	14	fveq2d 6892	. . . . . . 7 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → ((RSpan‘𝑃)‘{𝑗}) = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
16	15	rspceeqv 3632	. . . . . 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 584	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))
18	4	adantr 481	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑃 ∈ Ring)
19		eqid 2732	. . . . . . 7 ⊢ (LPIdeal‘𝑃) = (LPIdeal‘𝑃)
20	19, 12, 5	islpidl 20876	. . . . . 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 256	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ∈ (LPIdeal‘𝑃))
23	22	ex 413	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ∈ (LPIdeal‘𝑃)))
24	23	ssrdv 3987	. 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃))
25	19, 6	islpir2 20881	. 2 ⊢ (𝑃 ∈ LPIR ↔ (𝑃 ∈ Ring ∧ (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃)))
26	4, 24, 25	sylanbrc 583	1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ⊆ wss 3947 {csn 4627 ‘cfv 6540 Basecbs 17140 Ringcrg 20049 DivRingcdr 20307 LIdealclidl 20775 RSpancrsp 20776 LPIdealclpidl 20871 LPIRclpir 20872 Poly₁cpl1 21692 idlGen_1pcig1p 25638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-drng 20309 df-subrg 20353 df-lmod 20465 df-lss 20535 df-lsp 20575 df-sra 20777 df-rgmod 20778 df-lidl 20779 df-rsp 20780 df-lpidl 20873 df-lpir 20874 df-rlreg 20891 df-cnfld 20937 df-ascl 21401 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-mdeg 25561 df-deg1 25562 df-mon1 25639 df-uc1p 25640 df-q1p 25641 df-r1p 25642 df-ig1p 25643

This theorem is referenced by: ply1pid 25688

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