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Theorem ply1lpir 26148

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 20674	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		ply1lpir.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22193	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
5		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
6		eqid 2737	. . . . . . . . 9 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
7	5, 6	lidlss 21172	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ⊆ (Base‘𝑃))
8	7	adantl 481	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ⊆ (Base‘𝑃))
9		eqid 2737	. . . . . . . 8 ⊢ (idlGen_1p‘𝑅) = (idlGen_1p‘𝑅)
10	2, 9, 6	ig1pcl 26145	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen_1p‘𝑅)‘𝑖) ∈ 𝑖)
11	8, 10	sseldd 3935	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ((idlGen_1p‘𝑅)‘𝑖) ∈ (Base‘𝑃))
12		eqid 2737	. . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
13	2, 9, 6, 12	ig1prsp 26147	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
14		sneq 4591	. . . . . . . 8 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → {𝑗} = {((idlGen_1p‘𝑅)‘𝑖)})
15	14	fveq2d 6839	. . . . . . 7 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → ((RSpan‘𝑃)‘{𝑗}) = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
16	15	rspceeqv 3600	. . . . . 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 585	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))
18	4	adantr 480	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑃 ∈ Ring)
19		eqid 2737	. . . . . . 7 ⊢ (LPIdeal‘𝑃) = (LPIdeal‘𝑃)
20	19, 12, 5	islpidl 21285	. . . . . 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 257	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ∈ (LPIdeal‘𝑃))
23	22	ex 412	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ∈ (LPIdeal‘𝑃)))
24	23	ssrdv 3940	. 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃))
25	19, 6	islpir2 21290	. 2 ⊢ (𝑃 ∈ LPIR ↔ (𝑃 ∈ Ring ∧ (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃)))
26	4, 24, 25	sylanbrc 584	1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 ⊆ wss 3902 {csn 4581 ‘cfv 6493 Basecbs 17141 Ringcrg 20173 DivRingcdr 20667 LIdealclidl 21166 RSpancrsp 21167 LPIdealclpidl 21280 LPIRclpir 21281 Poly₁cpl1 22122 idlGen_1pcig1p 26096

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-pre-sup 11109 ax-addf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-ofr 7626 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-inf 9351 df-oi 9420 df-card 9856 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-z 12494 df-dec 12613 df-uz 12757 df-fz 13429 df-fzo 13576 df-seq 13930 df-hash 14259 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-starv 17197 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-unif 17205 df-hom 17206 df-cco 17207 df-0g 17366 df-gsum 17367 df-prds 17372 df-pws 17374 df-mre 17510 df-mrc 17511 df-acs 17513 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-submnd 18714 df-grp 18871 df-minusg 18872 df-sbg 18873 df-mulg 19003 df-subg 19058 df-ghm 19147 df-cntz 19251 df-cmn 19716 df-abl 19717 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20278 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-subrng 20484 df-subrg 20508 df-rlreg 20632 df-drng 20669 df-lmod 20818 df-lss 20888 df-lsp 20928 df-sra 21130 df-rgmod 21131 df-lidl 21168 df-rsp 21169 df-lpidl 21282 df-lpir 21283 df-cnfld 21315 df-ascl 21815 df-psr 21870 df-mvr 21871 df-mpl 21872 df-opsr 21874 df-psr1 22125 df-vr1 22126 df-ply1 22127 df-coe1 22128 df-mdeg 26021 df-deg1 26022 df-mon1 26097 df-uc1p 26098 df-q1p 26099 df-r1p 26100 df-ig1p 26101

This theorem is referenced by: ply1pid 26149

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