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Theorem ply1lpir 26414

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 20903	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		ply1lpir.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22478	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 18	. 2 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
5		eqid 2762	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑃)
6		eqid 2762	. . . . . . . . 9 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
7	5, 6	lidlss 21405	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ⊆ (Base‘𝑃))
8	7	adantl 487	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ⊆ (Base‘𝑃))
9		eqid 2762	. . . . . . . 8 ⊢ (idlGen_1p‘𝑅) = (idlGen_1p‘𝑅)
10	2, 9, 6	ig1pcl 26411	. . . . . . 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 2762	. . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
13	2, 9, 6, 12	ig1prsp 26413	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
14		sneq 4597	. . . . . . . 8 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → {𝑗} = {((idlGen_1p‘𝑅)‘𝑖)})
15	14	fveq2d 6886	. . . . . . 7 ⊢ (𝑗 = ((idlGen_1p‘𝑅)‘𝑖) → ((RSpan‘𝑃)‘{𝑗}) = ((RSpan‘𝑃)‘{((idlGen_1p‘𝑅)‘𝑖)}))
16	15	rspceeqv 3602	. . . . . 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 596	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗}))
18	4	adantr 486	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑃 ∈ Ring)
19		eqid 2762	. . . . . . 7 ⊢ (LPIdeal‘𝑃) = (LPIdeal‘𝑃)
20	19, 12, 5	islpidl 21562	. . . . . 6 ⊢ (𝑃 ∈ Ring → (𝑖 ∈ (LPIdeal‘𝑃) ↔ ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗})))
21	18, 20	syl 18	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → (𝑖 ∈ (LPIdeal‘𝑃) ↔ ∃𝑗 ∈ (Base‘𝑃)𝑖 = ((RSpan‘𝑃)‘{𝑗})))
22	17, 21	mpbird 260	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝑖 ∈ (LIdeal‘𝑃)) → 𝑖 ∈ (LPIdeal‘𝑃))
23	22	ex 418	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑖 ∈ (LIdeal‘𝑃) → 𝑖 ∈ (LPIdeal‘𝑃)))
24	23	ssrdv 3940	. 2 ⊢ (𝑅 ∈ DivRing → (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃))
25	19, 6	islpir2 21567	. 2 ⊢ (𝑃 ∈ LPIR ↔ (𝑃 ∈ Ring ∧ (LIdeal‘𝑃) ⊆ (LPIdeal‘𝑃)))
26	4, 24, 25	sylanbrc 595	1 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∃wrex 3088 ⊆ wss 3902 {csn 4587 ‘cfv 6537 Basecbs 17307 Ringcrg 20378 DivRingcdr 20896 LIdealclidl 21399 RSpancrsp 21400 LPIdealclpidl 21557 LPIRclpir 21558 Poly₁cpl1 22408 idlGen_1pcig1p 26362

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-fz 13566 df-fzo 13714 df-seq 14070 df-hash 14399 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-0g 17532 df-gsum 17533 df-prds 17538 df-pws 17540 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-mulg 19197 df-subg 19252 df-ghm 19347 df-cntz 19450 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-cring 20381 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-subrng 20714 df-subrg 20738 df-rlreg 20862 df-drng 20898 df-lmod 21052 df-lss 21122 df-lsp 21162 df-sra 21363 df-rgmod 21364 df-lidl 21401 df-rsp 21402 df-lpidl 21559 df-lpir 21560 df-cnfld 21592 df-ascl 22076 df-psr 22130 df-mvr 22131 df-mpl 22132 df-opsr 22134 df-psr1 22411 df-vr1 22412 df-ply1 22413 df-coe1 22414 df-mdeg 26287 df-deg1 26288 df-mon1 26363 df-uc1p 26364 df-q1p 26365 df-r1p 26366 df-ig1p 26367

This theorem is used by: ply1pid 26415

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