ig1prsp - Metamath Proof Explorer

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Theorem ig1prsp 26102

Description: Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)

**Hypotheses**
Ref	Expression
ig1pval.p	⊢ 𝑃 = (Poly₁‘𝑅)
ig1pval.g	⊢ 𝐺 = (idlGen_1p‘𝑅)
ig1pcl.u	⊢ 𝑈 = (LIdeal‘𝑃)
ig1prsp.k	⊢ 𝐾 = (RSpan‘𝑃)

**Assertion**
Ref	Expression
ig1prsp	⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝐼 = (𝐾‘{(𝐺‘𝐼)}))

Proof of Theorem ig1prsp Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ig1pval.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
2		ig1pval.g	. . 3 ⊢ 𝐺 = (idlGen_1p‘𝑅)
3		ig1pcl.u	. . 3 ⊢ 𝑈 = (LIdeal‘𝑃)
4	1, 2, 3	ig1pcl 26100	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼)
5		eqid 2727	. . . . 5 ⊢ (∥_r‘𝑃) = (∥_r‘𝑃)
6	1, 2, 3, 5	ig1pdvds 26101	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝐼)(∥_r‘𝑃)𝑥)
7	6	3expa 1116	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝐼)(∥_r‘𝑃)𝑥)
8	7	ralrimiva 3141	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ 𝐼 (𝐺‘𝐼)(∥_r‘𝑃)𝑥)
9		drngring 20620	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
10	1	ply1ring 22153	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
11	9, 10	syl 17	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
12	11	adantr 480	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝑃 ∈ Ring)
13		simpr 484	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ 𝑈)
14		eqid 2727	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
15	14, 3	lidlss 21097	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
16	15	adantl 481	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝐼 ⊆ (Base‘𝑃))
17	16, 4	sseldd 3979	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ (Base‘𝑃))
18		ig1prsp.k	. . . 4 ⊢ 𝐾 = (RSpan‘𝑃)
19	14, 3, 18, 5	lidldvgen 21213	. . 3 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ (𝐺‘𝐼) ∈ (Base‘𝑃)) → (𝐼 = (𝐾‘{(𝐺‘𝐼)}) ↔ ((𝐺‘𝐼) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐺‘𝐼)(∥_r‘𝑃)𝑥)))
20	12, 13, 17, 19	syl3anc 1369	. 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐼 = (𝐾‘{(𝐺‘𝐼)}) ↔ ((𝐺‘𝐼) ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐺‘𝐼)(∥_r‘𝑃)𝑥)))
21	4, 8, 20	mpbir2and 712	1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝐼 = (𝐾‘{(𝐺‘𝐼)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 {csn 4624 class class class wbr 5142 ‘cfv 6542 Basecbs 17171 Ringcrg 20164 ∥_rcdsr 20282 DivRingcdr 20613 LIdealclidl 21091 RSpancrsp 21092 Poly₁cpl1 22083 idlGen_1pcig1p 26052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rsp 21094 df-rlreg 21219 df-cnfld 21267 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-mdeg 25975 df-deg1 25976 df-mon1 26053 df-uc1p 26054 df-q1p 26055 df-r1p 26056 df-ig1p 26057

This theorem is referenced by: ply1lpir 26103 ply1annig1p 33311

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