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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unitpidl1 | Structured version Visualization version GIF version |
Description: The ideal 𝐼 generated by an element 𝑋 of an integral domain 𝑅 is the unit ideal 𝐵 iff 𝑋 is a ring unit. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
unitpidl1.1 | ⊢ 𝑈 = (Unit‘𝑅) |
unitpidl1.2 | ⊢ 𝐾 = (RSpan‘𝑅) |
unitpidl1.3 | ⊢ 𝐼 = (𝐾‘{𝑋}) |
unitpidl1.4 | ⊢ 𝐵 = (Base‘𝑅) |
unitpidl1.5 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
unitpidl1.6 | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
Ref | Expression |
---|---|
unitpidl1 | ⊢ (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitpidl1.6 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
2 | df-idom 21221 | . . . . . . 7 ⊢ IDomn = (CRing ∩ Domn) | |
3 | 1, 2 | eleqtrdi 2838 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (CRing ∩ Domn)) |
4 | 3 | elin1d 4194 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) |
5 | 4 | ad3antrrr 729 | . . . 4 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → 𝑅 ∈ CRing) |
6 | simplr 768 | . . . 4 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → 𝑦 ∈ 𝐵) | |
7 | unitpidl1.5 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 7 | ad3antrrr 729 | . . . 4 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝐵) |
9 | simpr 484 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) | |
10 | 1 | idomringd 21244 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
11 | unitpidl1.1 | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
12 | eqid 2727 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
13 | 11, 12 | 1unit 20302 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝑈) |
14 | 10, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝑈) |
15 | 14 | ad3antrrr 729 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → (1r‘𝑅) ∈ 𝑈) |
16 | 9, 15 | eqeltrrd 2829 | . . . 4 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → (𝑦(.r‘𝑅)𝑋) ∈ 𝑈) |
17 | eqid 2727 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
18 | unitpidl1.4 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
19 | 11, 17, 18 | unitmulclb 20309 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑦(.r‘𝑅)𝑋) ∈ 𝑈 ↔ (𝑦 ∈ 𝑈 ∧ 𝑋 ∈ 𝑈))) |
20 | 19 | simplbda 499 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ (𝑦(.r‘𝑅)𝑋) ∈ 𝑈) → 𝑋 ∈ 𝑈) |
21 | 5, 6, 8, 16, 20 | syl31anc 1371 | . . 3 ⊢ ((((𝜑 ∧ 𝐼 = 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) → 𝑋 ∈ 𝑈) |
22 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → 𝑅 ∈ Ring) |
23 | 7 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → 𝑋 ∈ 𝐵) |
24 | unitpidl1.3 | . . . . . . . 8 ⊢ 𝐼 = (𝐾‘{𝑋}) | |
25 | 7 | snssd 4808 | . . . . . . . . 9 ⊢ (𝜑 → {𝑋} ⊆ 𝐵) |
26 | unitpidl1.2 | . . . . . . . . . 10 ⊢ 𝐾 = (RSpan‘𝑅) | |
27 | eqid 2727 | . . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
28 | 26, 18, 27 | rspcl 21120 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅)) |
29 | 10, 25, 28 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (𝐾‘{𝑋}) ∈ (LIdeal‘𝑅)) |
30 | 24, 29 | eqeltrid 2832 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
31 | 30 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → 𝐼 ∈ (LIdeal‘𝑅)) |
32 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → 𝐼 = 𝐵) | |
33 | 27, 18, 12 | lidl1el 21111 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((1r‘𝑅) ∈ 𝐼 ↔ 𝐼 = 𝐵)) |
34 | 33 | biimpar 477 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝐼 = 𝐵) → (1r‘𝑅) ∈ 𝐼) |
35 | 22, 31, 32, 34 | syl21anc 837 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → (1r‘𝑅) ∈ 𝐼) |
36 | 35, 24 | eleqtrdi 2838 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → (1r‘𝑅) ∈ (𝐾‘{𝑋})) |
37 | 18, 17, 26 | rspsnel 33023 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((1r‘𝑅) ∈ (𝐾‘{𝑋}) ↔ ∃𝑦 ∈ 𝐵 (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋))) |
38 | 37 | biimpa 476 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ (1r‘𝑅) ∈ (𝐾‘{𝑋})) → ∃𝑦 ∈ 𝐵 (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) |
39 | 22, 23, 36, 38 | syl21anc 837 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → ∃𝑦 ∈ 𝐵 (1r‘𝑅) = (𝑦(.r‘𝑅)𝑋)) |
40 | 21, 39 | r19.29a 3157 | . 2 ⊢ ((𝜑 ∧ 𝐼 = 𝐵) → 𝑋 ∈ 𝑈) |
41 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
42 | 26, 18 | rspssid 21121 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ {𝑋} ⊆ 𝐵) → {𝑋} ⊆ (𝐾‘{𝑋})) |
43 | 10, 25, 42 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ (𝐾‘{𝑋})) |
44 | 43, 24 | sseqtrrdi 4029 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝐼) |
45 | snssg 4783 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐼 ↔ {𝑋} ⊆ 𝐼)) | |
46 | 45 | biimpar 477 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ {𝑋} ⊆ 𝐼) → 𝑋 ∈ 𝐼) |
47 | 7, 44, 46 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
48 | 47 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝐼) |
49 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring) |
50 | 30 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐼 ∈ (LIdeal‘𝑅)) |
51 | 18, 11, 41, 48, 49, 50 | lidlunitel 33074 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐼 = 𝐵) |
52 | 40, 51 | impbida 800 | 1 ⊢ (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ∩ cin 3943 ⊆ wss 3944 {csn 4624 ‘cfv 6542 (class class class)co 7414 Basecbs 17171 .rcmulr 17225 1rcur 20112 Ringcrg 20164 CRingccrg 20165 Unitcui 20283 LIdealclidl 21091 RSpancrsp 21092 Domncdomn 21216 IDomncidom 21217 |
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 |
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-op 4631 df-uni 4904 df-int 4945 df-iun 4993 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-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-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 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-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 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-subrg 20497 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rsp 21094 df-idom 21221 |
This theorem is referenced by: mxidlirredi 33120 mxidlirred 33121 |
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