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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringlsmss1 | Structured version Visualization version GIF version |
Description: The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.) |
Ref | Expression |
---|---|
ringlsmss.1 | ⊢ 𝐵 = (Base‘𝑅) |
ringlsmss.2 | ⊢ 𝐺 = (mulGrp‘𝑅) |
ringlsmss.3 | ⊢ × = (LSSum‘𝐺) |
ringlsmss1.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
ringlsmss1.2 | ⊢ (𝜑 → 𝐸 ⊆ 𝐵) |
ringlsmss1.3 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
Ref | Expression |
---|---|
ringlsmss1 | ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → 𝑎 = (𝑖(.r‘𝑅)𝑒)) | |
2 | ringlsmss1.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
3 | 2 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑅 ∈ CRing) |
4 | ringlsmss1.2 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ⊆ 𝐵) | |
5 | 4 | sselda 3979 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐵) |
6 | 5 | adantlr 714 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝐵) |
7 | ringlsmss1.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
8 | ringlsmss.1 | . . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑅) | |
9 | eqid 2728 | . . . . . . . . . . . . 13 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
10 | 8, 9 | lidlss 21102 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
11 | 7, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
12 | 11 | sselda 3979 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐵) |
13 | 12 | adantr 480 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑖 ∈ 𝐵) |
14 | eqid 2728 | . . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
15 | 8, 14 | crngcom 20185 | . . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑒 ∈ 𝐵 ∧ 𝑖 ∈ 𝐵) → (𝑒(.r‘𝑅)𝑖) = (𝑖(.r‘𝑅)𝑒)) |
16 | 3, 6, 13, 15 | syl3anc 1369 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑒(.r‘𝑅)𝑖) = (𝑖(.r‘𝑅)𝑒)) |
17 | crngring 20179 | . . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
18 | 2, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | 18 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑅 ∈ Ring) |
20 | 7 | ad2antrr 725 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝐼 ∈ (LIdeal‘𝑅)) |
21 | simplr 768 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → 𝑖 ∈ 𝐼) | |
22 | 9, 8, 14 | lidlmcl 21115 | . . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼)) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
23 | 19, 20, 6, 21, 22 | syl22anc 838 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑒(.r‘𝑅)𝑖) ∈ 𝐼) |
24 | 16, 23 | eqeltrrd 2830 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
25 | 24 | adantllr 718 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
26 | 25 | adantr 480 | . . . . 5 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → (𝑖(.r‘𝑅)𝑒) ∈ 𝐼) |
27 | 1, 26 | eqeltrd 2829 | . . . 4 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑒 ∈ 𝐸) ∧ 𝑎 = (𝑖(.r‘𝑅)𝑒)) → 𝑎 ∈ 𝐼) |
28 | ringlsmss.2 | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
29 | ringlsmss.3 | . . . . . 6 ⊢ × = (LSSum‘𝐺) | |
30 | 8, 14, 28, 29, 11, 4 | elringlsm 33097 | . . . . 5 ⊢ (𝜑 → (𝑎 ∈ (𝐼 × 𝐸) ↔ ∃𝑖 ∈ 𝐼 ∃𝑒 ∈ 𝐸 𝑎 = (𝑖(.r‘𝑅)𝑒))) |
31 | 30 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) → ∃𝑖 ∈ 𝐼 ∃𝑒 ∈ 𝐸 𝑎 = (𝑖(.r‘𝑅)𝑒)) |
32 | 27, 31 | r19.29vva 3209 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐼 × 𝐸)) → 𝑎 ∈ 𝐼) |
33 | 32 | ex 412 | . 2 ⊢ (𝜑 → (𝑎 ∈ (𝐼 × 𝐸) → 𝑎 ∈ 𝐼)) |
34 | 33 | ssrdv 3985 | 1 ⊢ (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 ⊆ wss 3945 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 .rcmulr 17228 LSSumclsm 19583 mulGrpcmgp 20068 Ringcrg 20167 CRingccrg 20168 LIdealclidl 21096 |
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 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-lsm 19585 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-cring 20170 df-subrg 20502 df-lmod 20739 df-lss 20810 df-sra 21052 df-rgmod 21053 df-lidl 21098 |
This theorem is referenced by: idlsrgmulrss1 33217 |
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