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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsrgmnd | Structured version Visualization version GIF version |
Description: The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgmnd.1 | ⊢ 𝑆 = (IDLsrg‘𝑅) |
Ref | Expression |
---|---|
idlsrgmnd | ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmnd.1 | . . 3 ⊢ 𝑆 = (IDLsrg‘𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | 1, 2 | idlsrgbas 32042 | . 2 ⊢ (𝑅 ∈ Ring → (LIdeal‘𝑅) = (Base‘𝑆)) |
4 | eqid 2738 | . . 3 ⊢ (LSSum‘𝑅) = (LSSum‘𝑅) | |
5 | 1, 4 | idlsrgplusg 32043 | . 2 ⊢ (𝑅 ∈ Ring → (LSSum‘𝑅) = (+g‘𝑆)) |
6 | eqid 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2738 | . . 3 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
8 | simp1 1137 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring) | |
9 | simp2 1138 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅)) | |
10 | simp3 1139 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅)) | |
11 | 6, 4, 7, 8, 9, 10 | lsmidl 31982 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑖(LSSum‘𝑅)𝑗) ∈ (LIdeal‘𝑅)) |
12 | 2 | lidlsubg 20614 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) → 𝑖 ∈ (SubGrp‘𝑅)) |
13 | 12 | 3ad2antr1 1189 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → 𝑖 ∈ (SubGrp‘𝑅)) |
14 | 2 | lidlsubg 20614 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑗 ∈ (SubGrp‘𝑅)) |
15 | 14 | 3ad2antr2 1190 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → 𝑗 ∈ (SubGrp‘𝑅)) |
16 | 2 | lidlsubg 20614 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (SubGrp‘𝑅)) |
17 | 16 | 3ad2antr3 1191 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → 𝑘 ∈ (SubGrp‘𝑅)) |
18 | 4 | lsmass 19386 | . . 3 ⊢ ((𝑖 ∈ (SubGrp‘𝑅) ∧ 𝑗 ∈ (SubGrp‘𝑅) ∧ 𝑘 ∈ (SubGrp‘𝑅)) → ((𝑖(LSSum‘𝑅)𝑗)(LSSum‘𝑅)𝑘) = (𝑖(LSSum‘𝑅)(𝑗(LSSum‘𝑅)𝑘))) |
19 | 13, 15, 17, 18 | syl3anc 1372 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → ((𝑖(LSSum‘𝑅)𝑗)(LSSum‘𝑅)𝑘) = (𝑖(LSSum‘𝑅)(𝑗(LSSum‘𝑅)𝑘))) |
20 | eqid 2738 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
21 | 2, 20 | lidl0 20618 | . 2 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
22 | 20, 4 | lsm02 19389 | . . 3 ⊢ (𝑖 ∈ (SubGrp‘𝑅) → ({(0g‘𝑅)} (LSSum‘𝑅)𝑖) = 𝑖) |
23 | 12, 22 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) → ({(0g‘𝑅)} (LSSum‘𝑅)𝑖) = 𝑖) |
24 | 20, 4 | lsm01 19388 | . . 3 ⊢ (𝑖 ∈ (SubGrp‘𝑅) → (𝑖(LSSum‘𝑅){(0g‘𝑅)}) = 𝑖) |
25 | 12, 24 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) → (𝑖(LSSum‘𝑅){(0g‘𝑅)}) = 𝑖) |
26 | 3, 5, 11, 19, 21, 23, 25 | ismndd 18514 | 1 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {csn 4585 ‘cfv 6492 (class class class)co 7350 Basecbs 17019 0gc0g 17257 Mndcmnd 18492 SubGrpcsubg 18857 LSSumclsm 19351 Ringcrg 19894 LIdealclidl 20560 RSpancrsp 20561 IDLsrgcidlsrg 32038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-fz 13355 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-0g 17259 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-submnd 18538 df-grp 18687 df-minusg 18688 df-sbg 18689 df-subg 18860 df-cntz 19032 df-lsm 19353 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-ring 19896 df-subrg 20149 df-lmod 20253 df-lss 20322 df-lsp 20362 df-sra 20562 df-rgmod 20563 df-lidl 20564 df-rsp 20565 df-idlsrg 32039 |
This theorem is referenced by: idlsrgcmnd 32053 |
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