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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 2759 | . . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
3 | 1, 2 | idlsrgbas 31155 | . 2 ⊢ (𝑅 ∈ Ring → (LIdeal‘𝑅) = (Base‘𝑆)) |
4 | eqid 2759 | . . 3 ⊢ (LSSum‘𝑅) = (LSSum‘𝑅) | |
5 | 1, 4 | idlsrgplusg 31156 | . 2 ⊢ (𝑅 ∈ Ring → (LSSum‘𝑅) = (+g‘𝑆)) |
6 | eqid 2759 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2759 | . . 3 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
8 | simp1 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Ring) | |
9 | simp2 1135 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑖 ∈ (LIdeal‘𝑅)) | |
10 | simp3 1136 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅)) | |
11 | 6, 4, 7, 8, 9, 10 | lsmidl 31095 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑖(LSSum‘𝑅)𝑗) ∈ (LIdeal‘𝑅)) |
12 | 2 | lidlsubg 20041 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) → 𝑖 ∈ (SubGrp‘𝑅)) |
13 | 12 | 3ad2antr1 1186 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → 𝑖 ∈ (SubGrp‘𝑅)) |
14 | 2 | lidlsubg 20041 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 𝑗 ∈ (SubGrp‘𝑅)) |
15 | 14 | 3ad2antr2 1187 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → 𝑗 ∈ (SubGrp‘𝑅)) |
16 | 2 | lidlsubg 20041 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)) → 𝑘 ∈ (SubGrp‘𝑅)) |
17 | 16 | 3ad2antr3 1188 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑖 ∈ (LIdeal‘𝑅) ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑘 ∈ (LIdeal‘𝑅))) → 𝑘 ∈ (SubGrp‘𝑅)) |
18 | 4 | lsmass 18847 | . . 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 1369 | . 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 2759 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
21 | 2, 20 | lidl0 20045 | . 2 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
22 | 20, 4 | lsm02 18850 | . . 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 18849 | . . 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 17984 | 1 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 {csn 4515 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 0gc0g 16756 Mndcmnd 17962 SubGrpcsubg 18325 LSSumclsm 18811 Ringcrg 19350 LIdealclidl 19995 RSpancrsp 19996 IDLsrgcidlsrg 31151 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-sca 16624 df-vsca 16625 df-ip 16626 df-tset 16627 df-ple 16628 df-0g 16758 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-grp 18157 df-minusg 18158 df-sbg 18159 df-subg 18328 df-cntz 18499 df-lsm 18813 df-cmn 18960 df-abl 18961 df-mgp 19293 df-ur 19305 df-ring 19352 df-subrg 19586 df-lmod 19689 df-lss 19757 df-lsp 19797 df-sra 19997 df-rgmod 19998 df-lidl 19999 df-rsp 20000 df-idlsrg 31152 |
This theorem is referenced by: idlsrgcmnd 31166 |
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