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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmidl | Structured version Visualization version GIF version |
Description: The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
Ref | Expression |
---|---|
lsmidl.1 | ⊢ 𝐵 = (Base‘𝑅) |
lsmidl.3 | ⊢ ⊕ = (LSSum‘𝑅) |
lsmidl.4 | ⊢ 𝐾 = (RSpan‘𝑅) |
lsmidl.5 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
lsmidl.6 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
lsmidl.7 | ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) |
Ref | Expression |
---|---|
lsmidl | ⊢ (𝜑 → (𝐼 ⊕ 𝐽) ∈ (LIdeal‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmidl.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | lsmidl.3 | . . 3 ⊢ ⊕ = (LSSum‘𝑅) | |
3 | lsmidl.4 | . . 3 ⊢ 𝐾 = (RSpan‘𝑅) | |
4 | lsmidl.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
5 | lsmidl.6 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
6 | lsmidl.7 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) | |
7 | 1, 2, 3, 4, 5, 6 | lsmidllsp 31007 | . 2 ⊢ (𝜑 → (𝐼 ⊕ 𝐽) = (𝐾‘(𝐼 ∪ 𝐽))) |
8 | rlmlmod 19970 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
9 | 4, 8 | syl 17 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
10 | eqid 2798 | . . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | 1, 10 | lidlss 19976 | . . . . 5 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
12 | 5, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
13 | 1, 10 | lidlss 19976 | . . . . 5 ⊢ (𝐽 ∈ (LIdeal‘𝑅) → 𝐽 ⊆ 𝐵) |
14 | 6, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ⊆ 𝐵) |
15 | 12, 14 | unssd 4113 | . . 3 ⊢ (𝜑 → (𝐼 ∪ 𝐽) ⊆ 𝐵) |
16 | rlmbas 19960 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
17 | 1, 16 | eqtri 2821 | . . . 4 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
18 | lidlval 19957 | . . . 4 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
19 | rspval 19958 | . . . . 5 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
20 | 3, 19 | eqtri 2821 | . . . 4 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
21 | 17, 18, 20 | lspcl 19741 | . . 3 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ (𝐼 ∪ 𝐽) ⊆ 𝐵) → (𝐾‘(𝐼 ∪ 𝐽)) ∈ (LIdeal‘𝑅)) |
22 | 9, 15, 21 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐾‘(𝐼 ∪ 𝐽)) ∈ (LIdeal‘𝑅)) |
23 | 7, 22 | eqeltrd 2890 | 1 ⊢ (𝜑 → (𝐼 ⊕ 𝐽) ∈ (LIdeal‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 LSSumclsm 18751 Ringcrg 19290 LModclmod 19627 LSpanclspn 19736 ringLModcrglmod 19934 LIdealclidl 19935 RSpancrsp 19936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 |
This theorem is referenced by: mxidlprm 31048 idlsrgmnd 31067 |
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