Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsrgmulrss1 | Structured version Visualization version GIF version |
Description: In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgmulrss1.1 | ⊢ 𝑆 = (IDLsrg‘𝑅) |
idlsrgmulrss1.2 | ⊢ 𝐵 = (LIdeal‘𝑅) |
idlsrgmulrss1.3 | ⊢ ⊗ = (.r‘𝑆) |
idlsrgmulrss1.4 | ⊢ · = (.r‘𝑅) |
idlsrgmulrss1.5 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
idlsrgmulrss1.6 | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
idlsrgmulrss1.7 | ⊢ (𝜑 → 𝐽 ∈ 𝐵) |
Ref | Expression |
---|---|
idlsrgmulrss1 | ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrss1.1 | . . 3 ⊢ 𝑆 = (IDLsrg‘𝑅) | |
2 | idlsrgmulrss1.2 | . . 3 ⊢ 𝐵 = (LIdeal‘𝑅) | |
3 | idlsrgmulrss1.3 | . . 3 ⊢ ⊗ = (.r‘𝑆) | |
4 | eqid 2734 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | eqid 2734 | . . 3 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅)) | |
6 | idlsrgmulrss1.5 | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | idlsrgmulrss1.6 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
8 | idlsrgmulrss1.7 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | idlsrgmulrval 31340 | . 2 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽))) |
10 | crngring 19546 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
11 | rlmlmod 20214 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
12 | 6, 10, 11 | 3syl 18 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
13 | eqid 2734 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13, 2 | lidlss 20220 | . . . . 5 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ⊆ (Base‘𝑅)) |
15 | 7, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑅)) |
16 | 13, 2 | lidlss 20220 | . . . . . 6 ⊢ (𝐽 ∈ 𝐵 → 𝐽 ⊆ (Base‘𝑅)) |
17 | 8, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑅)) |
18 | 7, 2 | eleqtrdi 2844 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
19 | 13, 4, 5, 6, 17, 18 | ringlsmss1 31270 | . . . 4 ⊢ (𝜑 → (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ 𝐼) |
20 | rlmbas 20204 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
21 | rspval 20202 | . . . . 5 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
22 | 20, 21 | lspss 19993 | . . . 4 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ⊆ (Base‘𝑅) ∧ (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ 𝐼) → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ ((RSpan‘𝑅)‘𝐼)) |
23 | 12, 15, 19, 22 | syl3anc 1373 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ ((RSpan‘𝑅)‘𝐼)) |
24 | 6, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
25 | eqid 2734 | . . . . 5 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
26 | 25, 2 | rspidlid 31256 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝐵) → ((RSpan‘𝑅)‘𝐼) = 𝐼) |
27 | 24, 7, 26 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘𝐼) = 𝐼) |
28 | 23, 27 | sseqtrd 3931 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ 𝐼) |
29 | 9, 28 | eqsstrd 3929 | 1 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 .rcmulr 16768 LSSumclsm 18995 mulGrpcmgp 19476 Ringcrg 19534 CRingccrg 19535 LModclmod 19871 ringLModcrglmod 20178 LIdealclidl 20179 RSpancrsp 20180 IDLsrgcidlsrg 31331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-lsm 18997 df-cmn 19144 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-subrg 19770 df-lmod 19873 df-lss 19941 df-lsp 19981 df-sra 20181 df-rgmod 20182 df-lidl 20183 df-rsp 20184 df-idlsrg 31332 |
This theorem is referenced by: idlsrgmulrssin 31344 zarclsun 31506 |
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