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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsrgmulrcl | Structured version Visualization version GIF version |
Description: Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgmulrval.1 | ⊢ 𝑆 = (IDLsrg‘𝑅) |
idlsrgmulrval.2 | ⊢ 𝐵 = (LIdeal‘𝑅) |
idlsrgmulrval.3 | ⊢ ⊗ = (.r‘𝑆) |
idlsrgmulrcl.1 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
idlsrgmulrcl.2 | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
idlsrgmulrcl.3 | ⊢ (𝜑 → 𝐽 ∈ 𝐵) |
Ref | Expression |
---|---|
idlsrgmulrcl | ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrval.1 | . . 3 ⊢ 𝑆 = (IDLsrg‘𝑅) | |
2 | idlsrgmulrval.2 | . . 3 ⊢ 𝐵 = (LIdeal‘𝑅) | |
3 | idlsrgmulrval.3 | . . 3 ⊢ ⊗ = (.r‘𝑆) | |
4 | eqid 2728 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | eqid 2728 | . . 3 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅)) | |
6 | idlsrgmulrcl.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | idlsrgmulrcl.2 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
8 | idlsrgmulrcl.3 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | idlsrgmulrval 33215 | . 2 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽))) |
10 | eqid 2728 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | 10, 2 | lidlss 21102 | . . . . 5 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ⊆ (Base‘𝑅)) |
12 | 7, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑅)) |
13 | 10, 2 | lidlss 21102 | . . . . 5 ⊢ (𝐽 ∈ 𝐵 → 𝐽 ⊆ (Base‘𝑅)) |
14 | 8, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑅)) |
15 | 10, 4, 5, 6, 12, 14 | ringlsmss 33099 | . . 3 ⊢ (𝜑 → (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ (Base‘𝑅)) |
16 | eqid 2728 | . . . 4 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
17 | 16, 10, 2 | rspcl 21125 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ∈ 𝐵) |
18 | 6, 15, 17 | syl2anc 583 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ∈ 𝐵) |
19 | 9, 18 | eqeltrd 2829 | 1 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 .rcmulr 17228 LSSumclsm 19583 mulGrpcmgp 20068 Ringcrg 20167 LIdealclidl 21096 RSpancrsp 21097 IDLsrgcidlsrg 33206 |
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-tp 4630 df-op 4632 df-uni 4905 df-int 4946 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-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 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-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 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-tset 17246 df-ple 17247 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-mgp 20069 df-ur 20116 df-ring 20169 df-subrg 20502 df-lmod 20739 df-lss 20810 df-lsp 20850 df-sra 21052 df-rgmod 21053 df-lidl 21098 df-rsp 21099 df-idlsrg 33207 |
This theorem is referenced by: zarclsun 33466 |
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