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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 π are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgmulrval.1 | β’ π = (IDLsrgβπ ) |
idlsrgmulrval.2 | β’ π΅ = (LIdealβπ ) |
idlsrgmulrval.3 | β’ β = (.rβπ) |
idlsrgmulrcl.1 | β’ (π β π β Ring) |
idlsrgmulrcl.2 | β’ (π β πΌ β π΅) |
idlsrgmulrcl.3 | β’ (π β π½ β π΅) |
Ref | Expression |
---|---|
idlsrgmulrcl | β’ (π β (πΌ β π½) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrval.1 | . . 3 β’ π = (IDLsrgβπ ) | |
2 | idlsrgmulrval.2 | . . 3 β’ π΅ = (LIdealβπ ) | |
3 | idlsrgmulrval.3 | . . 3 β’ β = (.rβπ) | |
4 | eqid 2725 | . . 3 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
5 | eqid 2725 | . . 3 β’ (LSSumβ(mulGrpβπ )) = (LSSumβ(mulGrpβπ )) | |
6 | idlsrgmulrcl.1 | . . 3 β’ (π β π β Ring) | |
7 | idlsrgmulrcl.2 | . . 3 β’ (π β πΌ β π΅) | |
8 | idlsrgmulrcl.3 | . . 3 β’ (π β π½ β π΅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | idlsrgmulrval 33241 | . 2 β’ (π β (πΌ β π½) = ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½))) |
10 | eqid 2725 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
11 | 10, 2 | lidlss 21110 | . . . . 5 β’ (πΌ β π΅ β πΌ β (Baseβπ )) |
12 | 7, 11 | syl 17 | . . . 4 β’ (π β πΌ β (Baseβπ )) |
13 | 10, 2 | lidlss 21110 | . . . . 5 β’ (π½ β π΅ β π½ β (Baseβπ )) |
14 | 8, 13 | syl 17 | . . . 4 β’ (π β π½ β (Baseβπ )) |
15 | 10, 4, 5, 6, 12, 14 | ringlsmss 33125 | . . 3 β’ (π β (πΌ(LSSumβ(mulGrpβπ ))π½) β (Baseβπ )) |
16 | eqid 2725 | . . . 4 β’ (RSpanβπ ) = (RSpanβπ ) | |
17 | 16, 10, 2 | rspcl 21133 | . . 3 β’ ((π β Ring β§ (πΌ(LSSumβ(mulGrpβπ ))π½) β (Baseβπ )) β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β π΅) |
18 | 6, 15, 17 | syl2anc 582 | . 2 β’ (π β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β π΅) |
19 | 9, 18 | eqeltrd 2825 | 1 β’ (π β (πΌ β π½) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3939 βcfv 6541 (class class class)co 7414 Basecbs 17177 .rcmulr 17231 LSSumclsm 19591 mulGrpcmgp 20076 Ringcrg 20175 LIdealclidl 21104 RSpancrsp 21105 IDLsrgcidlsrg 33232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-lsm 19593 df-mgp 20077 df-ur 20124 df-ring 20177 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-rsp 21107 df-idlsrg 33233 |
This theorem is referenced by: zarclsun 33500 |
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