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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsrgmulrss2 | Structured version Visualization version GIF version |
Description: The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgmulrss2.1 | β’ π = (IDLsrgβπ ) |
idlsrgmulrss2.2 | β’ π΅ = (LIdealβπ ) |
idlsrgmulrss2.3 | β’ β = (.rβπ) |
idlsrgmulrss2.5 | β’ Β· = (.rβπ ) |
idlsrgmulrss2.6 | β’ (π β π β Ring) |
idlsrgmulrss2.7 | β’ (π β πΌ β π΅) |
idlsrgmulrss2.8 | β’ (π β π½ β π΅) |
Ref | Expression |
---|---|
idlsrgmulrss2 | β’ (π β (πΌ β π½) β π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrss2.1 | . . 3 β’ π = (IDLsrgβπ ) | |
2 | idlsrgmulrss2.2 | . . 3 β’ π΅ = (LIdealβπ ) | |
3 | idlsrgmulrss2.3 | . . 3 β’ β = (.rβπ) | |
4 | eqid 2725 | . . 3 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
5 | eqid 2725 | . . 3 β’ (LSSumβ(mulGrpβπ )) = (LSSumβ(mulGrpβπ )) | |
6 | idlsrgmulrss2.6 | . . 3 β’ (π β π β Ring) | |
7 | idlsrgmulrss2.7 | . . 3 β’ (π β πΌ β π΅) | |
8 | idlsrgmulrss2.8 | . . 3 β’ (π β π½ β π΅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | idlsrgmulrval 33269 | . 2 β’ (π β (πΌ β π½) = ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½))) |
10 | rlmlmod 21100 | . . . . 5 β’ (π β Ring β (ringLModβπ ) β LMod) | |
11 | 6, 10 | syl 17 | . . . 4 β’ (π β (ringLModβπ ) β LMod) |
12 | eqid 2725 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
13 | 12, 2 | lidlss 21112 | . . . . 5 β’ (π½ β π΅ β π½ β (Baseβπ )) |
14 | 8, 13 | syl 17 | . . . 4 β’ (π β π½ β (Baseβπ )) |
15 | 12, 2 | lidlss 21112 | . . . . . 6 β’ (πΌ β π΅ β πΌ β (Baseβπ )) |
16 | 7, 15 | syl 17 | . . . . 5 β’ (π β πΌ β (Baseβπ )) |
17 | 8, 2 | eleqtrdi 2835 | . . . . 5 β’ (π β π½ β (LIdealβπ )) |
18 | 12, 4, 5, 6, 16, 17 | ringlsmss2 33155 | . . . 4 β’ (π β (πΌ(LSSumβ(mulGrpβπ ))π½) β π½) |
19 | rlmbas 21090 | . . . . 5 β’ (Baseβπ ) = (Baseβ(ringLModβπ )) | |
20 | rspval 21111 | . . . . 5 β’ (RSpanβπ ) = (LSpanβ(ringLModβπ )) | |
21 | 19, 20 | lspss 20872 | . . . 4 β’ (((ringLModβπ ) β LMod β§ π½ β (Baseβπ ) β§ (πΌ(LSSumβ(mulGrpβπ ))π½) β π½) β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β ((RSpanβπ )βπ½)) |
22 | 11, 14, 18, 21 | syl3anc 1368 | . . 3 β’ (π β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β ((RSpanβπ )βπ½)) |
23 | eqid 2725 | . . . . 5 β’ (RSpanβπ ) = (RSpanβπ ) | |
24 | 23, 2 | rspidlid 33135 | . . . 4 β’ ((π β Ring β§ π½ β π΅) β ((RSpanβπ )βπ½) = π½) |
25 | 6, 8, 24 | syl2anc 582 | . . 3 β’ (π β ((RSpanβπ )βπ½) = π½) |
26 | 22, 25 | sseqtrd 4013 | . 2 β’ (π β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β π½) |
27 | 9, 26 | eqsstrd 4011 | 1 β’ (π β (πΌ β π½) β π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3939 βcfv 6543 (class class class)co 7416 Basecbs 17179 .rcmulr 17233 LSSumclsm 19593 mulGrpcmgp 20078 Ringcrg 20177 LModclmod 20747 ringLModcrglmod 21061 LIdealclidl 21106 RSpancrsp 21107 IDLsrgcidlsrg 33260 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-lsm 19595 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-subrg 20512 df-lmod 20749 df-lss 20820 df-lsp 20860 df-sra 21062 df-rgmod 21063 df-lidl 21108 df-rsp 21109 df-idlsrg 33261 |
This theorem is referenced by: idlsrgmulrssin 33273 zarclsun 33528 |
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