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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringlsmss2 | Structured version Visualization version GIF version |
Description: The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
Ref | Expression |
---|---|
ringlsmss.1 | β’ π΅ = (Baseβπ ) |
ringlsmss.2 | β’ πΊ = (mulGrpβπ ) |
ringlsmss.3 | β’ Γ = (LSSumβπΊ) |
ringlsmss2.1 | β’ (π β π β Ring) |
ringlsmss2.2 | β’ (π β πΈ β π΅) |
ringlsmss2.3 | β’ (π β πΌ β (LIdealβπ )) |
Ref | Expression |
---|---|
ringlsmss2 | β’ (π β (πΈ Γ πΌ) β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . 5 β’ (((((π β§ π β (πΈ Γ πΌ)) β§ π β πΈ) β§ π β πΌ) β§ π = (π(.rβπ )π)) β π = (π(.rβπ )π)) | |
2 | ringlsmss2.1 | . . . . . . . . 9 β’ (π β π β Ring) | |
3 | 2 | ad2antrr 724 | . . . . . . . 8 β’ (((π β§ π β πΈ) β§ π β πΌ) β π β Ring) |
4 | ringlsmss2.3 | . . . . . . . . 9 β’ (π β πΌ β (LIdealβπ )) | |
5 | 4 | ad2antrr 724 | . . . . . . . 8 β’ (((π β§ π β πΈ) β§ π β πΌ) β πΌ β (LIdealβπ )) |
6 | ringlsmss2.2 | . . . . . . . . . 10 β’ (π β πΈ β π΅) | |
7 | 6 | sselda 3981 | . . . . . . . . 9 β’ ((π β§ π β πΈ) β π β π΅) |
8 | 7 | adantr 481 | . . . . . . . 8 β’ (((π β§ π β πΈ) β§ π β πΌ) β π β π΅) |
9 | simpr 485 | . . . . . . . 8 β’ (((π β§ π β πΈ) β§ π β πΌ) β π β πΌ) | |
10 | eqid 2732 | . . . . . . . . 9 β’ (LIdealβπ ) = (LIdealβπ ) | |
11 | ringlsmss.1 | . . . . . . . . 9 β’ π΅ = (Baseβπ ) | |
12 | eqid 2732 | . . . . . . . . 9 β’ (.rβπ ) = (.rβπ ) | |
13 | 10, 11, 12 | lidlmcl 20832 | . . . . . . . 8 β’ (((π β Ring β§ πΌ β (LIdealβπ )) β§ (π β π΅ β§ π β πΌ)) β (π(.rβπ )π) β πΌ) |
14 | 3, 5, 8, 9, 13 | syl22anc 837 | . . . . . . 7 β’ (((π β§ π β πΈ) β§ π β πΌ) β (π(.rβπ )π) β πΌ) |
15 | 14 | adantllr 717 | . . . . . 6 β’ ((((π β§ π β (πΈ Γ πΌ)) β§ π β πΈ) β§ π β πΌ) β (π(.rβπ )π) β πΌ) |
16 | 15 | adantr 481 | . . . . 5 β’ (((((π β§ π β (πΈ Γ πΌ)) β§ π β πΈ) β§ π β πΌ) β§ π = (π(.rβπ )π)) β (π(.rβπ )π) β πΌ) |
17 | 1, 16 | eqeltrd 2833 | . . . 4 β’ (((((π β§ π β (πΈ Γ πΌ)) β§ π β πΈ) β§ π β πΌ) β§ π = (π(.rβπ )π)) β π β πΌ) |
18 | ringlsmss.2 | . . . . . 6 β’ πΊ = (mulGrpβπ ) | |
19 | ringlsmss.3 | . . . . . 6 β’ Γ = (LSSumβπΊ) | |
20 | 11, 10 | lidlss 20825 | . . . . . . 7 β’ (πΌ β (LIdealβπ ) β πΌ β π΅) |
21 | 4, 20 | syl 17 | . . . . . 6 β’ (π β πΌ β π΅) |
22 | 11, 12, 18, 19, 6, 21 | elringlsm 32491 | . . . . 5 β’ (π β (π β (πΈ Γ πΌ) β βπ β πΈ βπ β πΌ π = (π(.rβπ )π))) |
23 | 22 | biimpa 477 | . . . 4 β’ ((π β§ π β (πΈ Γ πΌ)) β βπ β πΈ βπ β πΌ π = (π(.rβπ )π)) |
24 | 17, 23 | r19.29vva 3213 | . . 3 β’ ((π β§ π β (πΈ Γ πΌ)) β π β πΌ) |
25 | 24 | ex 413 | . 2 β’ (π β (π β (πΈ Γ πΌ) β π β πΌ)) |
26 | 25 | ssrdv 3987 | 1 β’ (π β (πΈ Γ πΌ) β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwrex 3070 β wss 3947 βcfv 6540 (class class class)co 7405 Basecbs 17140 .rcmulr 17194 LSSumclsm 19496 mulGrpcmgp 19981 Ringcrg 20049 LIdealclidl 20775 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-lsm 19498 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-lidl 20779 |
This theorem is referenced by: idlsrgmulrss2 32614 |
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