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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 | β’ π = (IDLsrgβπ ) |
idlsrgmulrss1.2 | β’ π΅ = (LIdealβπ ) |
idlsrgmulrss1.3 | β’ β = (.rβπ) |
idlsrgmulrss1.4 | β’ Β· = (.rβπ ) |
idlsrgmulrss1.5 | β’ (π β π β CRing) |
idlsrgmulrss1.6 | β’ (π β πΌ β π΅) |
idlsrgmulrss1.7 | β’ (π β π½ β π΅) |
Ref | Expression |
---|---|
idlsrgmulrss1 | β’ (π β (πΌ β π½) β πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrss1.1 | . . 3 β’ π = (IDLsrgβπ ) | |
2 | idlsrgmulrss1.2 | . . 3 β’ π΅ = (LIdealβπ ) | |
3 | idlsrgmulrss1.3 | . . 3 β’ β = (.rβπ) | |
4 | eqid 2733 | . . 3 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
5 | eqid 2733 | . . 3 β’ (LSSumβ(mulGrpβπ )) = (LSSumβ(mulGrpβπ )) | |
6 | idlsrgmulrss1.5 | . . 3 β’ (π β π β CRing) | |
7 | idlsrgmulrss1.6 | . . 3 β’ (π β πΌ β π΅) | |
8 | idlsrgmulrss1.7 | . . 3 β’ (π β π½ β π΅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | idlsrgmulrval 32306 | . 2 β’ (π β (πΌ β π½) = ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½))) |
10 | crngring 19984 | . . . . 5 β’ (π β CRing β π β Ring) | |
11 | rlmlmod 20719 | . . . . 5 β’ (π β Ring β (ringLModβπ ) β LMod) | |
12 | 6, 10, 11 | 3syl 18 | . . . 4 β’ (π β (ringLModβπ ) β LMod) |
13 | eqid 2733 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
14 | 13, 2 | lidlss 20725 | . . . . 5 β’ (πΌ β π΅ β πΌ β (Baseβπ )) |
15 | 7, 14 | syl 17 | . . . 4 β’ (π β πΌ β (Baseβπ )) |
16 | 13, 2 | lidlss 20725 | . . . . . 6 β’ (π½ β π΅ β π½ β (Baseβπ )) |
17 | 8, 16 | syl 17 | . . . . 5 β’ (π β π½ β (Baseβπ )) |
18 | 7, 2 | eleqtrdi 2844 | . . . . 5 β’ (π β πΌ β (LIdealβπ )) |
19 | 13, 4, 5, 6, 17, 18 | ringlsmss1 32232 | . . . 4 β’ (π β (πΌ(LSSumβ(mulGrpβπ ))π½) β πΌ) |
20 | rlmbas 20709 | . . . . 5 β’ (Baseβπ ) = (Baseβ(ringLModβπ )) | |
21 | rspval 20707 | . . . . 5 β’ (RSpanβπ ) = (LSpanβ(ringLModβπ )) | |
22 | 20, 21 | lspss 20489 | . . . 4 β’ (((ringLModβπ ) β LMod β§ πΌ β (Baseβπ ) β§ (πΌ(LSSumβ(mulGrpβπ ))π½) β πΌ) β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β ((RSpanβπ )βπΌ)) |
23 | 12, 15, 19, 22 | syl3anc 1372 | . . 3 β’ (π β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β ((RSpanβπ )βπΌ)) |
24 | 6, 10 | syl 17 | . . . 4 β’ (π β π β Ring) |
25 | eqid 2733 | . . . . 5 β’ (RSpanβπ ) = (RSpanβπ ) | |
26 | 25, 2 | rspidlid 32217 | . . . 4 β’ ((π β Ring β§ πΌ β π΅) β ((RSpanβπ )βπΌ) = πΌ) |
27 | 24, 7, 26 | syl2anc 585 | . . 3 β’ (π β ((RSpanβπ )βπΌ) = πΌ) |
28 | 23, 27 | sseqtrd 3988 | . 2 β’ (π β ((RSpanβπ )β(πΌ(LSSumβ(mulGrpβπ ))π½)) β πΌ) |
29 | 9, 28 | eqsstrd 3986 | 1 β’ (π β (πΌ β π½) β πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3914 βcfv 6500 (class class class)co 7361 Basecbs 17091 .rcmulr 17142 LSSumclsm 19424 mulGrpcmgp 19904 Ringcrg 19972 CRingccrg 19973 LModclmod 20365 ringLModcrglmod 20675 LIdealclidl 20676 RSpancrsp 20677 IDLsrgcidlsrg 32297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-subg 18933 df-lsm 19426 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-subrg 20262 df-lmod 20367 df-lss 20437 df-lsp 20477 df-sra 20678 df-rgmod 20679 df-lidl 20680 df-rsp 20681 df-idlsrg 32298 |
This theorem is referenced by: idlsrgmulrssin 32310 zarclsun 32515 |
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