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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsmidl | Structured version Visualization version GIF version |
Description: The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
Ref | Expression |
---|---|
lsmidl.1 | โข ๐ต = (Baseโ๐ ) |
lsmidl.3 | โข โ = (LSSumโ๐ ) |
lsmidl.4 | โข ๐พ = (RSpanโ๐ ) |
lsmidl.5 | โข (๐ โ ๐ โ Ring) |
lsmidl.6 | โข (๐ โ ๐ผ โ (LIdealโ๐ )) |
lsmidl.7 | โข (๐ โ ๐ฝ โ (LIdealโ๐ )) |
Ref | Expression |
---|---|
lsmidl | โข (๐ โ (๐ผ โ ๐ฝ) โ (LIdealโ๐ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmidl.1 | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | lsmidl.3 | . . 3 โข โ = (LSSumโ๐ ) | |
3 | lsmidl.4 | . . 3 โข ๐พ = (RSpanโ๐ ) | |
4 | lsmidl.5 | . . 3 โข (๐ โ ๐ โ Ring) | |
5 | lsmidl.6 | . . 3 โข (๐ โ ๐ผ โ (LIdealโ๐ )) | |
6 | lsmidl.7 | . . 3 โข (๐ โ ๐ฝ โ (LIdealโ๐ )) | |
7 | 1, 2, 3, 4, 5, 6 | lsmidllsp 32229 | . 2 โข (๐ โ (๐ผ โ ๐ฝ) = (๐พโ(๐ผ โช ๐ฝ))) |
8 | rlmlmod 20690 | . . . 4 โข (๐ โ Ring โ (ringLModโ๐ ) โ LMod) | |
9 | 4, 8 | syl 17 | . . 3 โข (๐ โ (ringLModโ๐ ) โ LMod) |
10 | eqid 2733 | . . . . . 6 โข (LIdealโ๐ ) = (LIdealโ๐ ) | |
11 | 1, 10 | lidlss 20696 | . . . . 5 โข (๐ผ โ (LIdealโ๐ ) โ ๐ผ โ ๐ต) |
12 | 5, 11 | syl 17 | . . . 4 โข (๐ โ ๐ผ โ ๐ต) |
13 | 1, 10 | lidlss 20696 | . . . . 5 โข (๐ฝ โ (LIdealโ๐ ) โ ๐ฝ โ ๐ต) |
14 | 6, 13 | syl 17 | . . . 4 โข (๐ โ ๐ฝ โ ๐ต) |
15 | 12, 14 | unssd 4147 | . . 3 โข (๐ โ (๐ผ โช ๐ฝ) โ ๐ต) |
16 | rlmbas 20680 | . . . . 5 โข (Baseโ๐ ) = (Baseโ(ringLModโ๐ )) | |
17 | 1, 16 | eqtri 2761 | . . . 4 โข ๐ต = (Baseโ(ringLModโ๐ )) |
18 | lidlval 20677 | . . . 4 โข (LIdealโ๐ ) = (LSubSpโ(ringLModโ๐ )) | |
19 | rspval 20678 | . . . . 5 โข (RSpanโ๐ ) = (LSpanโ(ringLModโ๐ )) | |
20 | 3, 19 | eqtri 2761 | . . . 4 โข ๐พ = (LSpanโ(ringLModโ๐ )) |
21 | 17, 18, 20 | lspcl 20452 | . . 3 โข (((ringLModโ๐ ) โ LMod โง (๐ผ โช ๐ฝ) โ ๐ต) โ (๐พโ(๐ผ โช ๐ฝ)) โ (LIdealโ๐ )) |
22 | 9, 15, 21 | syl2anc 585 | . 2 โข (๐ โ (๐พโ(๐ผ โช ๐ฝ)) โ (LIdealโ๐ )) |
23 | 7, 22 | eqeltrd 2834 | 1 โข (๐ โ (๐ผ โ ๐ฝ) โ (LIdealโ๐ )) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โช cun 3909 โ wss 3911 โcfv 6497 (class class class)co 7358 Basecbs 17088 LSSumclsm 19421 Ringcrg 19969 LModclmod 20336 LSpanclspn 20447 ringLModcrglmod 20646 LIdealclidl 20647 RSpancrsp 20648 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-cntz 19102 df-lsm 19423 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-subrg 20234 df-lmod 20338 df-lss 20408 df-lsp 20448 df-sra 20649 df-rgmod 20650 df-lidl 20651 df-rsp 20652 |
This theorem is referenced by: mxidlprm 32285 idlsrgmnd 32304 |
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