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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 | ⊢ 𝑆 = (IDLsrg‘𝑅) |
idlsrgmulrss2.2 | ⊢ 𝐵 = (LIdeal‘𝑅) |
idlsrgmulrss2.3 | ⊢ ⊗ = (.r‘𝑆) |
idlsrgmulrss2.5 | ⊢ · = (.r‘𝑅) |
idlsrgmulrss2.6 | ⊢ (𝜑 → 𝑅 ∈ Ring) |
idlsrgmulrss2.7 | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
idlsrgmulrss2.8 | ⊢ (𝜑 → 𝐽 ∈ 𝐵) |
Ref | Expression |
---|---|
idlsrgmulrss2 | ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrss2.1 | . . 3 ⊢ 𝑆 = (IDLsrg‘𝑅) | |
2 | idlsrgmulrss2.2 | . . 3 ⊢ 𝐵 = (LIdeal‘𝑅) | |
3 | idlsrgmulrss2.3 | . . 3 ⊢ ⊗ = (.r‘𝑆) | |
4 | eqid 2733 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | eqid 2733 | . . 3 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅)) | |
6 | idlsrgmulrss2.6 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | idlsrgmulrss2.7 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
8 | idlsrgmulrss2.8 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | idlsrgmulrval 32575 | . 2 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽))) |
10 | rlmlmod 20814 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
12 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | 12, 2 | lidlss 20820 | . . . . 5 ⊢ (𝐽 ∈ 𝐵 → 𝐽 ⊆ (Base‘𝑅)) |
14 | 8, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑅)) |
15 | 12, 2 | lidlss 20820 | . . . . . 6 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ⊆ (Base‘𝑅)) |
16 | 7, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑅)) |
17 | 8, 2 | eleqtrdi 2844 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) |
18 | 12, 4, 5, 6, 16, 17 | ringlsmss2 32472 | . . . 4 ⊢ (𝜑 → (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ 𝐽) |
19 | rlmbas 20804 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
20 | rspval 20802 | . . . . 5 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
21 | 19, 20 | lspss 20583 | . . . 4 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐽 ⊆ (Base‘𝑅) ∧ (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ 𝐽) → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ ((RSpan‘𝑅)‘𝐽)) |
22 | 11, 14, 18, 21 | syl3anc 1372 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ ((RSpan‘𝑅)‘𝐽)) |
23 | eqid 2733 | . . . . 5 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
24 | 23, 2 | rspidlid 32456 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ 𝐵) → ((RSpan‘𝑅)‘𝐽) = 𝐽) |
25 | 6, 8, 24 | syl2anc 585 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘𝐽) = 𝐽) |
26 | 22, 25 | sseqtrd 4021 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ 𝐽) |
27 | 9, 26 | eqsstrd 4019 | 1 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⊆ wss 3947 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 .rcmulr 17194 LSSumclsm 19495 mulGrpcmgp 19979 Ringcrg 20047 LModclmod 20459 ringLModcrglmod 20770 LIdealclidl 20771 RSpancrsp 20772 IDLsrgcidlsrg 32566 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 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 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-tp 4632 df-op 4634 df-uni 4908 df-int 4950 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 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 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-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 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-tset 17212 df-ple 17213 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 19497 df-mgp 19980 df-ur 19997 df-ring 20049 df-subrg 20349 df-lmod 20461 df-lss 20531 df-lsp 20571 df-sra 20773 df-rgmod 20774 df-lidl 20775 df-rsp 20776 df-idlsrg 32567 |
This theorem is referenced by: idlsrgmulrssin 32579 zarclsun 32788 |
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