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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 2735 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | eqid 2735 | . . 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 33517 | . 2 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽))) |
10 | rlmlmod 21228 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
12 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | 12, 2 | lidlss 21240 | . . . . 5 ⊢ (𝐽 ∈ 𝐵 → 𝐽 ⊆ (Base‘𝑅)) |
14 | 8, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑅)) |
15 | 12, 2 | lidlss 21240 | . . . . . 6 ⊢ (𝐼 ∈ 𝐵 → 𝐼 ⊆ (Base‘𝑅)) |
16 | 7, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑅)) |
17 | 8, 2 | eleqtrdi 2849 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅)) |
18 | 12, 4, 5, 6, 16, 17 | ringlsmss2 33405 | . . . 4 ⊢ (𝜑 → (𝐼(LSSum‘(mulGrp‘𝑅))𝐽) ⊆ 𝐽) |
19 | rlmbas 21218 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
20 | rspval 21239 | . . . . 5 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
21 | 19, 20 | lspss 21000 | . . . 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 1370 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ ((RSpan‘𝑅)‘𝐽)) |
23 | eqid 2735 | . . . . 5 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
24 | 23, 2 | rspidlid 33383 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ 𝐵) → ((RSpan‘𝑅)‘𝐽) = 𝐽) |
25 | 6, 8, 24 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((RSpan‘𝑅)‘𝐽) = 𝐽) |
26 | 22, 25 | sseqtrd 4036 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼(LSSum‘(mulGrp‘𝑅))𝐽)) ⊆ 𝐽) |
27 | 9, 26 | eqsstrd 4034 | 1 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 LSSumclsm 19667 mulGrpcmgp 20152 Ringcrg 20251 LModclmod 20875 ringLModcrglmod 21189 LIdealclidl 21234 RSpancrsp 21235 IDLsrgcidlsrg 33508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-idlsrg 33509 |
This theorem is referenced by: idlsrgmulrssin 33521 zarclsun 33831 |
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