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Mirrors > Home > MPE Home > Th. List > lidlmcl | Structured version Visualization version GIF version |
Description: An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.) |
Ref | Expression |
---|---|
lidlcl.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
lidlcl.b | ⊢ 𝐵 = (Base‘𝑅) |
lidlmcl.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
lidlmcl | ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlmcl.t | . . . 4 ⊢ · = (.r‘𝑅) | |
2 | rlmvsca 19570 | . . . 4 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
3 | 1, 2 | eqtri 2849 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
4 | 3 | oveqi 6923 | . 2 ⊢ (𝑋 · 𝑌) = (𝑋( ·𝑠 ‘(ringLMod‘𝑅))𝑌) |
5 | rlmlmod 19573 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
6 | 5 | ad2antrr 717 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (ringLMod‘𝑅) ∈ LMod) |
7 | simpr 479 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ 𝑈) | |
8 | lidlcl.u | . . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅) | |
9 | lidlval 19560 | . . . . . 6 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
10 | 8, 9 | eqtri 2849 | . . . . 5 ⊢ 𝑈 = (LSubSp‘(ringLMod‘𝑅)) |
11 | 7, 10 | syl6eleq 2916 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) |
12 | 11 | adantr 474 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) |
13 | lidlcl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
14 | rlmsca 19568 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
15 | 14 | fveq2d 6441 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
16 | 13, 15 | syl5eq 2873 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
17 | 16 | eleq2d 2892 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘(Scalar‘(ringLMod‘𝑅))))) |
18 | 17 | biimpa 470 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))) |
19 | 18 | ad2ant2r 753 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → 𝑋 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))) |
20 | simprr 789 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → 𝑌 ∈ 𝐼) | |
21 | eqid 2825 | . . . 4 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
22 | eqid 2825 | . . . 4 ⊢ ( ·𝑠 ‘(ringLMod‘𝑅)) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
23 | eqid 2825 | . . . 4 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
24 | eqid 2825 | . . . 4 ⊢ (LSubSp‘(ringLMod‘𝑅)) = (LSubSp‘(ringLMod‘𝑅)) | |
25 | 21, 22, 23, 24 | lssvscl 19321 | . . 3 ⊢ ((((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ (LSubSp‘(ringLMod‘𝑅))) ∧ (𝑋 ∈ (Base‘(Scalar‘(ringLMod‘𝑅))) ∧ 𝑌 ∈ 𝐼)) → (𝑋( ·𝑠 ‘(ringLMod‘𝑅))𝑌) ∈ 𝐼) |
26 | 6, 12, 19, 20, 25 | syl22anc 872 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋( ·𝑠 ‘(ringLMod‘𝑅))𝑌) ∈ 𝐼) |
27 | 4, 26 | syl5eqel 2910 | 1 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 .rcmulr 16313 Scalarcsca 16315 ·𝑠 cvsca 16316 Ringcrg 18908 LModclmod 19226 LSubSpclss 19295 ringLModcrglmod 19537 LIdealclidl 19538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-mgp 18851 df-ur 18863 df-ring 18910 df-subrg 19141 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-lidl 19542 |
This theorem is referenced by: lidl1el 19586 drngnidl 19597 2idlcpbl 19602 zringlpirlem3 20201 ig1peu 24337 ig1pdvds 24342 hbtlem2 38532 hbtlem4 38534 lidlmmgm 42786 |
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