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Mirrors > Home > MPE Home > Th. List > islidl | Structured version Visualization version GIF version |
Description: Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Ref | Expression |
---|---|
islidl.s | ⊢ 𝑈 = (LIdeal‘𝑅) |
islidl.b | ⊢ 𝐵 = (Base‘𝑅) |
islidl.p | ⊢ + = (+g‘𝑅) |
islidl.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
islidl | ⊢ (𝐼 ∈ 𝑈 ↔ (𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmsca2 19951 | . 2 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
2 | baseid 16521 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
3 | islidl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | strfvi 16515 | . 2 ⊢ 𝐵 = (Base‘( I ‘𝑅)) |
5 | rlmbas 19945 | . . 3 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
6 | 3, 5 | eqtri 2843 | . 2 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
7 | islidl.p | . . 3 ⊢ + = (+g‘𝑅) | |
8 | rlmplusg 19946 | . . 3 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
9 | 7, 8 | eqtri 2843 | . 2 ⊢ + = (+g‘(ringLMod‘𝑅)) |
10 | islidl.t | . . 3 ⊢ · = (.r‘𝑅) | |
11 | rlmvsca 19952 | . . 3 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | 10, 11 | eqtri 2843 | . 2 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
13 | islidl.s | . . 3 ⊢ 𝑈 = (LIdeal‘𝑅) | |
14 | lidlval 19942 | . . 3 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
15 | 13, 14 | eqtri 2843 | . 2 ⊢ 𝑈 = (LSubSp‘(ringLMod‘𝑅)) |
16 | 1, 4, 6, 9, 12, 15 | islss 19684 | 1 ⊢ (𝐼 ∈ 𝑈 ↔ (𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐼 ((𝑥 · 𝑎) + 𝑏) ∈ 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 ∀wral 3133 ⊆ wss 3919 ∅c0 4274 I cid 5440 ‘cfv 6336 (class class class)co 7137 ndxcnx 16458 Basecbs 16461 +gcplusg 16543 .rcmulr 16544 ·𝑠 cvsca 16547 LSubSpclss 19681 ringLModcrglmod 19919 LIdealclidl 19920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-ress 16469 df-plusg 16556 df-sca 16559 df-vsca 16560 df-ip 16561 df-lss 19682 df-sra 19922 df-rgmod 19923 df-lidl 19924 |
This theorem is referenced by: ssmxidllem 30983 hbtlem2 39811 2zlidl 44285 |
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