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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 20092 | . 2 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
2 | baseid 16646 | . . 3 ⊢ Base = Slot (Base‘ndx) | |
3 | islidl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | strfvi 16640 | . 2 ⊢ 𝐵 = (Base‘( I ‘𝑅)) |
5 | rlmbas 20086 | . . 3 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
6 | 3, 5 | eqtri 2761 | . 2 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
7 | islidl.p | . . 3 ⊢ + = (+g‘𝑅) | |
8 | rlmplusg 20087 | . . 3 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
9 | 7, 8 | eqtri 2761 | . 2 ⊢ + = (+g‘(ringLMod‘𝑅)) |
10 | islidl.t | . . 3 ⊢ · = (.r‘𝑅) | |
11 | rlmvsca 20093 | . . 3 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | 10, 11 | eqtri 2761 | . 2 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
13 | islidl.s | . . 3 ⊢ 𝑈 = (LIdeal‘𝑅) | |
14 | lidlval 20083 | . . 3 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
15 | 13, 14 | eqtri 2761 | . 2 ⊢ 𝑈 = (LSubSp‘(ringLMod‘𝑅)) |
16 | 1, 4, 6, 9, 12, 15 | islss 19825 | 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 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ⊆ wss 3843 ∅c0 4211 I cid 5428 ‘cfv 6339 (class class class)co 7170 ndxcnx 16583 Basecbs 16586 +gcplusg 16668 .rcmulr 16669 ·𝑠 cvsca 16672 LSubSpclss 19822 ringLModcrglmod 20060 LIdealclidl 20061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-sca 16684 df-vsca 16685 df-ip 16686 df-lss 19823 df-sra 20063 df-rgmod 20064 df-lidl 20065 |
This theorem is referenced by: intlidl 31174 rhmpreimaidl 31175 idlinsubrg 31180 rhmimaidl 31181 ssmxidllem 31213 hbtlem2 40521 2zlidl 45026 |
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