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Mirrors > Home > MPE Home > Th. List > rnglidl1 | Structured version Visualization version GIF version |
Description: The base set of every non-unital ring is an ideal. For unital rings, such ideals are called "unit ideals", see lidl1 21136. (Contributed by AV, 19-Feb-2025.) |
Ref | Expression |
---|---|
rnglidl0.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
rnglidl1.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
rnglidl1 | ⊢ (𝑅 ∈ Rng → 𝐵 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnglidl1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | eqimssi 4042 | . . 3 ⊢ 𝐵 ⊆ (Base‘𝑅) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑅 ∈ Rng → 𝐵 ⊆ (Base‘𝑅)) |
4 | rnggrp 20105 | . . 3 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
5 | 1 | grpbn0 18930 | . . 3 ⊢ (𝑅 ∈ Grp → 𝐵 ≠ ∅) |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑅 ∈ Rng → 𝐵 ≠ ∅) |
7 | eqid 2728 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
8 | 4 | adantr 479 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑅 ∈ Grp) |
9 | simpl 481 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑅 ∈ Rng) | |
10 | 1 | eqcomi 2737 | . . . . . . . . 9 ⊢ (Base‘𝑅) = 𝐵 |
11 | 10 | eleq2i 2821 | . . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ 𝐵) |
12 | 11 | biimpi 215 | . . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑅) → 𝑥 ∈ 𝐵) |
13 | 12 | 3ad2ant1 1130 | . . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑥 ∈ 𝐵) |
14 | 13 | adantl 480 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐵) |
15 | simpr2 1192 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵) | |
16 | eqid 2728 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
17 | 1, 16 | rngcl 20111 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐵) |
18 | 9, 14, 15, 17 | syl3anc 1368 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝐵) |
19 | simpr3 1193 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵) | |
20 | 1, 7, 8, 18, 19 | grpcld 18911 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ 𝐵) |
21 | 20 | ralrimivvva 3201 | . 2 ⊢ (𝑅 ∈ Rng → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ 𝐵) |
22 | rnglidl0.u | . . 3 ⊢ 𝑈 = (LIdeal‘𝑅) | |
23 | eqid 2728 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
24 | 22, 23, 7, 16 | islidl 21118 | . 2 ⊢ (𝐵 ∈ 𝑈 ↔ (𝐵 ⊆ (Base‘𝑅) ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)𝑧) ∈ 𝐵)) |
25 | 3, 6, 21, 24 | syl3anbrc 1340 | 1 ⊢ (𝑅 ∈ Rng → 𝐵 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ⊆ wss 3949 ∅c0 4326 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 Grpcgrp 18897 Rngcrng 20099 LIdealclidl 21109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-sca 17256 df-vsca 17257 df-ip 17258 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-abl 19745 df-mgp 20082 df-rng 20100 df-lss 20823 df-sra 21065 df-rgmod 21066 df-lidl 21111 |
This theorem is referenced by: lidl1 21136 |
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