Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lidl1el | Structured version Visualization version GIF version |
Description: An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.) |
Ref | Expression |
---|---|
lidlcl.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
lidlcl.b | ⊢ 𝐵 = (Base‘𝑅) |
lidl1el.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
lidl1el | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ( 1 ∈ 𝐼 ↔ 𝐼 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
2 | lidlcl.u | . . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅) | |
3 | 1, 2 | lidlss 20248 | . . . . 5 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ 𝐵) |
4 | 3 | ad2antlr 727 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 1 ∈ 𝐼) → 𝐼 ⊆ 𝐵) |
5 | eqid 2737 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | lidl1el.o | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
7 | 1, 5, 6 | ringridm 19590 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎(.r‘𝑅) 1 ) = 𝑎) |
8 | 7 | ad2ant2rl 749 | . . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ( 1 ∈ 𝐼 ∧ 𝑎 ∈ 𝐵)) → (𝑎(.r‘𝑅) 1 ) = 𝑎) |
9 | 2, 1, 5 | lidlmcl 20255 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑎 ∈ 𝐵 ∧ 1 ∈ 𝐼)) → (𝑎(.r‘𝑅) 1 ) ∈ 𝐼) |
10 | 9 | ancom2s 650 | . . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ( 1 ∈ 𝐼 ∧ 𝑎 ∈ 𝐵)) → (𝑎(.r‘𝑅) 1 ) ∈ 𝐼) |
11 | 8, 10 | eqeltrrd 2839 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ( 1 ∈ 𝐼 ∧ 𝑎 ∈ 𝐵)) → 𝑎 ∈ 𝐼) |
12 | 11 | expr 460 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 1 ∈ 𝐼) → (𝑎 ∈ 𝐵 → 𝑎 ∈ 𝐼)) |
13 | 12 | ssrdv 3907 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 1 ∈ 𝐼) → 𝐵 ⊆ 𝐼) |
14 | 4, 13 | eqssd 3918 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 1 ∈ 𝐼) → 𝐼 = 𝐵) |
15 | 14 | ex 416 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ( 1 ∈ 𝐼 → 𝐼 = 𝐵)) |
16 | 1, 6 | ringidcl 19586 | . . . 4 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) |
17 | 16 | adantr 484 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → 1 ∈ 𝐵) |
18 | eleq2 2826 | . . 3 ⊢ (𝐼 = 𝐵 → ( 1 ∈ 𝐼 ↔ 1 ∈ 𝐵)) | |
19 | 17, 18 | syl5ibrcom 250 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐼 = 𝐵 → 1 ∈ 𝐼)) |
20 | 15, 19 | impbid 215 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ( 1 ∈ 𝐼 ↔ 𝐼 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 .rcmulr 16803 1rcur 19516 Ringcrg 19562 LIdealclidl 20207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-mgp 19505 df-ur 19517 df-ring 19564 df-subrg 19798 df-lmod 19901 df-lss 19969 df-sra 20209 df-rgmod 20210 df-lidl 20211 |
This theorem is referenced by: rsp1 20262 drngnidl 20267 pridln1 31332 mxidln1 31352 ssmxidllem 31355 uzlidlring 45160 |
Copyright terms: Public domain | W3C validator |