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Mirrors > Home > MPE Home > Th. List > crngunit | Structured version Visualization version GIF version |
Description: Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.) |
Ref | Expression |
---|---|
crngunit.1 | ⊢ 𝑈 = (Unit‘𝑅) |
crngunit.2 | ⊢ 1 = (1r‘𝑅) |
crngunit.3 | ⊢ ∥ = (∥r‘𝑅) |
Ref | Expression |
---|---|
crngunit | ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2736 | . . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2736 | . . . . . . . . . . 11 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
4 | eqid 2736 | . . . . . . . . . . 11 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)) | |
5 | 1, 2, 3, 4 | crngoppr 19599 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋)) |
6 | 5 | 3expa 1120 | . . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋)) |
7 | 6 | eqcomd 2742 | . . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘(oppr‘𝑅))𝑋) = (𝑦(.r‘𝑅)𝑋)) |
8 | 7 | an32s 652 | . . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘(oppr‘𝑅))𝑋) = (𝑦(.r‘𝑅)𝑋)) |
9 | 8 | eqeq1d 2738 | . . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ (𝑦(.r‘𝑅)𝑋) = 1 )) |
10 | 9 | rexbidva 3205 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) → (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑋) = 1 )) |
11 | 10 | pm5.32da 582 | . . . 4 ⊢ (𝑅 ∈ CRing → ((𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ) ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑋) = 1 ))) |
12 | 3, 1 | opprbas 19601 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
13 | eqid 2736 | . . . . 5 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
14 | 12, 13, 4 | dvdsr 19618 | . . . 4 ⊢ (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 )) |
15 | crngunit.3 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
16 | 1, 15, 2 | dvdsr 19618 | . . . 4 ⊢ (𝑋 ∥ 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑋) = 1 )) |
17 | 11, 14, 16 | 3bitr4g 317 | . . 3 ⊢ (𝑅 ∈ CRing → (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ 𝑋 ∥ 1 )) |
18 | 17 | anbi2d 632 | . 2 ⊢ (𝑅 ∈ CRing → ((𝑋 ∥ 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 ) ↔ (𝑋 ∥ 1 ∧ 𝑋 ∥ 1 ))) |
19 | crngunit.1 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
20 | crngunit.2 | . . 3 ⊢ 1 = (1r‘𝑅) | |
21 | 19, 20, 15, 3, 13 | isunit 19629 | . 2 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 )) |
22 | pm4.24 567 | . 2 ⊢ (𝑋 ∥ 1 ↔ (𝑋 ∥ 1 ∧ 𝑋 ∥ 1 )) | |
23 | 18, 21, 22 | 3bitr4g 317 | 1 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 .rcmulr 16750 1rcur 19470 CRingccrg 19517 opprcoppr 19594 ∥rcdsr 19610 Unitcui 19611 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 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 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-mulr 16763 df-cmn 19126 df-mgp 19459 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 |
This theorem is referenced by: dvdsunit 19635 znunit 20482 matunitlindflem2 35460 |
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