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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 | β’ π = (Unitβπ ) |
crngunit.2 | β’ 1 = (1rβπ ) |
crngunit.3 | β’ β₯ = (β₯rβπ ) |
Ref | Expression |
---|---|
crngunit | β’ (π β CRing β (π β π β π β₯ 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . . . . . . . 11 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2737 | . . . . . . . . . . 11 β’ (.rβπ ) = (.rβπ ) | |
3 | eqid 2737 | . . . . . . . . . . 11 β’ (opprβπ ) = (opprβπ ) | |
4 | eqid 2737 | . . . . . . . . . . 11 β’ (.rβ(opprβπ )) = (.rβ(opprβπ )) | |
5 | 1, 2, 3, 4 | crngoppr 20054 | . . . . . . . . . 10 β’ ((π β CRing β§ π¦ β (Baseβπ ) β§ π β (Baseβπ )) β (π¦(.rβπ )π) = (π¦(.rβ(opprβπ ))π)) |
6 | 5 | 3expa 1119 | . . . . . . . . 9 β’ (((π β CRing β§ π¦ β (Baseβπ )) β§ π β (Baseβπ )) β (π¦(.rβπ )π) = (π¦(.rβ(opprβπ ))π)) |
7 | 6 | eqcomd 2743 | . . . . . . . 8 β’ (((π β CRing β§ π¦ β (Baseβπ )) β§ π β (Baseβπ )) β (π¦(.rβ(opprβπ ))π) = (π¦(.rβπ )π)) |
8 | 7 | an32s 651 | . . . . . . 7 β’ (((π β CRing β§ π β (Baseβπ )) β§ π¦ β (Baseβπ )) β (π¦(.rβ(opprβπ ))π) = (π¦(.rβπ )π)) |
9 | 8 | eqeq1d 2739 | . . . . . 6 β’ (((π β CRing β§ π β (Baseβπ )) β§ π¦ β (Baseβπ )) β ((π¦(.rβ(opprβπ ))π) = 1 β (π¦(.rβπ )π) = 1 )) |
10 | 9 | rexbidva 3174 | . . . . 5 β’ ((π β CRing β§ π β (Baseβπ )) β (βπ¦ β (Baseβπ )(π¦(.rβ(opprβπ ))π) = 1 β βπ¦ β (Baseβπ )(π¦(.rβπ )π) = 1 )) |
11 | 10 | pm5.32da 580 | . . . 4 β’ (π β CRing β ((π β (Baseβπ ) β§ βπ¦ β (Baseβπ )(π¦(.rβ(opprβπ ))π) = 1 ) β (π β (Baseβπ ) β§ βπ¦ β (Baseβπ )(π¦(.rβπ )π) = 1 ))) |
12 | 3, 1 | opprbas 20057 | . . . . 5 β’ (Baseβπ ) = (Baseβ(opprβπ )) |
13 | eqid 2737 | . . . . 5 β’ (β₯rβ(opprβπ )) = (β₯rβ(opprβπ )) | |
14 | 12, 13, 4 | dvdsr 20076 | . . . 4 β’ (π(β₯rβ(opprβπ )) 1 β (π β (Baseβπ ) β§ βπ¦ β (Baseβπ )(π¦(.rβ(opprβπ ))π) = 1 )) |
15 | crngunit.3 | . . . . 5 β’ β₯ = (β₯rβπ ) | |
16 | 1, 15, 2 | dvdsr 20076 | . . . 4 β’ (π β₯ 1 β (π β (Baseβπ ) β§ βπ¦ β (Baseβπ )(π¦(.rβπ )π) = 1 )) |
17 | 11, 14, 16 | 3bitr4g 314 | . . 3 β’ (π β CRing β (π(β₯rβ(opprβπ )) 1 β π β₯ 1 )) |
18 | 17 | anbi2d 630 | . 2 β’ (π β CRing β ((π β₯ 1 β§ π(β₯rβ(opprβπ )) 1 ) β (π β₯ 1 β§ π β₯ 1 ))) |
19 | crngunit.1 | . . 3 β’ π = (Unitβπ ) | |
20 | crngunit.2 | . . 3 β’ 1 = (1rβπ ) | |
21 | 19, 20, 15, 3, 13 | isunit 20087 | . 2 β’ (π β π β (π β₯ 1 β§ π(β₯rβ(opprβπ )) 1 )) |
22 | pm4.24 565 | . 2 β’ (π β₯ 1 β (π β₯ 1 β§ π β₯ 1 )) | |
23 | 18, 21, 22 | 3bitr4g 314 | 1 β’ (π β CRing β (π β π β π β₯ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3074 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Basecbs 17084 .rcmulr 17135 1rcur 19914 CRingccrg 19966 opprcoppr 20049 β₯rcdsr 20068 Unitcui 20069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-cmn 19565 df-mgp 19898 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-unit 20072 |
This theorem is referenced by: dvdsunit 20093 znunit 20973 matunitlindflem2 36078 |
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