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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 2726 | . . . . . . . . . . 11 β’ (Baseβπ ) = (Baseβπ ) | |
2 | eqid 2726 | . . . . . . . . . . 11 β’ (.rβπ ) = (.rβπ ) | |
3 | eqid 2726 | . . . . . . . . . . 11 β’ (opprβπ ) = (opprβπ ) | |
4 | eqid 2726 | . . . . . . . . . . 11 β’ (.rβ(opprβπ )) = (.rβ(opprβπ )) | |
5 | 1, 2, 3, 4 | crngoppr 20240 | . . . . . . . . . 10 β’ ((π β CRing β§ π¦ β (Baseβπ ) β§ π β (Baseβπ )) β (π¦(.rβπ )π) = (π¦(.rβ(opprβπ ))π)) |
6 | 5 | 3expa 1115 | . . . . . . . . 9 β’ (((π β CRing β§ π¦ β (Baseβπ )) β§ π β (Baseβπ )) β (π¦(.rβπ )π) = (π¦(.rβ(opprβπ ))π)) |
7 | 6 | eqcomd 2732 | . . . . . . . 8 β’ (((π β CRing β§ π¦ β (Baseβπ )) β§ π β (Baseβπ )) β (π¦(.rβ(opprβπ ))π) = (π¦(.rβπ )π)) |
8 | 7 | an32s 649 | . . . . . . 7 β’ (((π β CRing β§ π β (Baseβπ )) β§ π¦ β (Baseβπ )) β (π¦(.rβ(opprβπ ))π) = (π¦(.rβπ )π)) |
9 | 8 | eqeq1d 2728 | . . . . . 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 3170 | . . . . 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 578 | . . . 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 20243 | . . . . 5 β’ (Baseβπ ) = (Baseβ(opprβπ )) |
13 | eqid 2726 | . . . . 5 β’ (β₯rβ(opprβπ )) = (β₯rβ(opprβπ )) | |
14 | 12, 13, 4 | dvdsr 20264 | . . . 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 20264 | . . . 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 628 | . 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 20275 | . 2 β’ (π β π β (π β₯ 1 β§ π(β₯rβ(opprβπ )) 1 )) |
22 | pm4.24 563 | . 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 395 = wceq 1533 β wcel 2098 βwrex 3064 class class class wbr 5141 βcfv 6537 (class class class)co 7405 Basecbs 17153 .rcmulr 17207 1rcur 20086 CRingccrg 20139 opprcoppr 20235 β₯rcdsr 20256 Unitcui 20257 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-cmn 19702 df-mgp 20040 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 |
This theorem is referenced by: dvdsunit 20281 znunit 21458 matunitlindflem2 36998 |
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