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Mirrors > Home > MPE Home > Th. List > unitrinv | Structured version Visualization version GIF version |
Description: A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
unitinvcl.1 | β’ π = (Unitβπ ) |
unitinvcl.2 | β’ πΌ = (invrβπ ) |
unitinvcl.3 | β’ Β· = (.rβπ ) |
unitinvcl.4 | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
unitrinv | β’ ((π β Ring β§ π β π) β (π Β· (πΌβπ)) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitinvcl.1 | . . . 4 β’ π = (Unitβπ ) | |
2 | eqid 2728 | . . . 4 β’ ((mulGrpβπ ) βΎs π) = ((mulGrpβπ ) βΎs π) | |
3 | 1, 2 | unitgrp 20336 | . . 3 β’ (π β Ring β ((mulGrpβπ ) βΎs π) β Grp) |
4 | 1, 2 | unitgrpbas 20335 | . . . 4 β’ π = (Baseβ((mulGrpβπ ) βΎs π)) |
5 | 1 | fvexi 6916 | . . . . 5 β’ π β V |
6 | eqid 2728 | . . . . . . 7 β’ (mulGrpβπ ) = (mulGrpβπ ) | |
7 | unitinvcl.3 | . . . . . . 7 β’ Β· = (.rβπ ) | |
8 | 6, 7 | mgpplusg 20092 | . . . . . 6 β’ Β· = (+gβ(mulGrpβπ )) |
9 | 2, 8 | ressplusg 17280 | . . . . 5 β’ (π β V β Β· = (+gβ((mulGrpβπ ) βΎs π))) |
10 | 5, 9 | ax-mp 5 | . . . 4 β’ Β· = (+gβ((mulGrpβπ ) βΎs π)) |
11 | eqid 2728 | . . . 4 β’ (0gβ((mulGrpβπ ) βΎs π)) = (0gβ((mulGrpβπ ) βΎs π)) | |
12 | unitinvcl.2 | . . . . 5 β’ πΌ = (invrβπ ) | |
13 | 1, 2, 12 | invrfval 20342 | . . . 4 β’ πΌ = (invgβ((mulGrpβπ ) βΎs π)) |
14 | 4, 10, 11, 13 | grprinv 18961 | . . 3 β’ ((((mulGrpβπ ) βΎs π) β Grp β§ π β π) β (π Β· (πΌβπ)) = (0gβ((mulGrpβπ ) βΎs π))) |
15 | 3, 14 | sylan 578 | . 2 β’ ((π β Ring β§ π β π) β (π Β· (πΌβπ)) = (0gβ((mulGrpβπ ) βΎs π))) |
16 | unitinvcl.4 | . . . 4 β’ 1 = (1rβπ ) | |
17 | 1, 2, 16 | unitgrpid 20338 | . . 3 β’ (π β Ring β 1 = (0gβ((mulGrpβπ ) βΎs π))) |
18 | 17 | adantr 479 | . 2 β’ ((π β Ring β§ π β π) β 1 = (0gβ((mulGrpβπ ) βΎs π))) |
19 | 15, 18 | eqtr4d 2771 | 1 β’ ((π β Ring β§ π β π) β (π Β· (πΌβπ)) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 βcfv 6553 (class class class)co 7426 βΎs cress 17218 +gcplusg 17242 .rcmulr 17243 0gc0g 17430 Grpcgrp 18904 mulGrpcmgp 20088 1rcur 20135 Ringcrg 20187 Unitcui 20308 invrcinvr 20340 |
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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 7879 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 |
This theorem is referenced by: 1rinv 20348 0unit 20349 dvrid 20359 subrguss 20540 subrginv 20541 subrgunit 20543 drnginvrr 20664 matunit 22608 slesolinvbi 22611 nminvr 24614 nrginvrcnlem 24636 ply1divalg 26101 dchrn0 27211 dvrcan5 32972 fldhmf1 41601 invginvrid 47527 |
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