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Theorem ringidmlem 13158
Description: Lemma for ringlidm 13159 and ringridm 13160. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
rngidm.b  |-  B  =  ( Base `  R
)
rngidm.t  |-  .x.  =  ( .r `  R )
rngidm.u  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
ringidmlem  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .1.  .x.  X
)  =  X  /\  ( X  .x.  .1.  )  =  X ) )

Proof of Theorem ringidmlem
StepHypRef Expression
1 eqid 2177 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
21ringmgp 13138 . . 3  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
3 simpr 110 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  B )
4 rngidm.b . . . . . 6  |-  B  =  ( Base `  R
)
51, 4mgpbasg 13089 . . . . 5  |-  ( R  e.  Ring  ->  B  =  ( Base `  (mulGrp `  R ) ) )
65adantr 276 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  B  =  ( Base `  (mulGrp `  R ) ) )
73, 6eleqtrd 2256 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  e.  ( Base `  (mulGrp `  R ) ) )
8 eqid 2177 . . . 4  |-  ( Base `  (mulGrp `  R )
)  =  ( Base `  (mulGrp `  R )
)
9 eqid 2177 . . . 4  |-  ( +g  `  (mulGrp `  R )
)  =  ( +g  `  (mulGrp `  R )
)
10 eqid 2177 . . . 4  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
118, 9, 10mndlrid 12789 . . 3  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  X  e.  ( Base `  (mulGrp `  R ) ) )  ->  ( ( ( 0g `  (mulGrp `  R ) ) ( +g  `  (mulGrp `  R ) ) X )  =  X  /\  ( X ( +g  `  (mulGrp `  R ) ) ( 0g `  (mulGrp `  R ) ) )  =  X ) )
122, 7, 11syl2an2r 595 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( ( 0g `  (mulGrp `  R ) ) ( +g  `  (mulGrp `  R ) ) X )  =  X  /\  ( X ( +g  `  (mulGrp `  R ) ) ( 0g `  (mulGrp `  R ) ) )  =  X ) )
13 rngidm.t . . . . . . 7  |-  .x.  =  ( .r `  R )
141, 13mgpplusgg 13087 . . . . . 6  |-  ( R  e.  Ring  ->  .x.  =  ( +g  `  (mulGrp `  R ) ) )
1514adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .x.  =  ( +g  `  (mulGrp `  R ) ) )
16 rngidm.u . . . . . . 7  |-  .1.  =  ( 1r `  R )
171, 16ringidvalg 13097 . . . . . 6  |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  (mulGrp `  R ) ) )
1817adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  .1.  =  ( 0g `  (mulGrp `  R ) ) )
19 eqidd 2178 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  X  =  X )
2015, 18, 19oveq123d 5895 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (  .1.  .x.  X )  =  ( ( 0g `  (mulGrp `  R ) ) ( +g  `  (mulGrp `  R ) ) X ) )
2120eqeq1d 2186 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .1.  .x.  X
)  =  X  <->  ( ( 0g `  (mulGrp `  R
) ) ( +g  `  (mulGrp `  R )
) X )  =  X ) )
2215, 19, 18oveq123d 5895 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .x.  .1.  )  =  ( X ( +g  `  (mulGrp `  R )
) ( 0g `  (mulGrp `  R ) ) ) )
2322eqeq1d 2186 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( X  .x.  .1.  )  =  X  <->  ( X
( +g  `  (mulGrp `  R ) ) ( 0g `  (mulGrp `  R ) ) )  =  X ) )
2421, 23anbi12d 473 . 2  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( (  .1.  .x.  X )  =  X  /\  ( X  .x.  .1.  )  =  X
)  <->  ( ( ( 0g `  (mulGrp `  R ) ) ( +g  `  (mulGrp `  R ) ) X )  =  X  /\  ( X ( +g  `  (mulGrp `  R ) ) ( 0g `  (mulGrp `  R ) ) )  =  X ) ) )
2512, 24mpbird 167 1  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
(  .1.  .x.  X
)  =  X  /\  ( X  .x.  .1.  )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530   .rcmulr 12531   0gc0g 12695   Mndcmnd 12771  mulGrpcmgp 13083   1rcur 13095   Ringcrg 13132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-mgp 13084  df-ur 13096  df-ring 13134
This theorem is referenced by:  ringlidm  13159  ringridm  13160  ringid  13162  subrg1  13312
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