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Theorem ringidss 13165
Description: A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
ringidss.g  |-  M  =  ( (mulGrp `  R
)s 
A )
ringidss.b  |-  B  =  ( Base `  R
)
ringidss.u  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
ringidss  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )

Proof of Theorem ringidss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2177 . 2  |-  ( 0g
`  M )  =  ( 0g `  M
)
3 eqid 2177 . 2  |-  ( +g  `  M )  =  ( +g  `  M )
4 simp3 999 . . 3  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  e.  A )
5 ringidss.g . . . . 5  |-  M  =  ( (mulGrp `  R
)s 
A )
65a1i 9 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  M  =  ( (mulGrp `  R )s  A ) )
7 eqid 2177 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
8 ringidss.b . . . . . 6  |-  B  =  ( Base `  R
)
97, 8mgpbasg 13089 . . . . 5  |-  ( R  e.  Ring  ->  B  =  ( Base `  (mulGrp `  R ) ) )
1093ad2ant1 1018 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  B  =  ( Base `  (mulGrp `  R ) ) )
117mgpex 13088 . . . . 5  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  _V )
12113ad2ant1 1018 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  (mulGrp `  R )  e.  _V )
13 simp2 998 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  C_  B )
146, 10, 12, 13ressbas2d 12522 . . 3  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  =  ( Base `  M
) )
154, 14eleqtrd 2256 . 2  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  e.  ( Base `  M
) )
1614, 13eqsstrrd 3192 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( Base `  M )  C_  B )
1716sselda 3155 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
y  e.  B )
18 eqid 2177 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
197, 18mgpplusgg 13087 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) ) )
20193ad2ant1 1018 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( .r `  R )  =  ( +g  `  (mulGrp `  R ) ) )
21 basfn 12514 . . . . . . . . . 10  |-  Base  Fn  _V
22 simp1 997 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  R  e.  Ring )
2322elexd 2750 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  R  e.  _V )
24 funfvex 5532 . . . . . . . . . . 11  |-  ( ( Fun  Base  /\  R  e. 
dom  Base )  ->  ( Base `  R )  e. 
_V )
2524funfni 5316 . . . . . . . . . 10  |-  ( (
Base  Fn  _V  /\  R  e.  _V )  ->  ( Base `  R )  e. 
_V )
2621, 23, 25sylancr 414 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( Base `  R )  e. 
_V )
278, 26eqeltrid 2264 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  B  e.  _V )
2827, 13ssexd 4143 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  e.  _V )
296, 20, 28, 12ressplusgd 12581 . . . . . 6  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( .r `  R )  =  ( +g  `  M
) )
3029adantr 276 . . . . 5  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( .r `  R
)  =  ( +g  `  M ) )
3130oveqd 5891 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( .r
`  R ) y )  =  (  .1.  ( +g  `  M
) y ) )
32 ringidss.u . . . . . 6  |-  .1.  =  ( 1r `  R )
338, 18, 32ringlidm 13159 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  B )  ->  (  .1.  ( .r `  R
) y )  =  y )
34333ad2antl1 1159 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( .r
`  R ) y )  =  y )
3531, 34eqtr3d 2212 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( +g  `  M ) y )  =  y )
3617, 35syldan 282 . 2  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
(  .1.  ( +g  `  M ) y )  =  y )
3730oveqd 5891 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( .r
`  R )  .1.  )  =  ( y ( +g  `  M
)  .1.  ) )
388, 18, 32ringridm 13160 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  B )  ->  (
y ( .r `  R )  .1.  )  =  y )
39383ad2antl1 1159 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( .r
`  R )  .1.  )  =  y )
4037, 39eqtr3d 2212 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( +g  `  M )  .1.  )  =  y )
4117, 40syldan 282 . 2  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
( y ( +g  `  M )  .1.  )  =  y )
421, 2, 3, 15, 36, 41ismgmid2 12753 1  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129    Fn wfn 5211   ` cfv 5216  (class class class)co 5874   Basecbs 12456   ↾s cress 12457   +g cplusg 12530   .rcmulr 12531   0gc0g 12695  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-iress 12464  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:  unitgrpid  13240
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