MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngidss Unicode version

Theorem rngidss 15383
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
rngidss.g  |-  M  =  ( (mulGrp `  R
)s 
A )
rngidss.b  |-  B  =  ( Base `  R
)
rngidss.u  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
rngidss  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )

Proof of Theorem rngidss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2296 . 2  |-  ( 0g
`  M )  =  ( 0g `  M
)
3 eqid 2296 . 2  |-  ( +g  `  M )  =  ( +g  `  M )
4 simp3 957 . . 3  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  e.  A )
5 rngidss.g . . . . 5  |-  M  =  ( (mulGrp `  R
)s 
A )
6 eqid 2296 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
7 rngidss.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7mgpbas 15347 . . . . 5  |-  B  =  ( Base `  (mulGrp `  R ) )
95, 8ressbas2 13215 . . . 4  |-  ( A 
C_  B  ->  A  =  ( Base `  M
) )
1093ad2ant2 977 . . 3  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  =  ( Base `  M
) )
114, 10eleqtrd 2372 . 2  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  e.  ( Base `  M
) )
12 simp2 956 . . . . 5  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  C_  B )
1310, 12eqsstr3d 3226 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( Base `  M )  C_  B )
1413sselda 3193 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
y  e.  B )
15 fvex 5555 . . . . . . . 8  |-  ( Base `  M )  e.  _V
1610, 15syl6eqel 2384 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  e.  _V )
17 eqid 2296 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
186, 17mgpplusg 15345 . . . . . . . 8  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
195, 18ressplusg 13266 . . . . . . 7  |-  ( A  e.  _V  ->  ( .r `  R )  =  ( +g  `  M
) )
2016, 19syl 15 . . . . . 6  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( .r `  R )  =  ( +g  `  M
) )
2120adantr 451 . . . . 5  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( .r `  R
)  =  ( +g  `  M ) )
2221oveqd 5891 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( .r
`  R ) y )  =  (  .1.  ( +g  `  M
) y ) )
23 rngidss.u . . . . . 6  |-  .1.  =  ( 1r `  R )
247, 17, 23rnglidm 15380 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  B )  ->  (  .1.  ( .r `  R
) y )  =  y )
25243ad2antl1 1117 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( .r
`  R ) y )  =  y )
2622, 25eqtr3d 2330 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( +g  `  M ) y )  =  y )
2714, 26syldan 456 . 2  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
(  .1.  ( +g  `  M ) y )  =  y )
2821oveqd 5891 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( .r
`  R )  .1.  )  =  ( y ( +g  `  M
)  .1.  ) )
297, 17, 23rngridm 15381 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  B )  ->  (
y ( .r `  R )  .1.  )  =  y )
30293ad2antl1 1117 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( .r
`  R )  .1.  )  =  y )
3128, 30eqtr3d 2330 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( +g  `  M )  .1.  )  =  y )
3214, 31syldan 456 . 2  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
( y ( +g  `  M )  .1.  )  =  y )
331, 2, 3, 11, 27, 32ismgmid2 14406 1  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   .rcmulr 13225   0gc0g 13416  mulGrpcmgp 15341   Ringcrg 15353   1rcur 15355
This theorem is referenced by:  unitgrpid  15467  xrge0iifmhm  23336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mgp 15342  df-rng 15356  df-ur 15358
  Copyright terms: Public domain W3C validator