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Theorem rngidss 15682
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 2435 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2435 . 2  |-  ( 0g
`  M )  =  ( 0g `  M
)
3 eqid 2435 . 2  |-  ( +g  `  M )  =  ( +g  `  M )
4 simp3 959 . . 3  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  e.  A )
5 rngidss.g . . . . 5  |-  M  =  ( (mulGrp `  R
)s 
A )
6 eqid 2435 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
7 rngidss.b . . . . . 6  |-  B  =  ( Base `  R
)
86, 7mgpbas 15646 . . . . 5  |-  B  =  ( Base `  (mulGrp `  R ) )
95, 8ressbas2 13512 . . . 4  |-  ( A 
C_  B  ->  A  =  ( Base `  M
) )
1093ad2ant2 979 . . 3  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  =  ( Base `  M
) )
114, 10eleqtrd 2511 . 2  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  e.  ( Base `  M
) )
12 simp2 958 . . . . 5  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  C_  B )
1310, 12eqsstr3d 3375 . . . 4  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( Base `  M )  C_  B )
1413sselda 3340 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
y  e.  B )
15 fvex 5734 . . . . . . . 8  |-  ( Base `  M )  e.  _V
1610, 15syl6eqel 2523 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  A  e.  _V )
17 eqid 2435 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
186, 17mgpplusg 15644 . . . . . . . 8  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
195, 18ressplusg 13563 . . . . . . 7  |-  ( A  e.  _V  ->  ( .r `  R )  =  ( +g  `  M
) )
2016, 19syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  ( .r `  R )  =  ( +g  `  M
) )
2120adantr 452 . . . . 5  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( .r `  R
)  =  ( +g  `  M ) )
2221oveqd 6090 . . . 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 15679 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  B )  ->  (  .1.  ( .r `  R
) y )  =  y )
25243ad2antl1 1119 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( .r
`  R ) y )  =  y )
2622, 25eqtr3d 2469 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  (  .1.  ( +g  `  M ) y )  =  y )
2714, 26syldan 457 . 2  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  ( Base `  M ) )  -> 
(  .1.  ( +g  `  M ) y )  =  y )
2821oveqd 6090 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( .r
`  R )  .1.  )  =  ( y ( +g  `  M
)  .1.  ) )
297, 17, 23rngridm 15680 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  B )  ->  (
y ( .r `  R )  .1.  )  =  y )
30293ad2antl1 1119 . . . 4  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( .r
`  R )  .1.  )  =  y )
3128, 30eqtr3d 2469 . . 3  |-  ( ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  /\  y  e.  B )  ->  ( y ( +g  `  M )  .1.  )  =  y )
3214, 31syldan 457 . 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 14705 1  |-  ( ( R  e.  Ring  /\  A  C_  B  /\  .1.  e.  A )  ->  .1.  =  ( 0g `  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   .rcmulr 13522   0gc0g 13715  mulGrpcmgp 15640   Ringcrg 15652   1rcur 15654
This theorem is referenced by:  unitgrpid  15766  xrge0iifmhm  24317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-mgp 15641  df-rng 15655  df-ur 15657
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