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Theorem subrg1 15571
Description: A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypotheses
Ref Expression
subrg1.1  |-  S  =  ( Rs  A )
subrg1.2  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
subrg1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  =  ( 1r `  S ) )

Proof of Theorem subrg1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
21subrg1cl 15569 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  A
)
3 subrg1.1 . . . . 5  |-  S  =  ( Rs  A )
43subrgbas 15570 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
52, 4eleqtrd 2372 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  (
Base `  S )
)
6 eqid 2296 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
76subrgss 15562 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
84, 7eqsstr3d 3226 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
98sselda 3193 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  R )
)
10 subrgrcl 15566 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
11 eqid 2296 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
126, 11, 1rngidmlem 15379 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( ( 1r `  R ) ( .r
`  R ) x )  =  x  /\  ( x ( .r
`  R ) ( 1r `  R ) )  =  x ) )
1310, 12sylan 457 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) ( 1r
`  R ) )  =  x ) )
143, 11ressmulr 13277 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1514oveqd 5891 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( ( 1r `  R ) ( .r `  R ) x )  =  ( ( 1r `  R
) ( .r `  S ) x ) )
1615eqeq1d 2304 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  <->  ( ( 1r `  R ) ( .r `  S ) x )  =  x ) )
1714oveqd 5891 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) ( 1r `  R ) )  =  ( x ( .r
`  S ) ( 1r `  R ) ) )
1817eqeq1d 2304 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) ( 1r
`  R ) )  =  x  <->  ( x
( .r `  S
) ( 1r `  R ) )  =  x ) )
1916, 18anbi12d 691 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  ( (
( ( 1r `  R ) ( .r
`  R ) x )  =  x  /\  ( x ( .r
`  R ) ( 1r `  R ) )  =  x )  <-> 
( ( ( 1r
`  R ) ( .r `  S ) x )  =  x  /\  ( x ( .r `  S ) ( 1r `  R
) )  =  x ) ) )
2019biimpa 470 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) ( 1r
`  R ) )  =  x ) )  ->  ( ( ( 1r `  R ) ( .r `  S
) x )  =  x  /\  ( x ( .r `  S
) ( 1r `  R ) )  =  x ) )
2113, 20syldan 456 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )
229, 21syldan 456 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  S )
)  ->  ( (
( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )
2322ralrimiva 2639 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  A. x  e.  ( Base `  S
) ( ( ( 1r `  R ) ( .r `  S
) x )  =  x  /\  ( x ( .r `  S
) ( 1r `  R ) )  =  x ) )
243subrgrng 15564 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
25 eqid 2296 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
26 eqid 2296 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
27 eqid 2296 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
2825, 26, 27isrngid 15382 . . . 4  |-  ( S  e.  Ring  ->  ( ( ( 1r `  R
)  e.  ( Base `  S )  /\  A. x  e.  ( Base `  S ) ( ( ( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )  <-> 
( 1r `  S
)  =  ( 1r
`  R ) ) )
2924, 28syl 15 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( (
( 1r `  R
)  e.  ( Base `  S )  /\  A. x  e.  ( Base `  S ) ( ( ( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )  <-> 
( 1r `  S
)  =  ( 1r
`  R ) ) )
305, 23, 29mpbi2and 887 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  S )  =  ( 1r `  R ) )
31 subrg1.2 . 2  |-  .1.  =  ( 1r `  R )
3230, 31syl6reqr 2347 1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  =  ( 1r `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   .rcmulr 13225   Ringcrg 15353   1rcur 15355  SubRingcsubrg 15557
This theorem is referenced by:  subrguss  15576  subrginv  15577  subrgunit  15579  subsubrg  15587  sralmod  15955  subrgnzr  16035  ressascl  16099  mpl1  16204  subrgmvr  16221  gzrngunitlem  16452  prmirredlem  16462  mulgrhm  16476  mulgrhm2  16477  zrh1  16483  zlmlmod  16493  clm1  18587  lgsqrlem1  20596  qrng1  20787
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-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559
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