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Theorem subrg1 15878
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 2436 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
21subrg1cl 15876 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  A
)
3 subrg1.1 . . . . 5  |-  S  =  ( Rs  A )
43subrgbas 15877 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
52, 4eleqtrd 2512 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  (
Base `  S )
)
6 eqid 2436 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
76subrgss 15869 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
84, 7eqsstr3d 3383 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
98sselda 3348 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  R )
)
10 subrgrcl 15873 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
11 eqid 2436 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
126, 11, 1rngidmlem 15686 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( ( 1r `  R ) ( .r
`  R ) x )  =  x  /\  ( x ( .r
`  R ) ( 1r `  R ) )  =  x ) )
1310, 12sylan 458 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) ( 1r
`  R ) )  =  x ) )
143, 11ressmulr 13582 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1514oveqd 6098 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( ( 1r `  R ) ( .r `  R ) x )  =  ( ( 1r `  R
) ( .r `  S ) x ) )
1615eqeq1d 2444 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  <->  ( ( 1r `  R ) ( .r `  S ) x )  =  x ) )
1714oveqd 6098 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) ( 1r `  R ) )  =  ( x ( .r
`  S ) ( 1r `  R ) ) )
1817eqeq1d 2444 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) ( 1r
`  R ) )  =  x  <->  ( x
( .r `  S
) ( 1r `  R ) )  =  x ) )
1916, 18anbi12d 692 . . . . . . 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 471 . . . . . 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 457 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )
229, 21syldan 457 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  S )
)  ->  ( (
( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )
2322ralrimiva 2789 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  A. x  e.  ( Base `  S
) ( ( ( 1r `  R ) ( .r `  S
) x )  =  x  /\  ( x ( .r `  S
) ( 1r `  R ) )  =  x ) )
243subrgrng 15871 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
25 eqid 2436 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
26 eqid 2436 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
27 eqid 2436 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
2825, 26, 27isrngid 15689 . . . 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 16 . . 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 888 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  S )  =  ( 1r `  R ) )
31 subrg1.2 . 2  |-  .1.  =  ( 1r `  R )
3230, 31syl6reqr 2487 1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  =  ( 1r `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   .rcmulr 13530   Ringcrg 15660   1rcur 15662  SubRingcsubrg 15864
This theorem is referenced by:  subrguss  15883  subrginv  15884  subrgunit  15886  subsubrg  15894  sralmod  16258  subrgnzr  16338  ressascl  16402  mpl1  16507  subrgmvr  16524  gzrngunitlem  16763  prmirredlem  16773  mulgrhm  16787  mulgrhm2  16788  zrh1  16794  zlmlmod  16804  clm1  19098  lgsqrlem1  21125  qrng1  21316  subrgchr  24230  zzs1  24268  re1r  24274
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866
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