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Theorem subrg1 13312
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 subrg1.2 . 2  |-  .1.  =  ( 1r `  R )
2 eqid 2177 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
32subrg1cl 13310 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  A
)
4 subrg1.1 . . . . 5  |-  S  =  ( Rs  A )
54subrgbas 13311 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  A  =  ( Base `  S )
)
63, 5eleqtrd 2256 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  (
Base `  S )
)
7 eqid 2177 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
87subrgss 13303 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
95, 8eqsstrrd 3192 . . . . . 6  |-  ( A  e.  (SubRing `  R
)  ->  ( Base `  S )  C_  ( Base `  R ) )
109sselda 3155 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  R )
)
11 subrgrcl 13307 . . . . . . 7  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
12 eqid 2177 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
137, 12, 2ringidmlem 13158 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( ( 1r `  R ) ( .r
`  R ) x )  =  x  /\  ( x ( .r
`  R ) ( 1r `  R ) )  =  x ) )
1411, 13sylan 283 . . . . . 6  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  /\  (
x ( .r `  R ) ( 1r
`  R ) )  =  x ) )
154, 12ressmulrg 12597 . . . . . . . . . . 11  |-  ( ( A  e.  (SubRing `  R
)  /\  R  e.  Ring )  ->  ( .r `  R )  =  ( .r `  S ) )
1611, 15mpdan 421 . . . . . . . . . 10  |-  ( A  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1716oveqd 5891 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( ( 1r `  R ) ( .r `  R ) x )  =  ( ( 1r `  R
) ( .r `  S ) x ) )
1817eqeq1d 2186 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( (
( 1r `  R
) ( .r `  R ) x )  =  x  <->  ( ( 1r `  R ) ( .r `  S ) x )  =  x ) )
1916oveqd 5891 . . . . . . . . 9  |-  ( A  e.  (SubRing `  R
)  ->  ( x
( .r `  R
) ( 1r `  R ) )  =  ( x ( .r
`  S ) ( 1r `  R ) ) )
2019eqeq1d 2186 . . . . . . . 8  |-  ( A  e.  (SubRing `  R
)  ->  ( (
x ( .r `  R ) ( 1r
`  R ) )  =  x  <->  ( x
( .r `  S
) ( 1r `  R ) )  =  x ) )
2118, 20anbi12d 473 . . . . . . 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 ) ) )
2221biimpa 296 . . . . . 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 ) )
2314, 22syldan 282 . . . . 5  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( (
( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )
2410, 23syldan 282 . . . 4  |-  ( ( A  e.  (SubRing `  R
)  /\  x  e.  ( Base `  S )
)  ->  ( (
( 1r `  R
) ( .r `  S ) x )  =  x  /\  (
x ( .r `  S ) ( 1r
`  R ) )  =  x ) )
2524ralrimiva 2550 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  A. x  e.  ( Base `  S
) ( ( ( 1r `  R ) ( .r `  S
) x )  =  x  /\  ( x ( .r `  S
) ( 1r `  R ) )  =  x ) )
264subrgring 13305 . . . 4  |-  ( A  e.  (SubRing `  R
)  ->  S  e.  Ring )
27 eqid 2177 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
28 eqid 2177 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
29 eqid 2177 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
3027, 28, 29isringid 13161 . . . 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 ) ) )
3126, 30syl 14 . . 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 ) ) )
326, 25, 31mpbi2and 943 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( 1r `  S )  =  ( 1r `  R ) )
331, 32eqtr4id 2229 1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  =  ( 1r `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5216  (class class class)co 5874   Basecbs 12456   ↾s cress 12457   .rcmulr 12531   1rcur 13095   Ringcrg 13132  SubRingcsubrg 13298
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-lttrn 7924  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-subg 12983  df-mgp 13084  df-ur 13096  df-ring 13134  df-subrg 13300
This theorem is referenced by:  subrguss  13317  subrginv  13318  subrgunit  13320  subrgnzr  13323  subsubrg  13326  zring1  13382
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