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

Theorem subrgpsr 16179
Description: A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
subrgpsr.s  |-  S  =  ( I mPwSer  R )
subrgpsr.h  |-  H  =  ( Rs  T )
subrgpsr.u  |-  U  =  ( I mPwSer  H )
subrgpsr.b  |-  B  =  ( Base `  U
)
Assertion
Ref Expression
subrgpsr  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  e.  (SubRing `  S )
)

Proof of Theorem subrgpsr
Dummy variables  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgpsr.s . . . 4  |-  S  =  ( I mPwSer  R )
2 simpl 443 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  I  e.  V )
3 subrgrcl 15566 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
43adantl 452 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  R  e.  Ring )
51, 2, 4psrrng 16171 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  S  e.  Ring )
6 subrgpsr.u . . . . 5  |-  U  =  ( I mPwSer  H )
7 subrgpsr.h . . . . . . 7  |-  H  =  ( Rs  T )
87subrgrng 15564 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
98adantl 452 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  H  e.  Ring )
106, 2, 9psrrng 16171 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  U  e.  Ring )
11 subrgpsr.b . . . . . 6  |-  B  =  ( Base `  U
)
1211a1i 10 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( Base `  U
) )
13 eqid 2296 . . . . . 6  |-  ( Ss  B )  =  ( Ss  B )
14 simpr 447 . . . . . 6  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  e.  (SubRing `  R )
)
151, 7, 6, 11, 13, 14resspsrbas 16175 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( Base `  ( Ss  B ) ) )
161, 7, 6, 11, 13, 14resspsradd 16176 . . . . 5  |-  ( ( ( I  e.  V  /\  T  e.  (SubRing `  R ) )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  U
) y )  =  ( x ( +g  `  ( Ss  B ) ) y ) )
171, 7, 6, 11, 13, 14resspsrmul 16177 . . . . 5  |-  ( ( ( I  e.  V  /\  T  e.  (SubRing `  R ) )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( .r `  U ) y )  =  ( x ( .r `  ( Ss  B ) ) y ) )
1812, 15, 16, 17rngpropd 15388 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( U  e.  Ring  <->  ( Ss  B
)  e.  Ring )
)
1910, 18mpbid 201 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( Ss  B )  e.  Ring )
205, 19jca 518 . 2  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( S  e.  Ring  /\  ( Ss  B )  e.  Ring ) )
21 eqid 2296 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
2213, 21ressbasss 13216 . . . 4  |-  ( Base `  ( Ss  B ) )  C_  ( Base `  S )
2315sseq1d 3218 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( B  C_  ( Base `  S
)  <->  ( Base `  ( Ss  B ) )  C_  ( Base `  S )
) )
2422, 23mpbiri 224 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  C_  ( Base `  S
) )
25 eqid 2296 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2625subrg1cl 15569 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  T
)
27 subrgsubg 15567 . . . . . . . . . . . 12  |-  ( T  e.  (SubRing `  R
)  ->  T  e.  (SubGrp `  R ) )
28 eqid 2296 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
2928subg0cl 14645 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  T
)
3027, 29syl 15 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  e.  T
)
31 ifcl 3614 . . . . . . . . . . 11  |-  ( ( ( 1r `  R
)  e.  T  /\  ( 0g `  R )  e.  T )  ->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  T )
3226, 30, 31syl2anc 642 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  if (
x  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) )  e.  T
)
3332adantl 452 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  if ( x  =  (
I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  T )
347subrgbas 15570 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
3534adantl 452 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  =  ( Base `  H
) )
3633, 35eleqtrd 2372 . . . . . . . 8  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  if ( x  =  (
I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  ( Base `  H
) )
3736adantr 451 . . . . . . 7  |-  ( ( ( I  e.  V  /\  T  e.  (SubRing `  R ) )  /\  x  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } )  ->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  ( Base `  H ) )
38 eqid 2296 . . . . . . 7  |-  ( x  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) )  =  ( x  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) ) )
3937, 38fmptd 5700 . . . . . 6  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  (
x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) : { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } --> ( Base `  H ) )
40 eqid 2296 . . . . . . . 8  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
41 eqid 2296 . . . . . . . 8  |-  ( 1r
`  S )  =  ( 1r `  S
)
421, 2, 4, 40, 28, 25, 41psr1 16172 . . . . . . 7  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S )  =  ( x  e.  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } 
|->  if ( x  =  ( I  X.  {
0 } ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ) )
4342feq1d 5395 . . . . . 6  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  (
( 1r `  S
) : { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin } --> ( Base `  H )  <->  ( x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) ) ) : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  H
) ) )
4439, 43mpbird 223 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S ) : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  H
) )
45 fvex 5555 . . . . . 6  |-  ( Base `  H )  e.  _V
46 ovex 5899 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
4746rabex 4181 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
4845, 47elmap 6812 . . . . 5  |-  ( ( 1r `  S )  e.  ( ( Base `  H )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } )  <->  ( 1r `  S ) : {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } --> ( Base `  H
) )
4944, 48sylibr 203 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S )  e.  ( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) )
50 eqid 2296 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
516, 50, 40, 11, 2psrbas 16140 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( ( Base `  H )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
5249, 51eleqtrrd 2373 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S )  e.  B )
5324, 52jca 518 . 2  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( B  C_  ( Base `  S
)  /\  ( 1r `  S )  e.  B
) )
5421, 41issubrg 15561 . 2  |-  ( B  e.  (SubRing `  S
)  <->  ( ( S  e.  Ring  /\  ( Ss  B )  e.  Ring )  /\  ( B  C_  ( Base `  S )  /\  ( 1r `  S
)  e.  B ) ) )
5520, 53, 54sylanbrc 645 1  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  e.  (SubRing `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   ifcif 3578   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   0cc0 8753   NNcn 9762   NN0cn0 9981   Basecbs 13164   ↾s cress 13165   0gc0g 13416  SubGrpcsubg 14631   Ringcrg 15353   1rcur 15355  SubRingcsubrg 15557   mPwSer cmps 16103
This theorem is referenced by:  ressmplbas2  16215  subrgmpl  16220
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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-int 3879  df-iun 3923  df-iin 3924  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  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-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-psr 16114
  Copyright terms: Public domain W3C validator