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Theorem subrgpsr 17873
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 457 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  I  e.  V )
3 subrgrcl 17234 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
43adantl 466 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  R  e.  Ring )
51, 2, 4psrrng 17865 . . 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 17232 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
98adantl 466 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  H  e.  Ring )
106, 2, 9psrrng 17865 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  U  e.  Ring )
11 subrgpsr.b . . . . . 6  |-  B  =  ( Base `  U
)
1211a1i 11 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( Base `  U
) )
13 eqid 2467 . . . . . 6  |-  ( Ss  B )  =  ( Ss  B )
14 simpr 461 . . . . . 6  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  e.  (SubRing `  R )
)
151, 7, 6, 11, 13, 14resspsrbas 17869 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( Base `  ( Ss  B ) ) )
161, 7, 6, 11, 13, 14resspsradd 17870 . . . . 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 17871 . . . . 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 17031 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( U  e.  Ring  <->  ( Ss  B
)  e.  Ring )
)
1910, 18mpbid 210 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( Ss  B )  e.  Ring )
205, 19jca 532 . 2  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( S  e.  Ring  /\  ( Ss  B )  e.  Ring ) )
21 eqid 2467 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
2213, 21ressbasss 14547 . . . 4  |-  ( Base `  ( Ss  B ) )  C_  ( Base `  S )
2315, 22syl6eqss 3554 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  C_  ( Base `  S
) )
24 eqid 2467 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2524subrg1cl 17237 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  T
)
26 subrgsubg 17235 . . . . . . . . . . . 12  |-  ( T  e.  (SubRing `  R
)  ->  T  e.  (SubGrp `  R ) )
27 eqid 2467 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
2827subg0cl 16014 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  T
)
2926, 28syl 16 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  e.  T
)
30 ifcl 3981 . . . . . . . . . . 11  |-  ( ( ( 1r `  R
)  e.  T  /\  ( 0g `  R )  e.  T )  ->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  T )
3125, 29, 30syl2anc 661 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  if (
x  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) )  e.  T
)
3231adantl 466 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  if ( x  =  (
I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  T )
337subrgbas 17238 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
3433adantl 466 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  =  ( Base `  H
) )
3532, 34eleqtrd 2557 . . . . . . . 8  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  if ( x  =  (
I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  ( Base `  H
) )
3635adantr 465 . . . . . . 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 ) )
37 eqid 2467 . . . . . . 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
) ) )
3836, 37fmptd 6045 . . . . . 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 ) )
39 eqid 2467 . . . . . . . 8  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
40 eqid 2467 . . . . . . . 8  |-  ( 1r
`  S )  =  ( 1r `  S
)
411, 2, 4, 39, 27, 24, 40psr1 17866 . . . . . . 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 ) ) ) )
4241feq1d 5717 . . . . . 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
) ) )
4338, 42mpbird 232 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S ) : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  H
) )
44 fvex 5876 . . . . . 6  |-  ( Base `  H )  e.  _V
45 ovex 6309 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
4645rabex 4598 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
4744, 46elmap 7447 . . . . 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
) )
4843, 47sylibr 212 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S )  e.  ( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) )
49 eqid 2467 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
506, 49, 39, 11, 2psrbas 17829 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( ( Base `  H )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
5148, 50eleqtrrd 2558 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S )  e.  B )
5223, 51jca 532 . 2  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( B  C_  ( Base `  S
)  /\  ( 1r `  S )  e.  B
) )
5321, 40issubrg 17229 . 2  |-  ( B  e.  (SubRing `  S
)  <->  ( ( S  e.  Ring  /\  ( Ss  B )  e.  Ring )  /\  ( B  C_  ( Base `  S )  /\  ( 1r `  S
)  e.  B ) ) )
5420, 52, 53sylanbrc 664 1  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  e.  (SubRing `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   ifcif 3939   {csn 4027    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   "cima 5002   -->wf 5584   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   Fincfn 7516   0cc0 9492   NNcn 10536   NN0cn0 10795   Basecbs 14490   ↾s cress 14491   0gc0g 14695  SubGrpcsubg 16000   1rcur 16955   Ringcrg 17000  SubRingcsubrg 17225   mPwSer cmps 17799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-tset 14574  df-0g 14697  df-gsum 14698  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-mulg 15870  df-subg 16003  df-ghm 16070  df-cntz 16160  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-subrg 17227  df-psr 17804
This theorem is referenced by:  ressmplbas2  17916  subrgmpl  17921
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