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Theorem subrgpsr 16482
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 444 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  I  e.  V )
3 subrgrcl 15873 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
43adantl 453 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  R  e.  Ring )
51, 2, 4psrrng 16474 . . 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 15871 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
98adantl 453 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  H  e.  Ring )
106, 2, 9psrrng 16474 . . . 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 2436 . . . . . 6  |-  ( Ss  B )  =  ( Ss  B )
14 simpr 448 . . . . . 6  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  e.  (SubRing `  R )
)
151, 7, 6, 11, 13, 14resspsrbas 16478 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( Base `  ( Ss  B ) ) )
161, 7, 6, 11, 13, 14resspsradd 16479 . . . . 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 16480 . . . . 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 15695 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( U  e.  Ring  <->  ( Ss  B
)  e.  Ring )
)
1910, 18mpbid 202 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( Ss  B )  e.  Ring )
205, 19jca 519 . 2  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( S  e.  Ring  /\  ( Ss  B )  e.  Ring ) )
21 eqid 2436 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
2213, 21ressbasss 13521 . . . 4  |-  ( Base `  ( Ss  B ) )  C_  ( Base `  S )
2315, 22syl6eqss 3398 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  C_  ( Base `  S
) )
24 eqid 2436 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2524subrg1cl 15876 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  T
)
26 subrgsubg 15874 . . . . . . . . . . . 12  |-  ( T  e.  (SubRing `  R
)  ->  T  e.  (SubGrp `  R ) )
27 eqid 2436 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
2827subg0cl 14952 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  T
)
2926, 28syl 16 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  e.  T
)
30 ifcl 3775 . . . . . . . . . . 11  |-  ( ( ( 1r `  R
)  e.  T  /\  ( 0g `  R )  e.  T )  ->  if ( x  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  T )
3125, 29, 30syl2anc 643 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  if (
x  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) )  e.  T
)
3231adantl 453 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  if ( x  =  (
I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  T )
337subrgbas 15877 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
3433adantl 453 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  =  ( Base `  H
) )
3532, 34eleqtrd 2512 . . . . . . . 8  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  if ( x  =  (
I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) )  e.  ( Base `  H
) )
3635adantr 452 . . . . . . 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 2436 . . . . . . 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 5893 . . . . . 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 2436 . . . . . . . 8  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
40 eqid 2436 . . . . . . . 8  |-  ( 1r
`  S )  =  ( 1r `  S
)
411, 2, 4, 39, 27, 24, 40psr1 16475 . . . . . . 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 5580 . . . . . 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 224 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S ) : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  H
) )
44 fvex 5742 . . . . . 6  |-  ( Base `  H )  e.  _V
45 ovex 6106 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
4645rabex 4354 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
4744, 46elmap 7042 . . . . 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 204 . . . 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 2436 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
506, 49, 39, 11, 2psrbas 16443 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( ( Base `  H )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
5148, 50eleqtrrd 2513 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( 1r `  S )  e.  B )
5223, 51jca 519 . 2  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  ( B  C_  ( Base `  S
)  /\  ( 1r `  S )  e.  B
) )
5321, 40issubrg 15868 . 2  |-  ( B  e.  (SubRing `  S
)  <->  ( ( S  e.  Ring  /\  ( Ss  B )  e.  Ring )  /\  ( B  C_  ( Base `  S )  /\  ( 1r `  S
)  e.  B ) ) )
5420, 52, 53sylanbrc 646 1  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  e.  (SubRing `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   ifcif 3739   {csn 3814    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   0cc0 8990   NNcn 10000   NN0cn0 10221   Basecbs 13469   ↾s cress 13470   0gc0g 13723  SubGrpcsubg 14938   Ringcrg 15660   1rcur 15662  SubRingcsubrg 15864   mPwSer cmps 16406
This theorem is referenced by:  ressmplbas2  16518  subrgmpl  16523
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-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  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-int 4051  df-iun 4095  df-iin 4096  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-se 4542  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-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  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-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866  df-psr 16417
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