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Theorem subrgpsr 17496
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 16875 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
43adantl 466 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  R  e.  Ring )
51, 2, 4psrrng 17488 . . 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 16873 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
98adantl 466 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  H  e.  Ring )
106, 2, 9psrrng 17488 . . . 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 2443 . . . . . 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 17492 . . . . 5  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( Base `  ( Ss  B ) ) )
161, 7, 6, 11, 13, 14resspsradd 17493 . . . . 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 17494 . . . . 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 16681 . . . 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 2443 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
2213, 21ressbasss 14235 . . . 4  |-  ( Base `  ( Ss  B ) )  C_  ( Base `  S )
2315, 22syl6eqss 3411 . . 3  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  C_  ( Base `  S
) )
24 eqid 2443 . . . . . . . . . . . 12  |-  ( 1r
`  R )  =  ( 1r `  R
)
2524subrg1cl 16878 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 1r `  R )  e.  T
)
26 subrgsubg 16876 . . . . . . . . . . . 12  |-  ( T  e.  (SubRing `  R
)  ->  T  e.  (SubGrp `  R ) )
27 eqid 2443 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
2827subg0cl 15694 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  T
)
2926, 28syl 16 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  e.  T
)
30 ifcl 3836 . . . . . . . . . . 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 16879 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
3433adantl 466 . . . . . . . . 9  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  T  =  ( Base `  H
) )
3532, 34eleqtrd 2519 . . . . . . . 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 2443 . . . . . . 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 5872 . . . . . 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 2443 . . . . . . . 8  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
40 eqid 2443 . . . . . . . 8  |-  ( 1r
`  S )  =  ( 1r `  S
)
411, 2, 4, 39, 27, 24, 40psr1 17489 . . . . . . 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 5551 . . . . . 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 5706 . . . . . 6  |-  ( Base `  H )  e.  _V
45 ovex 6121 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
4645rabex 4448 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
4744, 46elmap 7246 . . . . 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 2443 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
506, 49, 39, 11, 2psrbas 17453 . . . 4  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  B  =  ( ( Base `  H )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
5148, 50eleqtrrd 2520 . . 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 16870 . 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 1369    e. wcel 1756   {crab 2724    C_ wss 3333   ifcif 3796   {csn 3882    e. cmpt 4355    X. cxp 4843   `'ccnv 4844   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   Fincfn 7315   0cc0 9287   NNcn 10327   NN0cn0 10584   Basecbs 14179   ↾s cress 14180   0gc0g 14383  SubGrpcsubg 15680   1rcur 16608   Ringcrg 16650  SubRingcsubrg 16866   mPwSer cmps 17423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-ofr 6326  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-tset 14262  df-0g 14385  df-gsum 14386  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-grp 15550  df-minusg 15551  df-mulg 15553  df-subg 15683  df-ghm 15750  df-cntz 15840  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-subrg 16868  df-psr 17428
This theorem is referenced by:  ressmplbas2  17539  subrgmpl  17544
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