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

Theorem resspsrvsca 16178
Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
resspsr.s  |-  S  =  ( I mPwSer  R )
resspsr.h  |-  H  =  ( Rs  T )
resspsr.u  |-  U  =  ( I mPwSer  H )
resspsr.b  |-  B  =  ( Base `  U
)
resspsr.p  |-  P  =  ( Ss  B )
resspsr.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
Assertion
Ref Expression
resspsrvsca  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( X ( .s `  P
) Y ) )

Proof of Theorem resspsrvsca
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 resspsr.u . . 3  |-  U  =  ( I mPwSer  H )
2 eqid 2296 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
3 eqid 2296 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4 resspsr.b . . 3  |-  B  =  ( Base `  U
)
5 eqid 2296 . . 3  |-  ( .r
`  H )  =  ( .r `  H
)
6 eqid 2296 . . 3  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 simprl 732 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  T )
8 resspsr.2 . . . . . 6  |-  ( ph  ->  T  e.  (SubRing `  R
) )
98adantr 451 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  e.  (SubRing `  R
) )
10 resspsr.h . . . . . 6  |-  H  =  ( Rs  T )
1110subrgbas 15570 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
129, 11syl 15 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  =  ( Base `  H ) )
137, 12eleqtrd 2372 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  ( Base `  H ) )
14 simprr 733 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  B )
151, 2, 3, 4, 5, 6, 13, 14psrvsca 16152 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { X } )  o F ( .r `  H
) Y ) )
16 resspsr.s . . . 4  |-  S  =  ( I mPwSer  R )
17 eqid 2296 . . . 4  |-  ( .s
`  S )  =  ( .s `  S
)
18 eqid 2296 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2296 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2296 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
2118subrgss 15562 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
229, 21syl 15 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  C_  ( Base `  R
) )
2322, 7sseldd 3194 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  ( Base `  R ) )
24 resspsr.p . . . . . . . 8  |-  P  =  ( Ss  B )
2524, 19ressbasss 13216 . . . . . . 7  |-  ( Base `  P )  C_  ( Base `  S )
2616, 10, 1, 4, 24, 8resspsrbas 16175 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  P ) )
2726sseq1d 3218 . . . . . . 7  |-  ( ph  ->  ( B  C_  ( Base `  S )  <->  ( Base `  P )  C_  ( Base `  S ) ) )
2825, 27mpbiri 224 . . . . . 6  |-  ( ph  ->  B  C_  ( Base `  S ) )
2928adantr 451 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
3029, 14sseldd 3194 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
3116, 17, 18, 19, 20, 6, 23, 30psrvsca 16152 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { X } )  o F ( .r `  R
) Y ) )
3210, 20ressmulr 13277 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  H ) )
33 ofeq 6096 . . . . 5  |-  ( ( .r `  R )  =  ( .r `  H )  ->  o F ( .r `  R )  =  o F ( .r `  H ) )
349, 32, 333syl 18 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  o F ( .r `  R )  =  o F ( .r `  H ) )
3534oveqd 5891 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( ( { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  X.  { X } )  o F ( .r `  R ) Y )  =  ( ( { f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin }  X.  { X }
)  o F ( .r `  H ) Y ) )
3631, 35eqtrd 2328 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { X } )  o F ( .r `  H
) Y ) )
37 fvex 5555 . . . . 5  |-  ( Base `  U )  e.  _V
384, 37eqeltri 2366 . . . 4  |-  B  e. 
_V
3924, 17ressvsca 13300 . . . 4  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
4038, 39mp1i 11 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( .s `  S
)  =  ( .s
`  P ) )
4140oveqd 5891 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( X ( .s `  P
) Y ) )
4215, 36, 413eqtr2d 2334 1  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( X ( .s `  P
) Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   {csn 3653    X. cxp 4703   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    o Fcof 6092    ^m cmap 6788   Fincfn 6879   NNcn 9762   NN0cn0 9981   Basecbs 13164   ↾s cress 13165   .rcmulr 13225   .scvsca 13228  SubRingcsubrg 15557   mPwSer cmps 16103
This theorem is referenced by:  ressmplvsca  16219
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-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-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-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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  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-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-subg 14634  df-rng 15356  df-subrg 15559  df-psr 16114
  Copyright terms: Public domain W3C validator