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Theorem resspsrvsca 17488
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 2441 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
3 eqid 2441 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4 resspsr.b . . 3  |-  B  =  ( Base `  U
)
5 eqid 2441 . . 3  |-  ( .r
`  H )  =  ( .r `  H
)
6 eqid 2441 . . 3  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 simprl 755 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  T )
8 resspsr.2 . . . . . 6  |-  ( ph  ->  T  e.  (SubRing `  R
) )
98adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  e.  (SubRing `  R
) )
10 resspsr.h . . . . . 6  |-  H  =  ( Rs  T )
1110subrgbas 16872 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
129, 11syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  =  ( Base `  H ) )
137, 12eleqtrd 2517 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  ( Base `  H ) )
14 simprr 756 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  B )
151, 2, 3, 4, 5, 6, 13, 14psrvsca 17460 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { X } )  oF ( .r `  H
) Y ) )
16 resspsr.s . . . 4  |-  S  =  ( I mPwSer  R )
17 eqid 2441 . . . 4  |-  ( .s
`  S )  =  ( .s `  S
)
18 eqid 2441 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2441 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2441 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
2118subrgss 16864 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
229, 21syl 16 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  C_  ( Base `  R
) )
2322, 7sseldd 3355 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  ( Base `  R ) )
24 resspsr.p . . . . . . . 8  |-  P  =  ( Ss  B )
2516, 10, 1, 4, 24, 8resspsrbas 17485 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  P ) )
2624, 19ressbasss 14228 . . . . . . 7  |-  ( Base `  P )  C_  ( Base `  S )
2725, 26syl6eqss 3404 . . . . . 6  |-  ( ph  ->  B  C_  ( Base `  S ) )
2827adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
2928, 14sseldd 3355 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
3016, 17, 18, 19, 20, 6, 23, 29psrvsca 17460 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { X } )  oF ( .r `  R
) Y ) )
3110, 20ressmulr 14289 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  H ) )
32 ofeq 6320 . . . . 5  |-  ( ( .r `  R )  =  ( .r `  H )  ->  oF ( .r `  R )  =  oF ( .r `  H ) )
339, 31, 323syl 20 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  oF ( .r
`  R )  =  oF ( .r
`  H ) )
3433oveqd 6106 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( ( { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  X.  { X } )  oF ( .r `  R ) Y )  =  ( ( { f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin }  X.  { X }
)  oF ( .r `  H ) Y ) )
3530, 34eqtrd 2473 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( ( { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  X.  { X } )  oF ( .r `  H
) Y ) )
36 fvex 5699 . . . . 5  |-  ( Base `  U )  e.  _V
374, 36eqeltri 2511 . . . 4  |-  B  e. 
_V
3824, 17ressvsca 14315 . . . 4  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
3937, 38mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( .s `  S
)  =  ( .s
`  P ) )
4039oveqd 6106 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( X ( .s `  P
) Y ) )
4115, 35, 403eqtr2d 2479 1  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( X ( .s `  P
) Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2717   _Vcvv 2970    C_ wss 3326   {csn 3875    X. cxp 4836   `'ccnv 4837   "cima 4841   ` cfv 5416  (class class class)co 6089    oFcof 6316    ^m cmap 7212   Fincfn 7308   NNcn 10320   NN0cn0 10577   Basecbs 14172   ↾s cress 14173   .rcmulr 14237   .scvsca 14240  SubRingcsubrg 16859   mPwSer cmps 17416
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-tset 14255  df-subg 15676  df-rng 16645  df-subrg 16861  df-psr 17421
This theorem is referenced by:  ressmplvsca  17536
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