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Theorem resspsrvsca 16482
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 2437 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
3 eqid 2437 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4 resspsr.b . . 3  |-  B  =  ( Base `  U
)
5 eqid 2437 . . 3  |-  ( .r
`  H )  =  ( .r `  H
)
6 eqid 2437 . . 3  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 simprl 734 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  T )
8 resspsr.2 . . . . . 6  |-  ( ph  ->  T  e.  (SubRing `  R
) )
98adantr 453 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  e.  (SubRing `  R
) )
10 resspsr.h . . . . . 6  |-  H  =  ( Rs  T )
1110subrgbas 15878 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
129, 11syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  =  ( Base `  H ) )
137, 12eleqtrd 2513 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  ( Base `  H ) )
14 simprr 735 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  B )
151, 2, 3, 4, 5, 6, 13, 14psrvsca 16456 . 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 2437 . . . 4  |-  ( .s
`  S )  =  ( .s `  S
)
18 eqid 2437 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2437 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2437 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
2118subrgss 15870 . . . . . 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 3350 . . . 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 16479 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  P ) )
2624, 19ressbasss 13522 . . . . . . 7  |-  ( Base `  P )  C_  ( Base `  S )
2725, 26syl6eqss 3399 . . . . . 6  |-  ( ph  ->  B  C_  ( Base `  S ) )
2827adantr 453 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
2928, 14sseldd 3350 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
3016, 17, 18, 19, 20, 6, 23, 29psrvsca 16456 . . 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 ) )
3110, 20ressmulr 13583 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  H ) )
32 ofeq 6308 . . . . 5  |-  ( ( .r `  R )  =  ( .r `  H )  ->  o F ( .r `  R )  =  o F ( .r `  H ) )
339, 31, 323syl 19 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  o F ( .r `  R )  =  o F ( .r `  H ) )
3433oveqd 6099 . . 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 ) )
3530, 34eqtrd 2469 . 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 ) )
36 fvex 5743 . . . . 5  |-  ( Base `  U )  e.  _V
374, 36eqeltri 2507 . . . 4  |-  B  e. 
_V
3824, 17ressvsca 13606 . . . 4  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
3937, 38mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( .s `  S
)  =  ( .s
`  P ) )
4039oveqd 6099 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( X ( .s `  P
) Y ) )
4115, 35, 403eqtr2d 2475 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 360    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957    C_ wss 3321   {csn 3815    X. cxp 4877   `'ccnv 4878   "cima 4882   ` cfv 5455  (class class class)co 6082    o Fcof 6304    ^m cmap 7019   Fincfn 7110   NNcn 10001   NN0cn0 10222   Basecbs 13470   ↾s cress 13471   .rcmulr 13531   .scvsca 13534  SubRingcsubrg 15865   mPwSer cmps 16407
This theorem is referenced by:  ressmplvsca  16523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-tset 13549  df-subg 14942  df-rng 15664  df-subrg 15867  df-psr 16418
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