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Theorem resspsrvsca 18200
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 2457 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
3 eqid 2457 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4 resspsr.b . . 3  |-  B  =  ( Base `  U
)
5 eqid 2457 . . 3  |-  ( .r
`  H )  =  ( .r `  H
)
6 eqid 2457 . . 3  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 simprl 756 . . . 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 17565 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
129, 11syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  =  ( Base `  H ) )
137, 12eleqtrd 2547 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  X  e.  ( Base `  H ) )
14 simprr 757 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  B )
151, 2, 3, 4, 5, 6, 13, 14psrvsca 18171 . 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 2457 . . . 4  |-  ( .s
`  S )  =  ( .s `  S
)
18 eqid 2457 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2457 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2457 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
2118subrgss 17557 . . . . . 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 3500 . . . 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 18197 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  P ) )
2624, 19ressbasss 14703 . . . . . . 7  |-  ( Base `  P )  C_  ( Base `  S )
2725, 26syl6eqss 3549 . . . . . 6  |-  ( ph  ->  B  C_  ( Base `  S ) )
2827adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
2928, 14sseldd 3500 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
3016, 17, 18, 19, 20, 6, 23, 29psrvsca 18171 . . 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 14769 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  H ) )
32 ofeq 6541 . . . . 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 6313 . . 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 2498 . 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 5882 . . . . 5  |-  ( Base `  U )  e.  _V
374, 36eqeltri 2541 . . . 4  |-  B  e. 
_V
3824, 17ressvsca 14795 . . . 4  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
3937, 38mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( .s `  S
)  =  ( .s
`  P ) )
4039oveqd 6313 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( X ( .s `  P
) Y ) )
4115, 35, 403eqtr2d 2504 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    C_ wss 3471   {csn 4032    X. cxp 5006   `'ccnv 5007   "cima 5011   ` cfv 5594  (class class class)co 6296    oFcof 6537    ^m cmap 7438   Fincfn 7535   NNcn 10556   NN0cn0 10816   Basecbs 14644   ↾s cress 14645   .rcmulr 14713   .scvsca 14716  SubRingcsubrg 17552   mPwSer cmps 18127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-tset 14731  df-subg 16325  df-ring 17327  df-subrg 17554  df-psr 18132
This theorem is referenced by:  ressmplvsca  18248
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