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Theorem ressmplvsca 16203
Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
ressmpl.s  |-  S  =  ( I mPoly  R )
ressmpl.h  |-  H  =  ( Rs  T )
ressmpl.u  |-  U  =  ( I mPoly  H )
ressmpl.b  |-  B  =  ( Base `  U
)
ressmpl.1  |-  ( ph  ->  I  e.  V )
ressmpl.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
ressmpl.p  |-  P  =  ( Ss  B )
Assertion
Ref Expression
ressmplvsca  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( X ( .s `  P
) Y ) )

Proof of Theorem ressmplvsca
StepHypRef Expression
1 ressmpl.u . . . . 5  |-  U  =  ( I mPoly  H )
2 eqid 2283 . . . . 5  |-  ( I mPwSer  H )  =  ( I mPwSer  H )
3 ressmpl.b . . . . 5  |-  B  =  ( Base `  U
)
4 eqid 2283 . . . . 5  |-  ( Base `  ( I mPwSer  H ) )  =  ( Base `  ( I mPwSer  H ) )
51, 2, 3, 4mplbasss 16177 . . . 4  |-  B  C_  ( Base `  ( I mPwSer  H ) )
65sseli 3176 . . 3  |-  ( Y  e.  B  ->  Y  e.  ( Base `  (
I mPwSer  H ) ) )
7 eqid 2283 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
8 ressmpl.h . . . 4  |-  H  =  ( Rs  T )
9 eqid 2283 . . . 4  |-  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) )  =  ( ( I mPwSer  R )s  (
Base `  ( I mPwSer  H ) ) )
10 ressmpl.2 . . . 4  |-  ( ph  ->  T  e.  (SubRing `  R
) )
117, 8, 2, 4, 9, 10resspsrvsca 16162 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  ( Base `  (
I mPwSer  H ) ) ) )  ->  ( X
( .s `  (
I mPwSer  H ) ) Y )  =  ( X ( .s `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) Y ) )
126, 11sylanr2 634 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  ( I mPwSer  H
) ) Y )  =  ( X ( .s `  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) ) ) Y ) )
13 fvex 5539 . . . . 5  |-  ( Base `  U )  e.  _V
143, 13eqeltri 2353 . . . 4  |-  B  e. 
_V
151, 2, 3mplval2 16176 . . . . 5  |-  U  =  ( ( I mPwSer  H
)s 
B )
16 eqid 2283 . . . . 5  |-  ( .s
`  ( I mPwSer  H
) )  =  ( .s `  ( I mPwSer  H ) )
1715, 16ressvsca 13284 . . . 4  |-  ( B  e.  _V  ->  ( .s `  ( I mPwSer  H
) )  =  ( .s `  U ) )
1814, 17ax-mp 8 . . 3  |-  ( .s
`  ( I mPwSer  H
) )  =  ( .s `  U )
1918oveqi 5871 . 2  |-  ( X ( .s `  (
I mPwSer  H ) ) Y )  =  ( X ( .s `  U
) Y )
20 fvex 5539 . . . . 5  |-  ( Base `  S )  e.  _V
21 ressmpl.s . . . . . . 7  |-  S  =  ( I mPoly  R )
22 eqid 2283 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2321, 7, 22mplval2 16176 . . . . . 6  |-  S  =  ( ( I mPwSer  R
)s  ( Base `  S
) )
24 eqid 2283 . . . . . 6  |-  ( .s
`  ( I mPwSer  R
) )  =  ( .s `  ( I mPwSer  R ) )
2523, 24ressvsca 13284 . . . . 5  |-  ( (
Base `  S )  e.  _V  ->  ( .s `  ( I mPwSer  R ) )  =  ( .s
`  S ) )
2620, 25ax-mp 8 . . . 4  |-  ( .s
`  ( I mPwSer  R
) )  =  ( .s `  S )
27 fvex 5539 . . . . 5  |-  ( Base `  ( I mPwSer  H ) )  e.  _V
289, 24ressvsca 13284 . . . . 5  |-  ( (
Base `  ( I mPwSer  H ) )  e.  _V  ->  ( .s `  (
I mPwSer  R ) )  =  ( .s `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) )
2927, 28ax-mp 8 . . . 4  |-  ( .s
`  ( I mPwSer  R
) )  =  ( .s `  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) ) )
30 ressmpl.p . . . . . 6  |-  P  =  ( Ss  B )
31 eqid 2283 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
3230, 31ressvsca 13284 . . . . 5  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
3314, 32ax-mp 8 . . . 4  |-  ( .s
`  S )  =  ( .s `  P
)
3426, 29, 333eqtr3i 2311 . . 3  |-  ( .s
`  ( ( I mPwSer  R )s  ( Base `  (
I mPwSer  H ) ) ) )  =  ( .s
`  P )
3534oveqi 5871 . 2  |-  ( X ( .s `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) Y )  =  ( X ( .s `  P ) Y )
3612, 19, 353eqtr3g 2338 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 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   .scvsca 13212  SubRingcsubrg 15541   mPwSer cmps 16087   mPoly cmpl 16089
This theorem is referenced by:  ressply1vsca  16310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-subg 14618  df-rng 15340  df-subrg 15543  df-psr 16098  df-mpl 16100
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