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Theorem ressmplvsca 16522
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 2436 . . . . 5  |-  ( I mPwSer  H )  =  ( I mPwSer  H )
3 ressmpl.b . . . . 5  |-  B  =  ( Base `  U
)
4 eqid 2436 . . . . 5  |-  ( Base `  ( I mPwSer  H ) )  =  ( Base `  ( I mPwSer  H ) )
51, 2, 3, 4mplbasss 16496 . . . 4  |-  B  C_  ( Base `  ( I mPwSer  H ) )
65sseli 3344 . . 3  |-  ( Y  e.  B  ->  Y  e.  ( Base `  (
I mPwSer  H ) ) )
7 eqid 2436 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
8 ressmpl.h . . . 4  |-  H  =  ( Rs  T )
9 eqid 2436 . . . 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 16481 . . 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 635 . 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 5742 . . . . 5  |-  ( Base `  U )  e.  _V
143, 13eqeltri 2506 . . . 4  |-  B  e. 
_V
151, 2, 3mplval2 16495 . . . . 5  |-  U  =  ( ( I mPwSer  H
)s 
B )
16 eqid 2436 . . . . 5  |-  ( .s
`  ( I mPwSer  H
) )  =  ( .s `  ( I mPwSer  H ) )
1715, 16ressvsca 13605 . . . 4  |-  ( B  e.  _V  ->  ( .s `  ( I mPwSer  H
) )  =  ( .s `  U ) )
1814, 17ax-mp 8 . . 3  |-  ( .s
`  ( I mPwSer  H
) )  =  ( .s `  U )
1918oveqi 6094 . 2  |-  ( X ( .s `  (
I mPwSer  H ) ) Y )  =  ( X ( .s `  U
) Y )
20 fvex 5742 . . . . 5  |-  ( Base `  S )  e.  _V
21 ressmpl.s . . . . . . 7  |-  S  =  ( I mPoly  R )
22 eqid 2436 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2321, 7, 22mplval2 16495 . . . . . 6  |-  S  =  ( ( I mPwSer  R
)s  ( Base `  S
) )
24 eqid 2436 . . . . . 6  |-  ( .s
`  ( I mPwSer  R
) )  =  ( .s `  ( I mPwSer  R ) )
2523, 24ressvsca 13605 . . . . 5  |-  ( (
Base `  S )  e.  _V  ->  ( .s `  ( I mPwSer  R ) )  =  ( .s
`  S ) )
2620, 25ax-mp 8 . . . 4  |-  ( .s
`  ( I mPwSer  R
) )  =  ( .s `  S )
27 fvex 5742 . . . . 5  |-  ( Base `  ( I mPwSer  H ) )  e.  _V
289, 24ressvsca 13605 . . . . 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 2436 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
3230, 31ressvsca 13605 . . . . 5  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
3314, 32ax-mp 8 . . . 4  |-  ( .s
`  S )  =  ( .s `  P
)
3426, 29, 333eqtr3i 2464 . . 3  |-  ( .s
`  ( ( I mPwSer  R )s  ( Base `  (
I mPwSer  H ) ) ) )  =  ( .s
`  P )
3534oveqi 6094 . 2  |-  ( X ( .s `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) Y )  =  ( X ( .s `  P ) Y )
3612, 19, 353eqtr3g 2491 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   .scvsca 13533  SubRingcsubrg 15864   mPwSer cmps 16406   mPoly cmpl 16408
This theorem is referenced by:  ressply1vsca  16626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-subg 14941  df-rng 15663  df-subrg 15866  df-psr 16417  df-mpl 16419
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