MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resspsrvsca Structured version   Unicode version

Theorem resspsrvsca 17872
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 2467 . . 3  |-  ( .s
`  U )  =  ( .s `  U
)
3 eqid 2467 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4 resspsr.b . . 3  |-  B  =  ( Base `  U
)
5 eqid 2467 . . 3  |-  ( .r
`  H )  =  ( .r `  H
)
6 eqid 2467 . . 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 17238 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
129, 11syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  T  =  ( Base `  H ) )
137, 12eleqtrd 2557 . . 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 17843 . 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 2467 . . . 4  |-  ( .s
`  S )  =  ( .s `  S
)
18 eqid 2467 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2467 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
20 eqid 2467 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
2118subrgss 17230 . . . . . 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 3505 . . . 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 17869 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  P ) )
2624, 19ressbasss 14547 . . . . . . 7  |-  ( Base `  P )  C_  ( Base `  S )
2725, 26syl6eqss 3554 . . . . . 6  |-  ( ph  ->  B  C_  ( Base `  S ) )
2827adantr 465 . . . . 5  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
2928, 14sseldd 3505 . . . 4  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
3016, 17, 18, 19, 20, 6, 23, 29psrvsca 17843 . . 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 14608 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  H ) )
32 ofeq 6526 . . . . 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 6301 . . 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 2508 . 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 5876 . . . . 5  |-  ( Base `  U )  e.  _V
374, 36eqeltri 2551 . . . 4  |-  B  e. 
_V
3824, 17ressvsca 14634 . . . 4  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
3937, 38mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( .s `  S
)  =  ( .s
`  P ) )
4039oveqd 6301 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  S ) Y )  =  ( X ( .s `  P
) Y ) )
4115, 35, 403eqtr2d 2514 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 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   {csn 4027    X. cxp 4997   `'ccnv 4998   "cima 5002   ` cfv 5588  (class class class)co 6284    oFcof 6522    ^m cmap 7420   Fincfn 7516   NNcn 10536   NN0cn0 10795   Basecbs 14490   ↾s cress 14491   .rcmulr 14556   .scvsca 14559  SubRingcsubrg 17225   mPwSer cmps 17799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-tset 14574  df-subg 16003  df-rng 17002  df-subrg 17227  df-psr 17804
This theorem is referenced by:  ressmplvsca  17920
  Copyright terms: Public domain W3C validator