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

Theorem resspsradd 16479
Description: A restricted power series algebra has the same addition 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
resspsradd  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )

Proof of Theorem resspsradd
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 resspsr.u . . 3  |-  U  =  ( I mPwSer  H )
2 resspsr.b . . 3  |-  B  =  ( Base `  U
)
3 eqid 2436 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
4 eqid 2436 . . 3  |-  ( +g  `  U )  =  ( +g  `  U )
5 simprl 733 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  B )
6 simprr 734 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  B )
71, 2, 3, 4, 5, 6psradd 16446 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X  o F ( +g  `  H
) Y ) )
8 resspsr.s . . . 4  |-  S  =  ( I mPwSer  R )
9 eqid 2436 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
10 eqid 2436 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
11 eqid 2436 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
12 fvex 5742 . . . . . . . 8  |-  ( Base `  R )  e.  _V
13 resspsr.2 . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubRing `  R
) )
14 resspsr.h . . . . . . . . . . 11  |-  H  =  ( Rs  T )
1514subrgbas 15877 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
1613, 15syl 16 . . . . . . . . 9  |-  ( ph  ->  T  =  ( Base `  H ) )
17 eqid 2436 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
1817subrgss 15869 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
1913, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  T  C_  ( Base `  R ) )
2016, 19eqsstr3d 3383 . . . . . . . 8  |-  ( ph  ->  ( Base `  H
)  C_  ( Base `  R ) )
21 mapss 7056 . . . . . . . 8  |-  ( ( ( Base `  R
)  e.  _V  /\  ( Base `  H )  C_  ( Base `  R
) )  ->  (
( Base `  H )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } )  C_  (
( Base `  R )  ^m  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } ) )
2212, 20, 21sylancr 645 . . . . . . 7  |-  ( ph  ->  ( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
2322adantr 452 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( ( Base `  H
)  ^m  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } ) 
C_  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
24 eqid 2436 . . . . . . 7  |-  ( Base `  H )  =  (
Base `  H )
25 eqid 2436 . . . . . . 7  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
26 reldmpsr 16428 . . . . . . . . . 10  |-  Rel  dom mPwSer
2726, 1, 2elbasov 13513 . . . . . . . . 9  |-  ( X  e.  B  ->  (
I  e.  _V  /\  H  e.  _V )
)
2827ad2antrl 709 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( I  e.  _V  /\  H  e.  _V )
)
2928simpld 446 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  I  e.  _V )
301, 24, 25, 2, 29psrbas 16443 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  =  ( ( Base `  H )  ^m  { f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
318, 17, 25, 9, 29psrbas 16443 . . . . . 6  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( Base `  S )  =  ( ( Base `  R )  ^m  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } ) )
3223, 30, 313sstr4d 3391 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  B  C_  ( Base `  S
) )
3332, 5sseldd 3349 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  X  e.  ( Base `  S ) )
3432, 6sseldd 3349 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  Y  e.  ( Base `  S ) )
358, 9, 10, 11, 33, 34psradd 16446 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  o F ( +g  `  R
) Y ) )
3614, 10ressplusg 13571 . . . . . . 7  |-  ( T  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  H ) )
3713, 36syl 16 . . . . . 6  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  H ) )
3837adantr 452 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  R )  =  ( +g  `  H
) )
39 ofeq 6307 . . . . 5  |-  ( ( +g  `  R )  =  ( +g  `  H
)  ->  o F
( +g  `  R )  =  o F ( +g  `  H ) )
4038, 39syl 16 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  o F ( +g  `  R
)  =  o F ( +g  `  H
) )
4140oveqd 6098 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X  o F ( +g  `  R
) Y )  =  ( X  o F ( +g  `  H
) Y ) )
4235, 41eqtrd 2468 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X  o F ( +g  `  H
) Y ) )
43 fvex 5742 . . . . 5  |-  ( Base `  U )  e.  _V
442, 43eqeltri 2506 . . . 4  |-  B  e. 
_V
45 resspsr.p . . . . 5  |-  P  =  ( Ss  B )
4645, 11ressplusg 13571 . . . 4  |-  ( B  e.  _V  ->  ( +g  `  S )  =  ( +g  `  P
) )
4744, 46mp1i 12 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( +g  `  S )  =  ( +g  `  P
) )
4847oveqd 6098 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  S ) Y )  =  ( X ( +g  `  P ) Y ) )
497, 42, 483eqtr2d 2474 1  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    C_ wss 3320   `'ccnv 4877   "cima 4881   ` cfv 5454  (class class class)co 6081    o Fcof 6303    ^m cmap 7018   Fincfn 7109   NNcn 10000   NN0cn0 10221   Basecbs 13469   ↾s cress 13470   +g cplusg 13529  SubRingcsubrg 15864   mPwSer cmps 16406
This theorem is referenced by:  subrgpsr  16482  ressmpladd  16520
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
  Copyright terms: Public domain W3C validator