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Theorem ressmpladd 16448
Description: A restricted polynomial algebra has the same addition 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
ressmpladd  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )

Proof of Theorem ressmpladd
StepHypRef Expression
1 ressmpl.u . . . . . 6  |-  U  =  ( I mPoly  H )
2 eqid 2388 . . . . . 6  |-  ( I mPwSer  H )  =  ( I mPwSer  H )
3 ressmpl.b . . . . . 6  |-  B  =  ( Base `  U
)
4 eqid 2388 . . . . . 6  |-  ( Base `  ( I mPwSer  H ) )  =  ( Base `  ( I mPwSer  H ) )
51, 2, 3, 4mplbasss 16424 . . . . 5  |-  B  C_  ( Base `  ( I mPwSer  H ) )
65sseli 3288 . . . 4  |-  ( X  e.  B  ->  X  e.  ( Base `  (
I mPwSer  H ) ) )
75sseli 3288 . . . 4  |-  ( Y  e.  B  ->  Y  e.  ( Base `  (
I mPwSer  H ) ) )
86, 7anim12i 550 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  e.  (
Base `  ( I mPwSer  H ) )  /\  Y  e.  ( Base `  (
I mPwSer  H ) ) ) )
9 eqid 2388 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
10 ressmpl.h . . . 4  |-  H  =  ( Rs  T )
11 eqid 2388 . . . 4  |-  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) )  =  ( ( I mPwSer  R )s  (
Base `  ( I mPwSer  H ) ) )
12 ressmpl.2 . . . 4  |-  ( ph  ->  T  e.  (SubRing `  R
) )
139, 10, 2, 4, 11, 12resspsradd 16407 . . 3  |-  ( (
ph  /\  ( X  e.  ( Base `  (
I mPwSer  H ) )  /\  Y  e.  ( Base `  ( I mPwSer  H ) ) ) )  -> 
( X ( +g  `  ( I mPwSer  H ) ) Y )  =  ( X ( +g  `  ( ( I mPwSer  R
)s  ( Base `  (
I mPwSer  H ) ) ) ) Y ) )
148, 13sylan2 461 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( +g  `  ( I mPwSer  H ) ) Y )  =  ( X ( +g  `  ( ( I mPwSer  R
)s  ( Base `  (
I mPwSer  H ) ) ) ) Y ) )
15 fvex 5683 . . . . 5  |-  ( Base `  U )  e.  _V
163, 15eqeltri 2458 . . . 4  |-  B  e. 
_V
171, 2, 3mplval2 16423 . . . . 5  |-  U  =  ( ( I mPwSer  H
)s 
B )
18 eqid 2388 . . . . 5  |-  ( +g  `  ( I mPwSer  H ) )  =  ( +g  `  ( I mPwSer  H ) )
1917, 18ressplusg 13499 . . . 4  |-  ( B  e.  _V  ->  ( +g  `  ( I mPwSer  H
) )  =  ( +g  `  U ) )
2016, 19ax-mp 8 . . 3  |-  ( +g  `  ( I mPwSer  H ) )  =  ( +g  `  U )
2120oveqi 6034 . 2  |-  ( X ( +g  `  (
I mPwSer  H ) ) Y )  =  ( X ( +g  `  U
) Y )
22 fvex 5683 . . . . 5  |-  ( Base `  S )  e.  _V
23 ressmpl.s . . . . . . 7  |-  S  =  ( I mPoly  R )
24 eqid 2388 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2523, 9, 24mplval2 16423 . . . . . 6  |-  S  =  ( ( I mPwSer  R
)s  ( Base `  S
) )
26 eqid 2388 . . . . . 6  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  R ) )
2725, 26ressplusg 13499 . . . . 5  |-  ( (
Base `  S )  e.  _V  ->  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  S ) )
2822, 27ax-mp 8 . . . 4  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  S )
29 fvex 5683 . . . . 5  |-  ( Base `  ( I mPwSer  H ) )  e.  _V
3011, 26ressplusg 13499 . . . . 5  |-  ( (
Base `  ( I mPwSer  H ) )  e.  _V  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) )
3129, 30ax-mp 8 . . . 4  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( ( I mPwSer  R
)s  ( Base `  (
I mPwSer  H ) ) ) )
32 ressmpl.p . . . . . 6  |-  P  =  ( Ss  B )
33 eqid 2388 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
3432, 33ressplusg 13499 . . . . 5  |-  ( B  e.  _V  ->  ( +g  `  S )  =  ( +g  `  P
) )
3516, 34ax-mp 8 . . . 4  |-  ( +g  `  S )  =  ( +g  `  P )
3628, 31, 353eqtr3i 2416 . . 3  |-  ( +g  `  ( ( I mPwSer  R
)s  ( Base `  (
I mPwSer  H ) ) ) )  =  ( +g  `  P )
3736oveqi 6034 . 2  |-  ( X ( +g  `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) Y )  =  ( X ( +g  `  P
) Y )
3814, 21, 373eqtr3g 2443 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 1649    e. wcel 1717   _Vcvv 2900   ` cfv 5395  (class class class)co 6021   Basecbs 13397   ↾s cress 13398   +g cplusg 13457  SubRingcsubrg 15792   mPwSer cmps 16334   mPoly cmpl 16336
This theorem is referenced by:  ressply1add  16552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-tset 13476  df-subg 14869  df-rng 15591  df-subrg 15794  df-psr 16345  df-mpl 16347
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