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

Theorem ressmplmul 16251
Description: A restricted polynomial algebra has the same 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
ressmplmul  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( .r
`  U ) Y )  =  ( X ( .r `  P
) Y ) )

Proof of Theorem ressmplmul
StepHypRef Expression
1 ressmpl.u . . . . . 6  |-  U  =  ( I mPoly  H )
2 eqid 2316 . . . . . 6  |-  ( I mPwSer  H )  =  ( I mPwSer  H )
3 ressmpl.b . . . . . 6  |-  B  =  ( Base `  U
)
4 eqid 2316 . . . . . 6  |-  ( Base `  ( I mPwSer  H ) )  =  ( Base `  ( I mPwSer  H ) )
51, 2, 3, 4mplbasss 16226 . . . . 5  |-  B  C_  ( Base `  ( I mPwSer  H ) )
65sseli 3210 . . . 4  |-  ( X  e.  B  ->  X  e.  ( Base `  (
I mPwSer  H ) ) )
75sseli 3210 . . . 4  |-  ( Y  e.  B  ->  Y  e.  ( Base `  (
I mPwSer  H ) ) )
86, 7anim12i 549 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  e.  (
Base `  ( I mPwSer  H ) )  /\  Y  e.  ( Base `  (
I mPwSer  H ) ) ) )
9 eqid 2316 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
10 ressmpl.h . . . 4  |-  H  =  ( Rs  T )
11 eqid 2316 . . . 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, 12resspsrmul 16210 . . 3  |-  ( (
ph  /\  ( X  e.  ( Base `  (
I mPwSer  H ) )  /\  Y  e.  ( Base `  ( I mPwSer  H ) ) ) )  -> 
( X ( .r
`  ( I mPwSer  H
) ) Y )  =  ( X ( .r `  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) ) ) Y ) )
148, 13sylan2 460 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( .r
`  ( I mPwSer  H
) ) Y )  =  ( X ( .r `  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) ) ) Y ) )
15 fvex 5577 . . . . 5  |-  ( Base `  U )  e.  _V
163, 15eqeltri 2386 . . . 4  |-  B  e. 
_V
171, 2, 3mplval2 16225 . . . . 5  |-  U  =  ( ( I mPwSer  H
)s 
B )
18 eqid 2316 . . . . 5  |-  ( .r
`  ( I mPwSer  H
) )  =  ( .r `  ( I mPwSer  H ) )
1917, 18ressmulr 13308 . . . 4  |-  ( B  e.  _V  ->  ( .r `  ( I mPwSer  H
) )  =  ( .r `  U ) )
2016, 19ax-mp 8 . . 3  |-  ( .r
`  ( I mPwSer  H
) )  =  ( .r `  U )
2120oveqi 5913 . 2  |-  ( X ( .r `  (
I mPwSer  H ) ) Y )  =  ( X ( .r `  U
) Y )
22 fvex 5577 . . . . 5  |-  ( Base `  S )  e.  _V
23 ressmpl.s . . . . . . 7  |-  S  =  ( I mPoly  R )
24 eqid 2316 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
2523, 9, 24mplval2 16225 . . . . . 6  |-  S  =  ( ( I mPwSer  R
)s  ( Base `  S
) )
26 eqid 2316 . . . . . 6  |-  ( .r
`  ( I mPwSer  R
) )  =  ( .r `  ( I mPwSer  R ) )
2725, 26ressmulr 13308 . . . . 5  |-  ( (
Base `  S )  e.  _V  ->  ( .r `  ( I mPwSer  R ) )  =  ( .r
`  S ) )
2822, 27ax-mp 8 . . . 4  |-  ( .r
`  ( I mPwSer  R
) )  =  ( .r `  S )
29 fvex 5577 . . . . 5  |-  ( Base `  ( I mPwSer  H ) )  e.  _V
3011, 26ressmulr 13308 . . . . 5  |-  ( (
Base `  ( I mPwSer  H ) )  e.  _V  ->  ( .r `  (
I mPwSer  R ) )  =  ( .r `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) )
3129, 30ax-mp 8 . . . 4  |-  ( .r
`  ( I mPwSer  R
) )  =  ( .r `  ( ( I mPwSer  R )s  ( Base `  ( I mPwSer  H ) ) ) )
32 ressmpl.p . . . . . 6  |-  P  =  ( Ss  B )
33 eqid 2316 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
3432, 33ressmulr 13308 . . . . 5  |-  ( B  e.  _V  ->  ( .r `  S )  =  ( .r `  P
) )
3516, 34ax-mp 8 . . . 4  |-  ( .r
`  S )  =  ( .r `  P
)
3628, 31, 353eqtr3i 2344 . . 3  |-  ( .r
`  ( ( I mPwSer  R )s  ( Base `  (
I mPwSer  H ) ) ) )  =  ( .r
`  P )
3736oveqi 5913 . 2  |-  ( X ( .r `  (
( I mPwSer  R )s  ( Base `  ( I mPwSer  H
) ) ) ) Y )  =  ( X ( .r `  P ) Y )
3814, 21, 373eqtr3g 2371 1  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( .r
`  U ) Y )  =  ( X ( .r `  P
) Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822   ` cfv 5292  (class class class)co 5900   Basecbs 13195   ↾s cress 13196   .rcmulr 13256  SubRingcsubrg 15590   mPwSer cmps 16136   mPoly cmpl 16138
This theorem is referenced by:  ressply1mul  16358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-ofr 6121  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-seq 11094  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-tset 13274  df-0g 13453  df-gsum 13454  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-subg 14667  df-mgp 15375  df-rng 15389  df-subrg 15592  df-psr 16147  df-mpl 16149
  Copyright terms: Public domain W3C validator