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Theorem mplbaspropd 16314
Description: Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
psrplusgpropd.b1  |-  ( ph  ->  B  =  ( Base `  R ) )
psrplusgpropd.b2  |-  ( ph  ->  B  =  ( Base `  S ) )
psrplusgpropd.p  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
Assertion
Ref Expression
mplbaspropd  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
Distinct variable groups:    ph, y, x   
x, B, y    y, R, x    y, S, x
Allowed substitution hints:    I( x, y)

Proof of Theorem mplbaspropd
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 psrplusgpropd.b1 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  R ) )
2 psrplusgpropd.b2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  S ) )
31, 2eqtr3d 2317 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
43psrbaspropd 16312 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
54adantr 451 . . . 4  |-  ( (
ph  /\  I  e.  _V )  ->  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  S ) ) )
6 psrplusgpropd.p . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
71, 2, 6grpidpropd 14399 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
87sneqd 3653 . . . . . . . 8  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  S ) } )
98difeq2d 3294 . . . . . . 7  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  S
) } ) )
109imaeq2d 5012 . . . . . 6  |-  ( ph  ->  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' a
" ( _V  \  { ( 0g `  S ) } ) ) )
1110eleq1d 2349 . . . . 5  |-  ( ph  ->  ( ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin  <->  ( `' a " ( _V  \  { ( 0g `  S ) } ) )  e.  Fin )
)
1211adantr 451 . . . 4  |-  ( (
ph  /\  I  e.  _V )  ->  ( ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin  <->  ( `' a " ( _V  \  { ( 0g
`  S ) } ) )  e.  Fin ) )
135, 12rabeqbidv 2783 . . 3  |-  ( (
ph  /\  I  e.  _V )  ->  { a  e.  ( Base `  (
I mPwSer  R ) )  |  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin }  =  { a  e.  (
Base `  ( I mPwSer  S ) )  |  ( `' a " ( _V  \  { ( 0g
`  S ) } ) )  e.  Fin } )
14 eqid 2283 . . . 4  |-  ( I mPoly 
R )  =  ( I mPoly  R )
15 eqid 2283 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
16 eqid 2283 . . . 4  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
17 eqid 2283 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
18 eqid 2283 . . . 4  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
1914, 15, 16, 17, 18mplbas 16174 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  { a  e.  ( Base `  (
I mPwSer  R ) )  |  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin }
20 eqid 2283 . . . 4  |-  ( I mPoly 
S )  =  ( I mPoly  S )
21 eqid 2283 . . . 4  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
22 eqid 2283 . . . 4  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
23 eqid 2283 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
24 eqid 2283 . . . 4  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
2520, 21, 22, 23, 24mplbas 16174 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  { a  e.  ( Base `  (
I mPwSer  S ) )  |  ( `' a "
( _V  \  {
( 0g `  S
) } ) )  e.  Fin }
2613, 19, 253eqtr4g 2340 . 2  |-  ( (
ph  /\  I  e.  _V )  ->  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  S ) ) )
27 reldmmpl 16172 . . . . . 6  |-  Rel  dom mPoly
2827ovprc1 5886 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  (/) )
2927ovprc1 5886 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  S )  =  (/) )
3028, 29eqtr4d 2318 . . . 4  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  ( I mPoly  S ) )
3130fveq2d 5529 . . 3  |-  ( -.  I  e.  _V  ->  (
Base `  ( I mPoly  R ) )  =  (
Base `  ( I mPoly  S ) ) )
3231adantl 452 . 2  |-  ( (
ph  /\  -.  I  e.  _V )  ->  ( Base `  ( I mPoly  R
) )  =  (
Base `  ( I mPoly  S ) ) )
3326, 32pm2.61dan 766 1  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   +g cplusg 13208   0gc0g 13400   mPwSer cmps 16087   mPoly cmpl 16089
This theorem is referenced by:  ply1baspropd  16321  mdegpropd  19470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-psr 16098  df-mpl 16100
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