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Theorem mplbaspropd 16524
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 2400 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
43psrbaspropd 16522 . . . . 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 14609 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
87sneqd 3742 . . . . . . . 8  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  S ) } )
98difeq2d 3381 . . . . . . 7  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  S
) } ) )
109imaeq2d 5115 . . . . . 6  |-  ( ph  ->  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' a
" ( _V  \  { ( 0g `  S ) } ) ) )
1110eleq1d 2432 . . . . 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 2868 . . 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 2366 . . . 4  |-  ( I mPoly 
R )  =  ( I mPoly  R )
15 eqid 2366 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
16 eqid 2366 . . . 4  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
17 eqid 2366 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
18 eqid 2366 . . . 4  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
1914, 15, 16, 17, 18mplbas 16384 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  { a  e.  ( Base `  (
I mPwSer  R ) )  |  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin }
20 eqid 2366 . . . 4  |-  ( I mPoly 
S )  =  ( I mPoly  S )
21 eqid 2366 . . . 4  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
22 eqid 2366 . . . 4  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
23 eqid 2366 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
24 eqid 2366 . . . 4  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
2520, 21, 22, 23, 24mplbas 16384 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  { a  e.  ( Base `  (
I mPwSer  S ) )  |  ( `' a "
( _V  \  {
( 0g `  S
) } ) )  e.  Fin }
2613, 19, 253eqtr4g 2423 . 2  |-  ( (
ph  /\  I  e.  _V )  ->  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  S ) ) )
27 reldmmpl 16382 . . . . . 6  |-  Rel  dom mPoly
2827ovprc1 6009 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  (/) )
2927ovprc1 6009 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  S )  =  (/) )
3028, 29eqtr4d 2401 . . . 4  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  ( I mPoly  S ) )
3130fveq2d 5636 . . 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 1647    e. wcel 1715   {crab 2632   _Vcvv 2873    \ cdif 3235   (/)c0 3543   {csn 3729   `'ccnv 4791   "cima 4795   ` cfv 5358  (class class class)co 5981   Fincfn 7006   Basecbs 13356   +g cplusg 13416   0gc0g 13610   mPwSer cmps 16297   mPoly cmpl 16299
This theorem is referenced by:  ply1baspropd  16531  mdegpropd  19685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-0g 13614  df-psr 16308  df-mpl 16310
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