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Theorem mplbaspropd 16585
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 2438 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
43psrbaspropd 16583 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
54adantr 452 . . . 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 14677 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
87sneqd 3787 . . . . . . . 8  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  S ) } )
98difeq2d 3425 . . . . . . 7  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  S
) } ) )
109imaeq2d 5162 . . . . . 6  |-  ( ph  ->  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' a
" ( _V  \  { ( 0g `  S ) } ) ) )
1110eleq1d 2470 . . . . 5  |-  ( ph  ->  ( ( `' a
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin  <->  ( `' a " ( _V  \  { ( 0g `  S ) } ) )  e.  Fin )
)
1211adantr 452 . . . 4  |-  ( (
ph  /\  I  e.  _V )  ->  ( ( `' a " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin  <->  ( `' a " ( _V  \  { ( 0g
`  S ) } ) )  e.  Fin ) )
135, 12rabeqbidv 2911 . . 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 2404 . . . 4  |-  ( I mPoly 
R )  =  ( I mPoly  R )
15 eqid 2404 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
16 eqid 2404 . . . 4  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
17 eqid 2404 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
18 eqid 2404 . . . 4  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
1914, 15, 16, 17, 18mplbas 16448 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  { a  e.  ( Base `  (
I mPwSer  R ) )  |  ( `' a "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin }
20 eqid 2404 . . . 4  |-  ( I mPoly 
S )  =  ( I mPoly  S )
21 eqid 2404 . . . 4  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
22 eqid 2404 . . . 4  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
23 eqid 2404 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
24 eqid 2404 . . . 4  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
2520, 21, 22, 23, 24mplbas 16448 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  { a  e.  ( Base `  (
I mPwSer  S ) )  |  ( `' a "
( _V  \  {
( 0g `  S
) } ) )  e.  Fin }
2613, 19, 253eqtr4g 2461 . 2  |-  ( (
ph  /\  I  e.  _V )  ->  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  S ) ) )
27 reldmmpl 16446 . . . . . 6  |-  Rel  dom mPoly
2827ovprc1 6068 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  (/) )
2927ovprc1 6068 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  S )  =  (/) )
3028, 29eqtr4d 2439 . . . 4  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  ( I mPoly  S ) )
3130fveq2d 5691 . . 3  |-  ( -.  I  e.  _V  ->  (
Base `  ( I mPoly  R ) )  =  (
Base `  ( I mPoly  S ) ) )
3231adantl 453 . 2  |-  ( (
ph  /\  -.  I  e.  _V )  ->  ( Base `  ( I mPoly  R
) )  =  (
Base `  ( I mPoly  S ) ) )
3326, 32pm2.61dan 767 1  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    \ cdif 3277   (/)c0 3588   {csn 3774   `'ccnv 4836   "cima 4840   ` cfv 5413  (class class class)co 6040   Fincfn 7068   Basecbs 13424   +g cplusg 13484   0gc0g 13678   mPwSer cmps 16361   mPoly cmpl 16363
This theorem is referenced by:  ply1baspropd  16592  mdegpropd  19960
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-0g 13682  df-psr 16372  df-mpl 16374
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