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Theorem mplbaspropd 18907
Description: Property deduction for polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Jul-2019.)
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 2507 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
43psrbaspropd 18905 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
54adantr 472 . . . 4  |-  ( (
ph  /\  I  e.  _V )  ->  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  S ) ) )
6 psrplusgpropd.p . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
71, 2, 6grpidpropd 16582 . . . . . 6  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
87breq2d 4407 . . . . 5  |-  ( ph  ->  ( a finSupp  ( 0g
`  R )  <->  a finSupp  ( 0g
`  S ) ) )
98adantr 472 . . . 4  |-  ( (
ph  /\  I  e.  _V )  ->  ( a finSupp 
( 0g `  R
)  <->  a finSupp  ( 0g `  S ) ) )
105, 9rabeqbidv 3026 . . 3  |-  ( (
ph  /\  I  e.  _V )  ->  { a  e.  ( Base `  (
I mPwSer  R ) )  |  a finSupp  ( 0g `  R ) }  =  { a  e.  (
Base `  ( I mPwSer  S ) )  |  a finSupp 
( 0g `  S
) } )
11 eqid 2471 . . . 4  |-  ( I mPoly 
R )  =  ( I mPoly  R )
12 eqid 2471 . . . 4  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
13 eqid 2471 . . . 4  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
14 eqid 2471 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
15 eqid 2471 . . . 4  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
1611, 12, 13, 14, 15mplbas 18730 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  { a  e.  ( Base `  (
I mPwSer  R ) )  |  a finSupp  ( 0g `  R ) }
17 eqid 2471 . . . 4  |-  ( I mPoly 
S )  =  ( I mPoly  S )
18 eqid 2471 . . . 4  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
19 eqid 2471 . . . 4  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
20 eqid 2471 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
21 eqid 2471 . . . 4  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
2217, 18, 19, 20, 21mplbas 18730 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  { a  e.  ( Base `  (
I mPwSer  S ) )  |  a finSupp  ( 0g `  S ) }
2310, 16, 223eqtr4g 2530 . 2  |-  ( (
ph  /\  I  e.  _V )  ->  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  S ) ) )
24 reldmmpl 18728 . . . . . 6  |-  Rel  dom mPoly
2524ovprc1 6339 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  (/) )
2624ovprc1 6339 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I mPoly  S )  =  (/) )
2725, 26eqtr4d 2508 . . . 4  |-  ( -.  I  e.  _V  ->  ( I mPoly  R )  =  ( I mPoly  S ) )
2827fveq2d 5883 . . 3  |-  ( -.  I  e.  _V  ->  (
Base `  ( I mPoly  R ) )  =  (
Base `  ( I mPoly  S ) ) )
2928adantl 473 . 2  |-  ( (
ph  /\  -.  I  e.  _V )  ->  ( Base `  ( I mPoly  R
) )  =  (
Base `  ( I mPoly  S ) ) )
3023, 29pm2.61dan 808 1  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   (/)c0 3722   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   finSupp cfsupp 7901   Basecbs 15199   +g cplusg 15268   0gc0g 15416   mPwSer cmps 18652   mPoly cmpl 18654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-tset 15287  df-0g 15418  df-psr 18657  df-mpl 18659
This theorem is referenced by:  ply1baspropd  18913  mdegpropd  23112
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