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Theorem mdegpropd 22569
Description: Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
Hypotheses
Ref Expression
mdegpropd.b1  |-  ( ph  ->  B  =  ( Base `  R ) )
mdegpropd.b2  |-  ( ph  ->  B  =  ( Base `  S ) )
mdegpropd.p  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
Assertion
Ref Expression
mdegpropd  |-  ( ph  ->  ( I mDeg  R )  =  ( I mDeg  S
) )
Distinct variable groups:    ph, x, y   
x, B, y    x, R, y    x, S, y
Allowed substitution hints:    I( x, y)

Proof of Theorem mdegpropd
Dummy variables  c 
a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegpropd.b1 . . . 4  |-  ( ph  ->  B  =  ( Base `  R ) )
2 mdegpropd.b2 . . . 4  |-  ( ph  ->  B  =  ( Base `  S ) )
3 mdegpropd.p . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
41, 2, 3mplbaspropd 18391 . . 3  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
51, 2, 3grpidpropd 16005 . . . . . 6  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
65oveq2d 6212 . . . . 5  |-  ( ph  ->  ( c supp  ( 0g
`  R ) )  =  ( c supp  ( 0g `  S ) ) )
76imaeq2d 5249 . . . 4  |-  ( ph  ->  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( c supp  ( 0g
`  R ) ) )  =  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( c supp  ( 0g `  S
) ) ) )
87supeq1d 7820 . . 3  |-  ( ph  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( c supp  ( 0g `  R
) ) ) , 
RR* ,  <  )  =  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( c supp  ( 0g `  S
) ) ) , 
RR* ,  <  ) )
94, 8mpteq12dv 4445 . 2  |-  ( ph  ->  ( c  e.  (
Base `  ( I mPoly  R ) )  |->  sup (
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( c supp  ( 0g
`  R ) ) ) ,  RR* ,  <  ) )  =  ( c  e.  ( Base `  (
I mPoly  S ) )  |->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( c supp  ( 0g `  S
) ) ) , 
RR* ,  <  ) ) )
10 eqid 2382 . . 3  |-  ( I mDeg 
R )  =  ( I mDeg  R )
11 eqid 2382 . . 3  |-  ( I mPoly 
R )  =  ( I mPoly  R )
12 eqid 2382 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
13 eqid 2382 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
14 eqid 2382 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
15 eqid 2382 . . 3  |-  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )  =  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )
1610, 11, 12, 13, 14, 15mdegfval 22545 . 2  |-  ( I mDeg 
R )  =  ( c  e.  ( Base `  ( I mPoly  R ) )  |->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( c supp  ( 0g `  R
) ) ) , 
RR* ,  <  ) )
17 eqid 2382 . . 3  |-  ( I mDeg 
S )  =  ( I mDeg  S )
18 eqid 2382 . . 3  |-  ( I mPoly 
S )  =  ( I mPoly  S )
19 eqid 2382 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
20 eqid 2382 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
2117, 18, 19, 20, 14, 15mdegfval 22545 . 2  |-  ( I mDeg 
S )  =  ( c  e.  ( Base `  ( I mPoly  S ) )  |->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( c supp  ( 0g `  S
) ) ) , 
RR* ,  <  ) )
229, 16, 213eqtr4g 2448 1  |-  ( ph  ->  ( I mDeg  R )  =  ( I mDeg  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   {crab 2736    |-> cmpt 4425   `'ccnv 4912   "cima 4916   ` cfv 5496  (class class class)co 6196   supp csupp 6817    ^m cmap 7338   Fincfn 7435   supcsup 7815   RR*cxr 9538    < clt 9539   NNcn 10452   NN0cn0 10712   Basecbs 14634   +g cplusg 14702   0gc0g 14847    gsumg cgsu 14848   mPoly cmpl 18115  ℂfldccnfld 18533   mDeg cmdg 22536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-tset 14721  df-0g 14849  df-psr 18118  df-mpl 18120  df-mdeg 22538
This theorem is referenced by:  deg1propd  22571
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