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Theorem mdegpropd 19486
Description: Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
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 16330 . . 3  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
51, 2, 3grpidpropd 14415 . . . . . . . 8  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
65sneqd 3666 . . . . . . 7  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  S ) } )
76difeq2d 3307 . . . . . 6  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  S
) } ) )
87imaeq2d 5028 . . . . 5  |-  ( ph  ->  ( `' c "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' c
" ( _V  \  { ( 0g `  S ) } ) ) )
98imaeq2d 5028 . . . 4  |-  ( ph  ->  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' c "
( _V  \  {
( 0g `  R
) } ) ) )  =  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' c " ( _V 
\  { ( 0g
`  S ) } ) ) ) )
109supeq1d 7215 . . 3  |-  ( ph  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' c " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  =  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' c " ( _V 
\  { ( 0g
`  S ) } ) ) ) , 
RR* ,  <  ) )
114, 10mpteq12dv 4114 . 2  |-  ( ph  ->  ( c  e.  (
Base `  ( I mPoly  R ) )  |->  sup (
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' c "
( _V  \  {
( 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 " ( _V 
\  { ( 0g
`  S ) } ) ) ) , 
RR* ,  <  ) ) )
12 eqid 2296 . . 3  |-  ( I mDeg 
R )  =  ( I mDeg  R )
13 eqid 2296 . . 3  |-  ( I mPoly 
R )  =  ( I mPoly  R )
14 eqid 2296 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
15 eqid 2296 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 eqid 2296 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
17 eqid 2296 . . 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 ) )
1812, 13, 14, 15, 16, 17mdegfval 19464 . 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 " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
19 eqid 2296 . . 3  |-  ( I mDeg 
S )  =  ( I mDeg  S )
20 eqid 2296 . . 3  |-  ( I mPoly 
S )  =  ( I mPoly  S )
21 eqid 2296 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
22 eqid 2296 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
2319, 20, 21, 22, 16, 17mdegfval 19464 . 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 " ( _V 
\  { ( 0g
`  S ) } ) ) ) , 
RR* ,  <  ) )
2411, 18, 233eqtr4g 2353 1  |-  ( ph  ->  ( I mDeg  R )  =  ( I mDeg  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    \ cdif 3162   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   supcsup 7209   RR*cxr 8882    < clt 8883   NNcn 9762   NN0cn0 9981   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   mPoly cmpl 16105  ℂfldccnfld 16393   mDeg cmdg 19455
This theorem is referenced by:  deg1propd  19488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-0g 13420  df-psr 16114  df-mpl 16116  df-mdeg 19457
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