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Theorem mdegpropd 20008
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 16631 . . 3  |-  ( ph  ->  ( Base `  (
I mPoly  R ) )  =  ( Base `  (
I mPoly  S ) ) )
51, 2, 3grpidpropd 14723 . . . . . . . 8  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  S ) )
65sneqd 3828 . . . . . . 7  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  S ) } )
76difeq2d 3466 . . . . . 6  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  S
) } ) )
87imaeq2d 5204 . . . . 5  |-  ( ph  ->  ( `' c "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' c
" ( _V  \  { ( 0g `  S ) } ) ) )
98imaeq2d 5204 . . . 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 7452 . . 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 4288 . 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 2437 . . 3  |-  ( I mDeg 
R )  =  ( I mDeg  R )
13 eqid 2437 . . 3  |-  ( I mPoly 
R )  =  ( I mPoly  R )
14 eqid 2437 . . 3  |-  ( Base `  ( I mPoly  R ) )  =  ( Base `  ( I mPoly  R ) )
15 eqid 2437 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 eqid 2437 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
17 eqid 2437 . . 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 19986 . 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 2437 . . 3  |-  ( I mDeg 
S )  =  ( I mDeg  S )
20 eqid 2437 . . 3  |-  ( I mPoly 
S )  =  ( I mPoly  S )
21 eqid 2437 . . 3  |-  ( Base `  ( I mPoly  S ) )  =  ( Base `  ( I mPoly  S ) )
22 eqid 2437 . . 3  |-  ( 0g
`  S )  =  ( 0g `  S
)
2319, 20, 21, 22, 16, 17mdegfval 19986 . 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 2494 1  |-  ( ph  ->  ( I mDeg  R )  =  ( I mDeg  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2710   _Vcvv 2957    \ cdif 3318   {csn 3815    e. cmpt 4267   `'ccnv 4878   "cima 4882   ` cfv 5455  (class class class)co 6082    ^m cmap 7019   Fincfn 7110   supcsup 7446   RR*cxr 9120    < clt 9121   NNcn 10001   NN0cn0 10222   Basecbs 13470   +g cplusg 13530   0gc0g 13724    gsumg cgsu 13725   mPoly cmpl 16409  ℂfldccnfld 16704   mDeg cmdg 19977
This theorem is referenced by:  deg1propd  20010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-tset 13549  df-0g 13728  df-psr 16418  df-mpl 16420  df-mdeg 19979
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