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Theorem deg1fval 20003
Description: Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypothesis
Ref Expression
deg1fval.d  |-  D  =  ( deg1  `  R )
Assertion
Ref Expression
deg1fval  |-  D  =  ( 1o mDeg  R )

Proof of Theorem deg1fval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 deg1fval.d . 2  |-  D  =  ( deg1  `  R )
2 oveq2 6089 . . . 4  |-  ( r  =  R  ->  ( 1o mDeg  r )  =  ( 1o mDeg  R ) )
3 df-deg1 19979 . . . 4  |- deg1  =  (
r  e.  _V  |->  ( 1o mDeg  r ) )
4 ovex 6106 . . . 4  |-  ( 1o mDeg  R )  e.  _V
52, 3, 4fvmpt 5806 . . 3  |-  ( R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
6 fvprc 5722 . . . 4  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  (/) )
7 reldmmdeg 19980 . . . . 5  |-  Rel  dom mDeg
87ovprc2 6110 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mDeg  R )  =  (/) )
96, 8eqtr4d 2471 . . 3  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
105, 9pm2.61i 158 . 2  |-  ( deg1  `  R
)  =  ( 1o mDeg  R )
111, 10eqtri 2456 1  |-  D  =  ( 1o mDeg  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454  (class class class)co 6081   1oc1o 6717   mDeg cmdg 19976   deg1 cdg1 19977
This theorem is referenced by:  deg1xrf  20004  deg1cl  20006  deg1propd  20009  deg1z  20010  deg1nn0cl  20011  deg1ldg  20015  deg1leb  20018  deg1val  20019  deg1addle  20024  deg1vscale  20027  deg1vsca  20028  deg1mulle2  20032  deg1le0  20034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-mdeg 19978  df-deg1 19979
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