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Theorem deg1fval 21694
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 6211 . . . 4  |-  ( r  =  R  ->  ( 1o mDeg  r )  =  ( 1o mDeg  R ) )
3 df-deg1 21668 . . . 4  |- deg1  =  (
r  e.  _V  |->  ( 1o mDeg  r ) )
4 ovex 6228 . . . 4  |-  ( 1o mDeg  R )  e.  _V
52, 3, 4fvmpt 5886 . . 3  |-  ( R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
6 fvprc 5796 . . . 4  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  (/) )
7 reldmmdeg 21669 . . . . 5  |-  Rel  dom mDeg
87ovprc2 6232 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mDeg  R )  =  (/) )
96, 8eqtr4d 2498 . . 3  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
105, 9pm2.61i 164 . 2  |-  ( deg1  `  R
)  =  ( 1o mDeg  R )
111, 10eqtri 2483 1  |-  D  =  ( 1o mDeg  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3748   ` cfv 5529  (class class class)co 6203   1oc1o 7026   mDeg cmdg 21665   deg1 cdg1 21666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-mdeg 21667  df-deg1 21668
This theorem is referenced by:  deg1xrf  21695  deg1cl  21697  deg1propd  21700  deg1z  21701  deg1nn0cl  21702  deg1ldg  21706  deg1leb  21709  deg1val  21710  deg1valOLD  21711  deg1addle  21716  deg1vscale  21719  deg1vsca  21720  deg1mulle2  21724  deg1le0  21726
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