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Theorem deg1fval 22325
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 6302 . . . 4  |-  ( r  =  R  ->  ( 1o mDeg  r )  =  ( 1o mDeg  R ) )
3 df-deg1 22299 . . . 4  |- deg1  =  (
r  e.  _V  |->  ( 1o mDeg  r ) )
4 ovex 6319 . . . 4  |-  ( 1o mDeg  R )  e.  _V
52, 3, 4fvmpt 5956 . . 3  |-  ( R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
6 fvprc 5865 . . . 4  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  (/) )
7 reldmmdeg 22300 . . . . 5  |-  Rel  dom mDeg
87ovprc2 6323 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mDeg  R )  =  (/) )
96, 8eqtr4d 2511 . . 3  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
105, 9pm2.61i 164 . 2  |-  ( deg1  `  R
)  =  ( 1o mDeg  R )
111, 10eqtri 2496 1  |-  D  =  ( 1o mDeg  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   ` cfv 5593  (class class class)co 6294   1oc1o 7133   mDeg cmdg 22296   deg1 cdg1 22297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-mdeg 22298  df-deg1 22299
This theorem is referenced by:  deg1xrf  22326  deg1cl  22328  deg1propd  22331  deg1z  22332  deg1nn0cl  22333  deg1ldg  22337  deg1leb  22340  deg1val  22341  deg1valOLD  22342  deg1addle  22347  deg1vscale  22350  deg1vsca  22351  deg1mulle2  22355  deg1le0  22357
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