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Theorem deg1fval 22346
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 6285 . . . 4  |-  ( r  =  R  ->  ( 1o mDeg  r )  =  ( 1o mDeg  R ) )
3 df-deg1 22320 . . . 4  |- deg1  =  (
r  e.  _V  |->  ( 1o mDeg  r ) )
4 ovex 6305 . . . 4  |-  ( 1o mDeg  R )  e.  _V
52, 3, 4fvmpt 5937 . . 3  |-  ( R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
6 fvprc 5846 . . . 4  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  (/) )
7 reldmmdeg 22321 . . . . 5  |-  Rel  dom mDeg
87ovprc2 6309 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mDeg  R )  =  (/) )
96, 8eqtr4d 2485 . . 3  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
105, 9pm2.61i 164 . 2  |-  ( deg1  `  R
)  =  ( 1o mDeg  R )
111, 10eqtri 2470 1  |-  D  =  ( 1o mDeg  R )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1381    e. wcel 1802   _Vcvv 3093   (/)c0 3767   ` cfv 5574  (class class class)co 6277   1oc1o 7121   mDeg cmdg 22317   deg1 cdg1 22318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-iota 5537  df-fun 5576  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-mdeg 22319  df-deg1 22320
This theorem is referenced by:  deg1xrf  22347  deg1cl  22349  deg1propd  22352  deg1z  22353  deg1nn0cl  22354  deg1ldg  22358  deg1leb  22361  deg1val  22362  deg1valOLD  22363  deg1addle  22368  deg1vscale  22371  deg1vsca  22372  deg1mulle2  22376  deg1le0  22378
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