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Theorem deg1fval 19466
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 5866 . . . 4  |-  ( r  =  R  ->  ( 1o mDeg  r )  =  ( 1o mDeg  R ) )
3 df-deg1 19442 . . . 4  |- deg1  =  (
r  e.  _V  |->  ( 1o mDeg  r ) )
4 ovex 5883 . . . 4  |-  ( 1o mDeg  R )  e.  _V
52, 3, 4fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
6 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  (/) )
7 reldmmdeg 19443 . . . . 5  |-  Rel  dom mDeg
87ovprc2 5887 . . . 4  |-  ( -.  R  e.  _V  ->  ( 1o mDeg  R )  =  (/) )
96, 8eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  ( deg1  `  R )  =  ( 1o mDeg  R ) )
105, 9pm2.61i 156 . 2  |-  ( deg1  `  R
)  =  ( 1o mDeg  R )
111, 10eqtri 2303 1  |-  D  =  ( 1o mDeg  R )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255  (class class class)co 5858   1oc1o 6472   mDeg cmdg 19439   deg1 cdg1 19440
This theorem is referenced by:  deg1xrf  19467  deg1cl  19469  deg1propd  19472  deg1z  19473  deg1nn0cl  19474  deg1ldg  19478  deg1leb  19481  deg1val  19482  deg1addle  19487  deg1vscale  19490  deg1vsca  19491  deg1mulle2  19495  deg1le0  19497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-mdeg 19441  df-deg1 19442
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