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Theorem deg1fval 23885
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 = ( deg1𝑅)
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 = ( deg1𝑅)
2 oveq2 6698 . . . 4 (𝑟 = 𝑅 → (1𝑜 mDeg 𝑟) = (1𝑜 mDeg 𝑅))
3 df-deg1 23861 . . . 4 deg1 = (𝑟 ∈ V ↦ (1𝑜 mDeg 𝑟))
4 ovex 6718 . . . 4 (1𝑜 mDeg 𝑅) ∈ V
52, 3, 4fvmpt 6321 . . 3 (𝑅 ∈ V → ( deg1𝑅) = (1𝑜 mDeg 𝑅))
6 fvprc 6223 . . . 4 𝑅 ∈ V → ( deg1𝑅) = ∅)
7 reldmmdeg 23862 . . . . 5 Rel dom mDeg
87ovprc2 6725 . . . 4 𝑅 ∈ V → (1𝑜 mDeg 𝑅) = ∅)
96, 8eqtr4d 2688 . . 3 𝑅 ∈ V → ( deg1𝑅) = (1𝑜 mDeg 𝑅))
105, 9pm2.61i 176 . 2 ( deg1𝑅) = (1𝑜 mDeg 𝑅)
111, 10eqtri 2673 1 𝐷 = (1𝑜 mDeg 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  cfv 5926  (class class class)co 6690  1𝑜c1o 7598   mDeg cmdg 23858   deg1 cdg1 23859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-mdeg 23860  df-deg1 23861
This theorem is referenced by:  deg1xrf  23886  deg1cl  23888  deg1propd  23891  deg1z  23892  deg1nn0cl  23893  deg1ldg  23897  deg1leb  23900  deg1val  23901  deg1addle  23906  deg1vscale  23909  deg1vsca  23910  deg1mulle2  23914  deg1le0  23916
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