Theorem deg1fval 25245

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.

Step	Hyp	Ref	Expression
1		deg1fval.d	. 2 ⊢ 𝐷 = ( deg1 ‘𝑅)
2		oveq2 7283	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 25218	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7308	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6875	. . 3 ⊢ (𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6766	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = ∅)
7		reldmmdeg 25219	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7315	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2781	. . 3 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 182	. 2 ⊢ ( deg1 ‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2766	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   mDeg cmdg 25215   deg₁ cdg1 25216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-mdeg 25217  df-deg1 25218

This theorem is referenced by:  deg1xrf  25246  deg1cl  25248  deg1propd  25251  deg1z  25252  deg1nn0cl  25253  deg1ldg  25257  deg1leb  25260  deg1val  25261  deg1addle  25266  deg1vscale  25269  deg1vsca  25270  deg1mulle2  25274  deg1le0  25276