Theorem deg1fval 26001

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 7361	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 25977	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7386	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6934	. . 3 ⊢ (𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6818	. . . 4 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = ∅)
7		reldmmdeg 25978	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7393	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2767	. . 3 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 182	. 2 ⊢ (deg1‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2752	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  ‘cfv 6486  (class class class)co 7353  1_oc1o 8388   mDeg cmdg 25974  deg₁cdg1 25975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-mdeg 25976  df-deg1 25977

This theorem is referenced by:  deg1xrf  26002  deg1cl  26004  deg1propd  26007  deg1z  26008  deg1nn0cl  26009  deg1ldg  26013  deg1leb  26016  deg1val  26017  deg1addle  26022  deg1vscale  26025  deg1vsca  26026  deg1mulle2  26030  deg1le0  26032