Theorem deg1fval 26053

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 7376	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 26029	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7401	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6949	. . 3 ⊢ (𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6834	. . . 4 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = ∅)
7		reldmmdeg 26030	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7408	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2775	. . 3 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 182	. 2 ⊢ (deg1‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2760	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ‘cfv 6500  (class class class)co 7368  1_oc1o 8400   mDeg cmdg 26026  deg₁cdg1 26027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-mdeg 26028  df-deg1 26029

This theorem is referenced by:  deg1xrf  26054  deg1cl  26056  deg1propd  26059  deg1z  26060  deg1nn0cl  26061  deg1ldg  26065  deg1leb  26068  deg1val  26069  deg1addle  26074  deg1vscale  26077  deg1vsca  26078  deg1mulle2  26082  deg1le0  26084