Theorem deg1fval 26312

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 7425	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 26288	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7450	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6990	. . 3 ⊢ (𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6874	. . . 4 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = ∅)
7		reldmmdeg 26289	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7457	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2800	. . 3 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 184	. 2 ⊢ (deg1‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2785	1 ⊢ 𝐷 = (1o mDeg 𝑅)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  ‘cfv 6537  (class class class)co 7417  1_oc1o 8452   mDeg cmdg 26285  deg₁cdg1 26286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-mdeg 26287  df-deg1 26288

This theorem is used by:  deg1xrf  26313  deg1cl  26315  deg1propd  26318  deg1z  26319  deg1nn0cl  26320  deg1ldg  26324  deg1leb  26327  deg1val  26328  deg1addle  26333  deg1vscale  26336  deg1vsca  26337  deg1mulle2  26341  deg1le0  26343