Theorem deg1fval 26120

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 7400	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 26096	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7425	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6971	. . 3 ⊢ (𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6855	. . . 4 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = ∅)
7		reldmmdeg 26097	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7432	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2799	. . 3 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 183	. 2 ⊢ (deg1‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2784	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ∅c0 4285  ‘cfv 6517  (class class class)co 7392  1_oc1o 8425   mDeg cmdg 26093  deg₁cdg1 26094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-mdeg 26095  df-deg1 26096

This theorem is referenced by:  deg1xrf  26121  deg1cl  26123  deg1propd  26126  deg1z  26127  deg1nn0cl  26128  deg1ldg  26132  deg1leb  26135  deg1val  26136  deg1addle  26141  deg1vscale  26144  deg1vsca  26145  deg1mulle2  26149  deg1le0  26151