Theorem deg1fval 25150

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 7263	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 25123	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7288	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6857	. . 3 ⊢ (𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6748	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = ∅)
7		reldmmdeg 25124	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7295	. . . 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 2108  Vcvv 3422  ∅c0 4253  ‘cfv 6418  (class class class)co 7255  1_oc1o 8260   mDeg cmdg 25120   deg₁ cdg1 25121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-mdeg 25122  df-deg1 25123

This theorem is referenced by:  deg1xrf  25151  deg1cl  25153  deg1propd  25156  deg1z  25157  deg1nn0cl  25158  deg1ldg  25162  deg1leb  25165  deg1val  25166  deg1addle  25171  deg1vscale  25174  deg1vsca  25175  deg1mulle2  25179  deg1le0  25181