Theorem deg1fval 25351

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 7345	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 25324	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7370	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6931	. . 3 ⊢ (𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6817	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = ∅)
7		reldmmdeg 25325	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7377	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2779	. . 3 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 182	. 2 ⊢ ( deg1 ‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2764	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2105  Vcvv 3441  ∅c0 4269  ‘cfv 6479  (class class class)co 7337  1_oc1o 8360   mDeg cmdg 25321   deg₁ cdg1 25322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-mdeg 25323  df-deg1 25324

This theorem is referenced by:  deg1xrf  25352  deg1cl  25354  deg1propd  25357  deg1z  25358  deg1nn0cl  25359  deg1ldg  25363  deg1leb  25366  deg1val  25367  deg1addle  25372  deg1vscale  25375  deg1vsca  25376  deg1mulle2  25380  deg1le0  25382