Theorem deg1fval 24666

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 7156	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 24642	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7181	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6761	. . 3 ⊢ (𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6656	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = ∅)
7		reldmmdeg 24643	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7188	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2857	. . 3 ⊢ (¬ 𝑅 ∈ V → ( deg1 ‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 184	. 2 ⊢ ( deg1 ‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2842	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1531   ∈ wcel 2108  Vcvv 3493  ∅c0 4289  ‘cfv 6348  (class class class)co 7148  1_oc1o 8087   mDeg cmdg 24639   deg₁ cdg1 24640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-mdeg 24641  df-deg1 24642

This theorem is referenced by:  deg1xrf  24667  deg1cl  24669  deg1propd  24672  deg1z  24673  deg1nn0cl  24674  deg1ldg  24678  deg1leb  24681  deg1val  24682  deg1addle  24687  deg1vscale  24690  deg1vsca  24691  deg1mulle2  24695  deg1le0  24697