Theorem deg1fval 26198

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 7408	. . . 4 ⊢ (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3		df-deg1 26174	. . . 4 ⊢ deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4		ovex 7433	. . . 4 ⊢ (1o mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6979	. . 3 ⊢ (𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
6		fvprc 6863	. . . 4 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = ∅)
7		reldmmdeg 26175	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 7440	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2803	. . 3 ⊢ (¬ 𝑅 ∈ V → (deg1‘𝑅) = (1o mDeg 𝑅))
10	5, 9	pm2.61i 184	. 2 ⊢ (deg1‘𝑅) = (1o mDeg 𝑅)
11	1, 10	eqtri 2788	1 ⊢ 𝐷 = (1o mDeg 𝑅)

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  ‘cfv 6525  (class class class)co 7400  1_oc1o 8434   mDeg cmdg 26171  deg₁cdg1 26172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-mdeg 26173  df-deg1 26174

This theorem is referenced by:  deg1xrf  26199  deg1cl  26201  deg1propd  26204  deg1z  26205  deg1nn0cl  26206  deg1ldg  26210  deg1leb  26213  deg1val  26214  deg1addle  26219  deg1vscale  26222  deg1vsca  26223  deg1mulle2  26227  deg1le0  26229