deg1fval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > deg1fval		Structured version Visualization version GIF version

Theorem deg1fval 23644

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 = (1_𝑜 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 6557	. . . 4 ⊢ (𝑟 = 𝑅 → (1_𝑜 mDeg 𝑟) = (1_𝑜 mDeg 𝑅))
3		df-deg1 23620	. . . 4 ⊢ deg₁ = (𝑟 ∈ V ↦ (1_𝑜 mDeg 𝑟))
4		ovex 6577	. . . 4 ⊢ (1_𝑜 mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6191	. . 3 ⊢ (𝑅 ∈ V → ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅))
6		fvprc 6097	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg₁ ‘𝑅) = ∅)
7		reldmmdeg 23621	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 6583	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1_𝑜 mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2647	. . 3 ⊢ (¬ 𝑅 ∈ V → ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅))
10	5, 9	pm2.61i 175	. 2 ⊢ ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅)
11	1, 10	eqtri 2632	1 ⊢ 𝐷 = (1_𝑜 mDeg 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 1_𝑜c1o 7440 mDeg cmdg 23617 deg₁ cdg1 23618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-mdeg 23619 df-deg1 23620

This theorem is referenced by: deg1xrf 23645 deg1cl 23647 deg1propd 23650 deg1z 23651 deg1nn0cl 23652 deg1ldg 23656 deg1leb 23659 deg1val 23660 deg1addle 23665 deg1vscale 23668 deg1vsca 23669 deg1mulle2 23673 deg1le0 23675

W3C validator