deg1fval - Metamath Proof Explorer

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Theorem deg1fval 23840

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 6658	. . . 4 ⊢ (𝑟 = 𝑅 → (1_𝑜 mDeg 𝑟) = (1_𝑜 mDeg 𝑅))
3		df-deg1 23816	. . . 4 ⊢ deg₁ = (𝑟 ∈ V ↦ (1_𝑜 mDeg 𝑟))
4		ovex 6678	. . . 4 ⊢ (1_𝑜 mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6282	. . 3 ⊢ (𝑅 ∈ V → ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅))
6		fvprc 6185	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg₁ ‘𝑅) = ∅)
7		reldmmdeg 23817	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 6685	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1_𝑜 mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2659	. . 3 ⊢ (¬ 𝑅 ∈ V → ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅))
10	5, 9	pm2.61i 176	. 2 ⊢ ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅)
11	1, 10	eqtri 2644	1 ⊢ 𝐷 = (1_𝑜 mDeg 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 1_𝑜c1o 7553 mDeg cmdg 23813 deg₁ cdg1 23814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-mdeg 23815 df-deg1 23816

This theorem is referenced by: deg1xrf 23841 deg1cl 23843 deg1propd 23846 deg1z 23847 deg1nn0cl 23848 deg1ldg 23852 deg1leb 23855 deg1val 23856 deg1addle 23861 deg1vscale 23864 deg1vsca 23865 deg1mulle2 23869 deg1le0 23871

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