deg1fval - Metamath Proof Explorer

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Theorem deg1fval 23885

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 6698	. . . 4 ⊢ (𝑟 = 𝑅 → (1_𝑜 mDeg 𝑟) = (1_𝑜 mDeg 𝑅))
3		df-deg1 23861	. . . 4 ⊢ deg₁ = (𝑟 ∈ V ↦ (1_𝑜 mDeg 𝑟))
4		ovex 6718	. . . 4 ⊢ (1_𝑜 mDeg 𝑅) ∈ V
5	2, 3, 4	fvmpt 6321	. . 3 ⊢ (𝑅 ∈ V → ( deg₁ ‘𝑅) = (1_𝑜 mDeg 𝑅))
6		fvprc 6223	. . . 4 ⊢ (¬ 𝑅 ∈ V → ( deg₁ ‘𝑅) = ∅)
7		reldmmdeg 23862	. . . . 5 ⊢ Rel dom mDeg
8	7	ovprc2 6725	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1_𝑜 mDeg 𝑅) = ∅)
9	6, 8	eqtr4d 2688	. . 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 2673	1 ⊢ 𝐷 = (1_𝑜 mDeg 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∅c0 3948 ‘cfv 5926 (class class class)co 6690 1_𝑜c1o 7598 mDeg cmdg 23858 deg₁ cdg1 23859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-mdeg 23860 df-deg1 23861

This theorem is referenced by: deg1xrf 23886 deg1cl 23888 deg1propd 23891 deg1z 23892 deg1nn0cl 23893 deg1ldg 23897 deg1leb 23900 deg1val 23901 deg1addle 23906 deg1vscale 23909 deg1vsca 23910 deg1mulle2 23914 deg1le0 23916

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