evl1fval1 - Metamath Proof Explorer

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Theorem evl1fval1 22218

Description: Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)

**Hypotheses**
Ref	Expression
evl1fval1.q	⊢ 𝑄 = (eval₁‘𝑅)
evl1fval1.b	⊢ 𝐵 = (Base‘𝑅)

**Assertion**
Ref	Expression
evl1fval1	⊢ 𝑄 = (𝑅 evalSub₁ 𝐵)

**Proof of Theorem evl1fval1**
Step	Hyp	Ref	Expression
1		evl1fval1.q	. . 3 ⊢ 𝑄 = (eval₁‘𝑅)
2		evl1fval1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	evl1fval1lem 22217	. 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
4		fvprc 6850	. . . 4 ⊢ (¬ 𝑅 ∈ V → (eval₁‘𝑅) = ∅)
5	1, 4	eqtrid 2776	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
6		reldmevls1 22204	. . . 4 ⊢ Rel dom evalSub₁
7	6	ovprc1 7426	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub₁ 𝐵) = ∅)
8	5, 7	eqtr4d 2767	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
9	3, 8	pm2.61i 182	1 ⊢ 𝑄 = (𝑅 evalSub₁ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ∅c0 4296 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 evalSub₁ ces1 22200 eval₁ce1 22201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-evls 21981 df-evl 21982 df-evls1 22202 df-evl1 22203

This theorem is referenced by: evls1scasrng 22226 evls1varsrng 22227 evl1gsumadd 22245 evl1varpw 22248 ressply1evl 22257 evl1maprhm 22266 evl1fpws 33533 cos9thpiminply 33778

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