evl1fval1 - Metamath Proof Explorer

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Theorem evl1fval1 22541

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 22540	. 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
4		fvprc 6877	. . . 4 ⊢ (¬ 𝑅 ∈ V → (eval₁‘𝑅) = ∅)
5	1, 4	eqtrid 2812	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
6		reldmevls1 22527	. . . 4 ⊢ Rel dom evalSub₁
7	6	ovprc1 7458	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub₁ 𝐵) = ∅)
8	5, 7	eqtr4d 2803	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
9	3, 8	pm2.61i 184	1 ⊢ 𝑄 = (𝑅 evalSub₁ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ∅c0 4286 ‘cfv 6540 (class class class)co 7419 Basecbs 17291 evalSub₁ ces1 22523 eval₁ce1 22524

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-evls 22275 df-evl 22276 df-evls1 22525 df-evl1 22526

This theorem is used by: evls1scasrng 22549 evls1varsrng 22550 evl1gsumadd 22568 evl1varpw 22571 ressply1evl 22580 evl1maprhm 22589 evl1fpws 33918 cos9thpiminply 34242

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