evl1fval1 - Metamath Proof Explorer

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Theorem evl1fval1 22658

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 22657	. 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
4		fvprc 6882	. . . 4 ⊢ (¬ 𝑅 ∈ V → (eval₁‘𝑅) = ∅)
5	1, 4	eqtrid 2807	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
6		reldmevls1 22644	. . . 4 ⊢ Rel dom evalSub₁
7	6	ovprc1 7463	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub₁ 𝐵) = ∅)
8	5, 7	eqtr4d 2798	. 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 1569 ∈ wcel 2144 Vcvv 3450 ∅c0 4277 ‘cfv 6543 (class class class)co 7424 Basecbs 17396 evalSub₁ ces1 22640 eval₁ce1 22641

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7427 df-oprab 7428 df-mpo 7429 df-evls 22392 df-evl 22393 df-evls1 22642 df-evl1 22643

This theorem is used by: evls1scasrng 22666 evls1varsrng 22667 evl1gsumadd 22685 evl1varpw 22688 ressply1evl 22697 evl1maprhm 22706 evl1fpws 34146 cos9thpiminply 34470

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