evl1fval1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > evl1fval1		Structured version Visualization version GIF version

Theorem evl1fval1 22642

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 22641	. 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
4		fvprc 6875	. . . 4 ⊢ (¬ 𝑅 ∈ V → (eval₁‘𝑅) = ∅)
5	1, 4	eqtrid 2808	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
6		reldmevls1 22628	. . . 4 ⊢ Rel dom evalSub₁
7	6	ovprc1 7457	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub₁ 𝐵) = ∅)
8	5, 7	eqtr4d 2799	. 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 2145 Vcvv 3451 ∅c0 4279 ‘cfv 6537 (class class class)co 7418 Basecbs 17380 evalSub₁ ces1 22624 eval₁ce1 22625

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 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 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7421 df-oprab 7422 df-mpo 7423 df-evls 22376 df-evl 22377 df-evls1 22626 df-evl1 22627

This theorem is used by: evls1scasrng 22650 evls1varsrng 22651 evl1gsumadd 22669 evl1varpw 22672 ressply1evl 22681 evl1maprhm 22690 evl1fpws 34089 cos9thpiminply 34413

W3C validator