evl1fval1 - Metamath Proof Explorer

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Theorem evl1fval1 22309

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 22308	. 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub₁ 𝐵))
4		fvprc 6827	. . . 4 ⊢ (¬ 𝑅 ∈ V → (eval₁‘𝑅) = ∅)
5	1, 4	eqtrid 2784	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
6		reldmevls1 22295	. . . 4 ⊢ Rel dom evalSub₁
7	6	ovprc1 7400	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub₁ 𝐵) = ∅)
8	5, 7	eqtr4d 2775	. 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 1542 ∈ wcel 2114 Vcvv 3430 ∅c0 4274 ‘cfv 6493 (class class class)co 7361 Basecbs 17173 evalSub₁ ces1 22291 eval₁ce1 22292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-evls 22065 df-evl 22066 df-evls1 22293 df-evl1 22294

This theorem is referenced by: evls1scasrng 22317 evls1varsrng 22318 evl1gsumadd 22336 evl1varpw 22339 ressply1evl 22348 evl1maprhm 22357 evl1fpws 33642 cos9thpiminply 33951

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