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Mirrors > Home > MPE Home > Th. List > evl1fval1 | Structured version Visualization version GIF version |
Description: Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.) |
Ref | Expression |
---|---|
evl1fval1.q | ⊢ 𝑄 = (eval1‘𝑅) |
evl1fval1.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
evl1fval1 | ⊢ 𝑄 = (𝑅 evalSub1 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1fval1.q | . . 3 ⊢ 𝑄 = (eval1‘𝑅) | |
2 | evl1fval1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | evl1fval1lem 22258 | . 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵)) |
4 | fvprc 6884 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅) | |
5 | 1, 4 | eqtrid 2777 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅) |
6 | reldmevls1 22245 | . . . 4 ⊢ Rel dom evalSub1 | |
7 | 6 | ovprc1 7455 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅) |
8 | 5, 7 | eqtr4d 2768 | . 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 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 evalSub1 ces1 22241 eval1ce1 22242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-evls 22025 df-evl 22026 df-evls1 22243 df-evl1 22244 |
This theorem is referenced by: evls1scasrng 22267 evls1varsrng 22268 evl1gsumadd 22286 evl1varpw 22289 ressply1evl 22298 evl1maprhm 22307 |
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