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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 22204 | . 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵)) |
4 | fvprc 6877 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅) | |
5 | 1, 4 | eqtrid 2778 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅) |
6 | reldmevls1 22191 | . . . 4 ⊢ Rel dom evalSub1 | |
7 | 6 | ovprc1 7444 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅) |
8 | 5, 7 | eqtr4d 2769 | . 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 3468 ∅c0 4317 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 evalSub1 ces1 22187 eval1ce1 22188 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-evls 21977 df-evl 21978 df-evls1 22189 df-evl1 22190 |
This theorem is referenced by: evls1scasrng 22213 evls1varsrng 22214 evl1gsumadd 22232 evl1varpw 22235 ressply1evl 33156 |
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