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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 22350 | . 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵)) |
4 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅) | |
5 | 1, 4 | eqtrid 2787 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅) |
6 | reldmevls1 22337 | . . . 4 ⊢ Rel dom evalSub1 | |
7 | 6 | ovprc1 7470 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅) |
8 | 5, 7 | eqtr4d 2778 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 evalSub1 ces1 22333 eval1ce1 22334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-evls 22116 df-evl 22117 df-evls1 22335 df-evl1 22336 |
This theorem is referenced by: evls1scasrng 22359 evls1varsrng 22360 evl1gsumadd 22378 evl1varpw 22381 ressply1evl 22390 evl1maprhm 22399 evl1fpws 33570 |
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