![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22169 | . 2 ⊢ (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵)) |
4 | fvprc 6883 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅) | |
5 | 1, 4 | eqtrid 2783 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅) |
6 | reldmevls1 22156 | . . . 4 ⊢ Rel dom evalSub1 | |
7 | 6 | ovprc1 7451 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅) |
8 | 5, 7 | eqtr4d 2774 | . 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 1540 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 evalSub1 ces1 22152 eval1ce1 22153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7415 df-oprab 7416 df-mpo 7417 df-evls 21946 df-evl 21947 df-evls1 22154 df-evl1 22155 |
This theorem is referenced by: evls1scasrng 22178 evls1varsrng 22179 evl1gsumadd 22197 evl1varpw 22200 ressply1evl 33087 |
Copyright terms: Public domain | W3C validator |