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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsvval | Structured version Visualization version GIF version |
Description: Give a formula for the evaluation of a polynomial. (Contributed by SN, 9-Feb-2025.) |
Ref | Expression |
---|---|
evlsvval.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsvval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑈) |
evlsvval.b | ⊢ 𝐵 = (Base‘𝑃) |
evlsvval.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
evlsvval.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsvval.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsvval.t | ⊢ 𝑇 = (𝑆 ↑s (𝐾 ↑m 𝐼)) |
evlsvval.m | ⊢ 𝑀 = (mulGrp‘𝑇) |
evlsvval.w | ⊢ ↑ = (.g‘𝑀) |
evlsvval.x | ⊢ · = (.r‘𝑇) |
evlsvval.f | ⊢ 𝐹 = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑m 𝐼) × {𝑥})) |
evlsvval.g | ⊢ 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑m 𝐼) ↦ (𝑎‘𝑥))) |
evlsvval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsvval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsvval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsvval.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
evlsvval | ⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6880 | . . . . . 6 ⊢ (𝑝 = 𝐴 → (𝑝‘𝑏) = (𝐴‘𝑏)) | |
2 | 1 | fveq2d 6885 | . . . . 5 ⊢ (𝑝 = 𝐴 → (𝐹‘(𝑝‘𝑏)) = (𝐹‘(𝐴‘𝑏))) |
3 | 2 | oveq1d 7416 | . . . 4 ⊢ (𝑝 = 𝐴 → ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))) = ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))) |
4 | 3 | mpteq2dv 5240 | . . 3 ⊢ (𝑝 = 𝐴 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))) |
5 | 4 | oveq2d 7417 | . 2 ⊢ (𝑝 = 𝐴 → (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))) = (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) |
6 | evlsvval.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
7 | evlsvval.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑈) | |
8 | evlsvval.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
9 | evlsvval.d | . . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
10 | evlsvval.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
11 | evlsvval.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
12 | evlsvval.t | . . 3 ⊢ 𝑇 = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
13 | evlsvval.m | . . 3 ⊢ 𝑀 = (mulGrp‘𝑇) | |
14 | evlsvval.w | . . 3 ⊢ ↑ = (.g‘𝑀) | |
15 | evlsvval.x | . . 3 ⊢ · = (.r‘𝑇) | |
16 | eqid 2724 | . . 3 ⊢ (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) = (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) | |
17 | evlsvval.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑m 𝐼) × {𝑥})) | |
18 | evlsvval.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑m 𝐼) ↦ (𝑎‘𝑥))) | |
19 | evlsvval.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
20 | evlsvval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
21 | evlsvval.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
22 | 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 | evlsval3 41620 | . 2 ⊢ (𝜑 → 𝑄 = (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))))) |
23 | evlsvval.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
24 | ovexd 7436 | . 2 ⊢ (𝜑 → (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))) ∈ V) | |
25 | 5, 22, 23, 24 | fvmptd4 41546 | 1 ⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 {csn 4620 ↦ cmpt 5221 × cxp 5664 ◡ccnv 5665 “ cima 5669 ‘cfv 6533 (class class class)co 7401 ∘f cof 7661 ↑m cmap 8816 Fincfn 8935 ℕcn 12209 ℕ0cn0 12469 Basecbs 17143 ↾s cress 17172 .rcmulr 17197 Σg cgsu 17385 ↑s cpws 17391 .gcmg 18985 mulGrpcmgp 20029 CRingccrg 20129 SubRingcsubrg 20459 mPoly cmpl 21768 evalSub ces 21943 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-lsp 20809 df-assa 21716 df-asp 21717 df-ascl 21718 df-psr 21771 df-mvr 21772 df-mpl 21773 df-evls 21945 |
This theorem is referenced by: evlsvvval 41624 evlsevl 41632 |
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