![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > evlvvval | Structured version Visualization version GIF version |
Description: Give a formula for the evaluation of a polynomial given assignments from variables to values. (Contributed by SN, 5-Mar-2025.) |
Ref | Expression |
---|---|
evlvvval.q | β’ π = (πΌ eval π ) |
evlvvval.p | β’ π = (πΌ mPoly π ) |
evlvvval.b | β’ π΅ = (Baseβπ) |
evlvvval.d | β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} |
evlvvval.k | β’ πΎ = (Baseβπ ) |
evlvvval.m | β’ π = (mulGrpβπ ) |
evlvvval.w | β’ β = (.gβπ) |
evlvvval.x | β’ Β· = (.rβπ ) |
evlvvval.i | β’ (π β πΌ β π) |
evlvvval.r | β’ (π β π β CRing) |
evlvvval.f | β’ (π β πΉ β π΅) |
evlvvval.a | β’ (π β π΄ β (πΎ βm πΌ)) |
Ref | Expression |
---|---|
evlvvval | β’ (π β ((πβπΉ)βπ΄) = (π Ξ£g (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π β πΌ β¦ ((πβπ) β (π΄βπ)))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 β’ ((πΌ evalSub π )β(Baseβπ )) = ((πΌ evalSub π )β(Baseβπ )) | |
2 | evlvvval.q | . . . 4 β’ π = (πΌ eval π ) | |
3 | eqid 2726 | . . . 4 β’ (πΌ mPoly (π βΎs (Baseβπ ))) = (πΌ mPoly (π βΎs (Baseβπ ))) | |
4 | eqid 2726 | . . . 4 β’ (π βΎs (Baseβπ )) = (π βΎs (Baseβπ )) | |
5 | eqid 2726 | . . . 4 β’ (Baseβ(πΌ mPoly (π βΎs (Baseβπ )))) = (Baseβ(πΌ mPoly (π βΎs (Baseβπ )))) | |
6 | evlvvval.i | . . . 4 β’ (π β πΌ β π) | |
7 | evlvvval.r | . . . 4 β’ (π β π β CRing) | |
8 | 7 | crngringd 20151 | . . . . 5 β’ (π β π β Ring) |
9 | eqid 2726 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
10 | 9 | subrgid 20475 | . . . . 5 β’ (π β Ring β (Baseβπ ) β (SubRingβπ )) |
11 | 8, 10 | syl 17 | . . . 4 β’ (π β (Baseβπ ) β (SubRingβπ )) |
12 | evlvvval.f | . . . . 5 β’ (π β πΉ β π΅) | |
13 | 9 | ressid 17198 | . . . . . . . . . 10 β’ (π β CRing β (π βΎs (Baseβπ )) = π ) |
14 | 7, 13 | syl 17 | . . . . . . . . 9 β’ (π β (π βΎs (Baseβπ )) = π ) |
15 | 14 | oveq2d 7421 | . . . . . . . 8 β’ (π β (πΌ mPoly (π βΎs (Baseβπ ))) = (πΌ mPoly π )) |
16 | evlvvval.p | . . . . . . . 8 β’ π = (πΌ mPoly π ) | |
17 | 15, 16 | eqtr4di 2784 | . . . . . . 7 β’ (π β (πΌ mPoly (π βΎs (Baseβπ ))) = π) |
18 | 17 | fveq2d 6889 | . . . . . 6 β’ (π β (Baseβ(πΌ mPoly (π βΎs (Baseβπ )))) = (Baseβπ)) |
19 | evlvvval.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
20 | 18, 19 | eqtr4di 2784 | . . . . 5 β’ (π β (Baseβ(πΌ mPoly (π βΎs (Baseβπ )))) = π΅) |
21 | 12, 20 | eleqtrrd 2830 | . . . 4 β’ (π β πΉ β (Baseβ(πΌ mPoly (π βΎs (Baseβπ ))))) |
22 | 1, 2, 3, 4, 5, 6, 7, 11, 21 | evlsevl 41697 | . . 3 β’ (π β (((πΌ evalSub π )β(Baseβπ ))βπΉ) = (πβπΉ)) |
23 | 22 | fveq1d 6887 | . 2 β’ (π β ((((πΌ evalSub π )β(Baseβπ ))βπΉ)βπ΄) = ((πβπΉ)βπ΄)) |
24 | evlvvval.d | . . 3 β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} | |
25 | evlvvval.k | . . 3 β’ πΎ = (Baseβπ ) | |
26 | evlvvval.m | . . 3 β’ π = (mulGrpβπ ) | |
27 | evlvvval.w | . . 3 β’ β = (.gβπ) | |
28 | evlvvval.x | . . 3 β’ Β· = (.rβπ ) | |
29 | evlvvval.a | . . 3 β’ (π β π΄ β (πΎ βm πΌ)) | |
30 | 1, 3, 5, 4, 24, 25, 26, 27, 28, 6, 7, 11, 21, 29 | evlsvvval 41689 | . 2 β’ (π β ((((πΌ evalSub π )β(Baseβπ ))βπΉ)βπ΄) = (π Ξ£g (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π β πΌ β¦ ((πβπ) β (π΄βπ)))))))) |
31 | 23, 30 | eqtr3d 2768 | 1 β’ (π β ((πβπΉ)βπ΄) = (π Ξ£g (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π β πΌ β¦ ((πβπ) β (π΄βπ)))))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3426 β¦ cmpt 5224 β‘ccnv 5668 β cima 5672 βcfv 6537 (class class class)co 7405 βm cmap 8822 Fincfn 8941 βcn 12216 β0cn0 12476 Basecbs 17153 βΎs cress 17182 .rcmulr 17207 Ξ£g cgsu 17395 .gcmg 18995 mulGrpcmgp 20039 Ringcrg 20138 CRingccrg 20139 SubRingcsubrg 20469 mPoly cmpl 21800 evalSub ces 21975 eval cevl 21976 |
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 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 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-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 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-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-lsp 20819 df-assa 21748 df-asp 21749 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-evls 21977 df-evl 21978 |
This theorem is referenced by: selvvvval 41711 evlselv 41713 |
Copyright terms: Public domain | W3C validator |