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Mirrors > Home > MPE Home > Th. List > evl1gsummon | Structured version Visualization version GIF version |
Description: Value of a univariate polynomial evaluation mapping an additive group sum of a multiple of an exponentiation of a variable to a group sum of the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.) |
Ref | Expression |
---|---|
evl1gsummon.q | β’ π = (eval1βπ ) |
evl1gsummon.k | β’ πΎ = (Baseβπ ) |
evl1gsummon.w | β’ π = (Poly1βπ ) |
evl1gsummon.b | β’ π΅ = (Baseβπ) |
evl1gsummon.x | β’ π = (var1βπ ) |
evl1gsummon.h | β’ π» = (mulGrpβπ ) |
evl1gsummon.e | β’ πΈ = (.gβπ») |
evl1gsummon.g | β’ πΊ = (mulGrpβπ) |
evl1gsummon.p | β’ β = (.gβπΊ) |
evl1gsummon.t1 | β’ Γ = ( Β·π βπ) |
evl1gsummon.t2 | β’ Β· = (.rβπ ) |
evl1gsummon.r | β’ (π β π β CRing) |
evl1gsummon.a | β’ (π β βπ₯ β π π΄ β πΎ) |
evl1gsummon.m | β’ (π β π β β0) |
evl1gsummon.f | β’ (π β π β Fin) |
evl1gsummon.n | β’ (π β βπ₯ β π π β β0) |
evl1gsummon.c | β’ (π β πΆ β πΎ) |
Ref | Expression |
---|---|
evl1gsummon | β’ (π β ((πβ(π Ξ£g (π₯ β π β¦ (π΄ Γ (π β π)))))βπΆ) = (π Ξ£g (π₯ β π β¦ (π΄ Β· (ππΈπΆ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsummon.q | . . 3 β’ π = (eval1βπ ) | |
2 | evl1gsummon.k | . . 3 β’ πΎ = (Baseβπ ) | |
3 | evl1gsummon.w | . . 3 β’ π = (Poly1βπ ) | |
4 | eqid 2725 | . . 3 β’ (π βs πΎ) = (π βs πΎ) | |
5 | evl1gsummon.b | . . 3 β’ π΅ = (Baseβπ) | |
6 | evl1gsummon.r | . . 3 β’ (π β π β CRing) | |
7 | crngring 20184 | . . . . . . 7 β’ (π β CRing β π β Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 β’ (π β π β Ring) |
9 | 3 | ply1lmod 22174 | . . . . . 6 β’ (π β Ring β π β LMod) |
10 | 8, 9 | syl 17 | . . . . 5 β’ (π β π β LMod) |
11 | 10 | adantr 479 | . . . 4 β’ ((π β§ π₯ β π) β π β LMod) |
12 | evl1gsummon.a | . . . . . 6 β’ (π β βπ₯ β π π΄ β πΎ) | |
13 | 12 | r19.21bi 3239 | . . . . 5 β’ ((π β§ π₯ β π) β π΄ β πΎ) |
14 | 3 | ply1sca 22175 | . . . . . . . . 9 β’ (π β CRing β π = (Scalarβπ)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 β’ (π β π = (Scalarβπ)) |
16 | 15 | fveq2d 6894 | . . . . . . 7 β’ (π β (Baseβπ ) = (Baseβ(Scalarβπ))) |
17 | 2, 16 | eqtrid 2777 | . . . . . 6 β’ (π β πΎ = (Baseβ(Scalarβπ))) |
18 | 17 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β π) β πΎ = (Baseβ(Scalarβπ))) |
19 | 13, 18 | eleqtrd 2827 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β (Baseβ(Scalarβπ))) |
20 | evl1gsummon.g | . . . . . 6 β’ πΊ = (mulGrpβπ) | |
21 | 20, 5 | mgpbas 20079 | . . . . 5 β’ π΅ = (BaseβπΊ) |
22 | evl1gsummon.p | . . . . 5 β’ β = (.gβπΊ) | |
23 | 3 | ply1ring 22170 | . . . . . . . 8 β’ (π β Ring β π β Ring) |
24 | 8, 23 | syl 17 | . . . . . . 7 β’ (π β π β Ring) |
25 | 20 | ringmgp 20178 | . . . . . . 7 β’ (π β Ring β πΊ β Mnd) |
26 | 24, 25 | syl 17 | . . . . . 6 β’ (π β πΊ β Mnd) |
27 | 26 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β π) β πΊ β Mnd) |
28 | evl1gsummon.n | . . . . . 6 β’ (π β βπ₯ β π π β β0) | |
29 | 28 | r19.21bi 3239 | . . . . 5 β’ ((π β§ π₯ β π) β π β β0) |
30 | 8 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β π) β π β Ring) |
31 | evl1gsummon.x | . . . . . . 7 β’ π = (var1βπ ) | |
32 | 31, 3, 5 | vr1cl 22140 | . . . . . 6 β’ (π β Ring β π β π΅) |
33 | 30, 32 | syl 17 | . . . . 5 β’ ((π β§ π₯ β π) β π β π΅) |
34 | 21, 22, 27, 29, 33 | mulgnn0cld 19049 | . . . 4 β’ ((π β§ π₯ β π) β (π β π) β π΅) |
35 | eqid 2725 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
36 | evl1gsummon.t1 | . . . . 5 β’ Γ = ( Β·π βπ) | |
37 | eqid 2725 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
38 | 5, 35, 36, 37 | lmodvscl 20760 | . . . 4 β’ ((π β LMod β§ π΄ β (Baseβ(Scalarβπ)) β§ (π β π) β π΅) β (π΄ Γ (π β π)) β π΅) |
39 | 11, 19, 34, 38 | syl3anc 1368 | . . 3 β’ ((π β§ π₯ β π) β (π΄ Γ (π β π)) β π΅) |
40 | evl1gsummon.m | . . 3 β’ (π β π β β0) | |
41 | evl1gsummon.f | . . 3 β’ (π β π β Fin) | |
42 | evl1gsummon.c | . . 3 β’ (π β πΆ β πΎ) | |
43 | 1, 2, 3, 4, 5, 6, 39, 40, 41, 42 | evl1gsumaddval 22282 | . 2 β’ (π β ((πβ(π Ξ£g (π₯ β π β¦ (π΄ Γ (π β π)))))βπΆ) = (π Ξ£g (π₯ β π β¦ ((πβ(π΄ Γ (π β π)))βπΆ)))) |
44 | 6 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β π) β π β CRing) |
45 | 42 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β π) β πΆ β πΎ) |
46 | evl1gsummon.h | . . . . 5 β’ π» = (mulGrpβπ ) | |
47 | evl1gsummon.e | . . . . 5 β’ πΈ = (.gβπ») | |
48 | evl1gsummon.t2 | . . . . 5 β’ Β· = (.rβπ ) | |
49 | 1, 3, 20, 31, 2, 22, 44, 29, 36, 13, 45, 46, 47, 48 | evl1scvarpwval 22287 | . . . 4 β’ ((π β§ π₯ β π) β ((πβ(π΄ Γ (π β π)))βπΆ) = (π΄ Β· (ππΈπΆ))) |
50 | 49 | mpteq2dva 5244 | . . 3 β’ (π β (π₯ β π β¦ ((πβ(π΄ Γ (π β π)))βπΆ)) = (π₯ β π β¦ (π΄ Β· (ππΈπΆ)))) |
51 | 50 | oveq2d 7429 | . 2 β’ (π β (π Ξ£g (π₯ β π β¦ ((πβ(π΄ Γ (π β π)))βπΆ))) = (π Ξ£g (π₯ β π β¦ (π΄ Β· (ππΈπΆ))))) |
52 | 43, 51 | eqtrd 2765 | 1 β’ (π β ((πβ(π Ξ£g (π₯ β π β¦ (π΄ Γ (π β π)))))βπΆ) = (π Ξ£g (π₯ β π β¦ (π΄ Β· (ππΈπΆ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3051 β wss 3941 β¦ cmpt 5227 βcfv 6543 (class class class)co 7413 Fincfn 8957 β0cn0 12497 Basecbs 17174 .rcmulr 17228 Scalarcsca 17230 Β·π cvsca 17231 Ξ£g cgsu 17416 βs cpws 17422 Mndcmnd 18688 .gcmg 19022 mulGrpcmgp 20073 Ringcrg 20172 CRingccrg 20173 LModclmod 20742 var1cv1 22098 Poly1cpl1 22099 eval1ce1 22237 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-gsum 17418 df-prds 17423 df-pws 17425 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-assa 21786 df-asp 21787 df-ascl 21788 df-psr 21841 df-mvr 21842 df-mpl 21843 df-opsr 21845 df-evls 22020 df-evl 22021 df-psr1 22102 df-vr1 22103 df-ply1 22104 df-evl1 22239 |
This theorem is referenced by: (None) |
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