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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 2727 | . . 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 20169 | . . . . . . 7 β’ (π β CRing β π β Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 β’ (π β π β Ring) |
9 | 3 | ply1lmod 22144 | . . . . . 6 β’ (π β Ring β π β LMod) |
10 | 8, 9 | syl 17 | . . . . 5 β’ (π β π β LMod) |
11 | 10 | adantr 480 | . . . 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 3243 | . . . . 5 β’ ((π β§ π₯ β π) β π΄ β πΎ) |
14 | 3 | ply1sca 22145 | . . . . . . . . 9 β’ (π β CRing β π = (Scalarβπ)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 β’ (π β π = (Scalarβπ)) |
16 | 15 | fveq2d 6895 | . . . . . . 7 β’ (π β (Baseβπ ) = (Baseβ(Scalarβπ))) |
17 | 2, 16 | eqtrid 2779 | . . . . . 6 β’ (π β πΎ = (Baseβ(Scalarβπ))) |
18 | 17 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β πΎ = (Baseβ(Scalarβπ))) |
19 | 13, 18 | eleqtrd 2830 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β (Baseβ(Scalarβπ))) |
20 | evl1gsummon.g | . . . . . 6 β’ πΊ = (mulGrpβπ) | |
21 | 20, 5 | mgpbas 20064 | . . . . 5 β’ π΅ = (BaseβπΊ) |
22 | evl1gsummon.p | . . . . 5 β’ β = (.gβπΊ) | |
23 | 3 | ply1ring 22140 | . . . . . . . 8 β’ (π β Ring β π β Ring) |
24 | 8, 23 | syl 17 | . . . . . . 7 β’ (π β π β Ring) |
25 | 20 | ringmgp 20163 | . . . . . . 7 β’ (π β Ring β πΊ β Mnd) |
26 | 24, 25 | syl 17 | . . . . . 6 β’ (π β πΊ β Mnd) |
27 | 26 | adantr 480 | . . . . 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 3243 | . . . . 5 β’ ((π β§ π₯ β π) β π β β0) |
30 | 8 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π) β π β Ring) |
31 | evl1gsummon.x | . . . . . . 7 β’ π = (var1βπ ) | |
32 | 31, 3, 5 | vr1cl 22110 | . . . . . 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 19034 | . . . 4 β’ ((π β§ π₯ β π) β (π β π) β π΅) |
35 | eqid 2727 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
36 | evl1gsummon.t1 | . . . . 5 β’ Γ = ( Β·π βπ) | |
37 | eqid 2727 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
38 | 5, 35, 36, 37 | lmodvscl 20743 | . . . 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 1369 | . . 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 22252 | . 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 480 | . . . . 5 β’ ((π β§ π₯ β π) β π β CRing) |
45 | 42 | adantr 480 | . . . . 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 22257 | . . . 4 β’ ((π β§ π₯ β π) β ((πβ(π΄ Γ (π β π)))βπΆ) = (π΄ Β· (ππΈπΆ))) |
50 | 49 | mpteq2dva 5242 | . . 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 7430 | . 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 2767 | 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 395 = wceq 1534 β wcel 2099 βwral 3056 β wss 3944 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 Fincfn 8953 β0cn0 12488 Basecbs 17165 .rcmulr 17219 Scalarcsca 17221 Β·π cvsca 17222 Ξ£g cgsu 17407 βs cpws 17413 Mndcmnd 18679 .gcmg 19007 mulGrpcmgp 20058 Ringcrg 20157 CRingccrg 20158 LModclmod 20725 var1cv1 22069 Poly1cpl1 22070 eval1ce1 22207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-cring 20160 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-lmod 20727 df-lss 20798 df-lsp 20838 df-assa 21767 df-asp 21768 df-ascl 21769 df-psr 21822 df-mvr 21823 df-mpl 21824 df-opsr 21826 df-evls 21996 df-evl 21997 df-psr1 22073 df-vr1 22074 df-ply1 22075 df-evl1 22209 |
This theorem is referenced by: (None) |
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