![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2737 | . . 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 19983 | . . . . . . 7 β’ (π β CRing β π β Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 β’ (π β π β Ring) |
9 | 3 | ply1lmod 21639 | . . . . . 6 β’ (π β Ring β π β LMod) |
10 | 8, 9 | syl 17 | . . . . 5 β’ (π β π β LMod) |
11 | 10 | adantr 482 | . . . 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 3237 | . . . . 5 β’ ((π β§ π₯ β π) β π΄ β πΎ) |
14 | 3 | ply1sca 21640 | . . . . . . . . 9 β’ (π β CRing β π = (Scalarβπ)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 β’ (π β π = (Scalarβπ)) |
16 | 15 | fveq2d 6851 | . . . . . . 7 β’ (π β (Baseβπ ) = (Baseβ(Scalarβπ))) |
17 | 2, 16 | eqtrid 2789 | . . . . . 6 β’ (π β πΎ = (Baseβ(Scalarβπ))) |
18 | 17 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β π) β πΎ = (Baseβ(Scalarβπ))) |
19 | 13, 18 | eleqtrd 2840 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β (Baseβ(Scalarβπ))) |
20 | evl1gsummon.g | . . . . . 6 β’ πΊ = (mulGrpβπ) | |
21 | 20, 5 | mgpbas 19909 | . . . . 5 β’ π΅ = (BaseβπΊ) |
22 | evl1gsummon.p | . . . . 5 β’ β = (.gβπΊ) | |
23 | 3 | ply1ring 21635 | . . . . . . . 8 β’ (π β Ring β π β Ring) |
24 | 8, 23 | syl 17 | . . . . . . 7 β’ (π β π β Ring) |
25 | 20 | ringmgp 19977 | . . . . . . 7 β’ (π β Ring β πΊ β Mnd) |
26 | 24, 25 | syl 17 | . . . . . 6 β’ (π β πΊ β Mnd) |
27 | 26 | adantr 482 | . . . . 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 3237 | . . . . 5 β’ ((π β§ π₯ β π) β π β β0) |
30 | 8 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β π) β π β Ring) |
31 | evl1gsummon.x | . . . . . . 7 β’ π = (var1βπ ) | |
32 | 31, 3, 5 | vr1cl 21604 | . . . . . 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 18904 | . . . 4 β’ ((π β§ π₯ β π) β (π β π) β π΅) |
35 | eqid 2737 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
36 | evl1gsummon.t1 | . . . . 5 β’ Γ = ( Β·π βπ) | |
37 | eqid 2737 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
38 | 5, 35, 36, 37 | lmodvscl 20355 | . . . 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 1372 | . . 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 21741 | . 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 482 | . . . . 5 β’ ((π β§ π₯ β π) β π β CRing) |
45 | 42 | adantr 482 | . . . . 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 21746 | . . . 4 β’ ((π β§ π₯ β π) β ((πβ(π΄ Γ (π β π)))βπΆ) = (π΄ Β· (ππΈπΆ))) |
50 | 49 | mpteq2dva 5210 | . . 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 7378 | . 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 2777 | 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 397 = wceq 1542 β wcel 2107 βwral 3065 β wss 3915 β¦ cmpt 5193 βcfv 6501 (class class class)co 7362 Fincfn 8890 β0cn0 12420 Basecbs 17090 .rcmulr 17141 Scalarcsca 17143 Β·π cvsca 17144 Ξ£g cgsu 17329 βs cpws 17335 Mndcmnd 18563 .gcmg 18879 mulGrpcmgp 19903 Ringcrg 19971 CRingccrg 19972 LModclmod 20338 var1cv1 21563 Poly1cpl1 21564 eval1ce1 21696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofr 7623 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-hom 17164 df-cco 17165 df-0g 17330 df-gsum 17331 df-prds 17336 df-pws 17338 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-ghm 19013 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-srg 19925 df-ring 19973 df-cring 19974 df-rnghom 20155 df-subrg 20236 df-lmod 20340 df-lss 20409 df-lsp 20449 df-assa 21275 df-asp 21276 df-ascl 21277 df-psr 21327 df-mvr 21328 df-mpl 21329 df-opsr 21331 df-evls 21498 df-evl 21499 df-psr1 21567 df-vr1 21568 df-ply1 21569 df-evl1 21698 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |