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Mirrors > Home > MPE Home > Th. List > evl1gsumadd | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (additive) group sums to group sums. Remark: the proof would be shorter if the theorem is proved directly instead of using evls1gsumadd 21531. (Contributed by AV, 15-Sep-2019.) |
Ref | Expression |
---|---|
evl1gsumadd.q | β’ π = (eval1βπ ) |
evl1gsumadd.k | β’ πΎ = (Baseβπ ) |
evl1gsumadd.w | β’ π = (Poly1βπ ) |
evl1gsumadd.p | β’ π = (π βs πΎ) |
evl1gsumadd.b | β’ π΅ = (Baseβπ) |
evl1gsumadd.r | β’ (π β π β CRing) |
evl1gsumadd.y | β’ ((π β§ π₯ β π) β π β π΅) |
evl1gsumadd.n | β’ (π β π β β0) |
evl1gsumadd.0 | β’ 0 = (0gβπ) |
evl1gsumadd.f | β’ (π β (π₯ β π β¦ π) finSupp 0 ) |
Ref | Expression |
---|---|
evl1gsumadd | β’ (π β (πβ(π Ξ£g (π₯ β π β¦ π))) = (π Ξ£g (π₯ β π β¦ (πβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsumadd.q | . . . . 5 β’ π = (eval1βπ ) | |
2 | evl1gsumadd.k | . . . . 5 β’ πΎ = (Baseβπ ) | |
3 | 1, 2 | evl1fval1 21538 | . . . 4 β’ π = (π evalSub1 πΎ) |
4 | 3 | a1i 11 | . . 3 β’ (π β π = (π evalSub1 πΎ)) |
5 | 4 | fveq1d 6802 | . 2 β’ (π β (πβ(π Ξ£g (π₯ β π β¦ π))) = ((π evalSub1 πΎ)β(π Ξ£g (π₯ β π β¦ π)))) |
6 | evl1gsumadd.w | . . . . 5 β’ π = (Poly1βπ ) | |
7 | evl1gsumadd.r | . . . . . . . 8 β’ (π β π β CRing) | |
8 | 2 | ressid 16995 | . . . . . . . 8 β’ (π β CRing β (π βΎs πΎ) = π ) |
9 | 7, 8 | syl 17 | . . . . . . 7 β’ (π β (π βΎs πΎ) = π ) |
10 | 9 | eqcomd 2742 | . . . . . 6 β’ (π β π = (π βΎs πΎ)) |
11 | 10 | fveq2d 6804 | . . . . 5 β’ (π β (Poly1βπ ) = (Poly1β(π βΎs πΎ))) |
12 | 6, 11 | eqtrid 2788 | . . . 4 β’ (π β π = (Poly1β(π βΎs πΎ))) |
13 | 12 | fvoveq1d 7325 | . . 3 β’ (π β ((π evalSub1 πΎ)β(π Ξ£g (π₯ β π β¦ π))) = ((π evalSub1 πΎ)β((Poly1β(π βΎs πΎ)) Ξ£g (π₯ β π β¦ π)))) |
14 | eqid 2736 | . . . 4 β’ (π evalSub1 πΎ) = (π evalSub1 πΎ) | |
15 | eqid 2736 | . . . 4 β’ (Poly1β(π βΎs πΎ)) = (Poly1β(π βΎs πΎ)) | |
16 | eqid 2736 | . . . 4 β’ (0gβ(Poly1β(π βΎs πΎ))) = (0gβ(Poly1β(π βΎs πΎ))) | |
17 | eqid 2736 | . . . 4 β’ (π βΎs πΎ) = (π βΎs πΎ) | |
18 | evl1gsumadd.p | . . . 4 β’ π = (π βs πΎ) | |
19 | eqid 2736 | . . . 4 β’ (Baseβ(Poly1β(π βΎs πΎ))) = (Baseβ(Poly1β(π βΎs πΎ))) | |
20 | crngring 19836 | . . . . 5 β’ (π β CRing β π β Ring) | |
21 | 2 | subrgid 20067 | . . . . 5 β’ (π β Ring β πΎ β (SubRingβπ )) |
22 | 7, 20, 21 | 3syl 18 | . . . 4 β’ (π β πΎ β (SubRingβπ )) |
23 | evl1gsumadd.y | . . . . 5 β’ ((π β§ π₯ β π) β π β π΅) | |
24 | evl1gsumadd.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
25 | 12 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β π) β π = (Poly1β(π βΎs πΎ))) |
26 | 25 | fveq2d 6804 | . . . . . 6 β’ ((π β§ π₯ β π) β (Baseβπ) = (Baseβ(Poly1β(π βΎs πΎ)))) |
27 | 24, 26 | eqtrid 2788 | . . . . 5 β’ ((π β§ π₯ β π) β π΅ = (Baseβ(Poly1β(π βΎs πΎ)))) |
28 | 23, 27 | eleqtrd 2839 | . . . 4 β’ ((π β§ π₯ β π) β π β (Baseβ(Poly1β(π βΎs πΎ)))) |
29 | evl1gsumadd.n | . . . 4 β’ (π β π β β0) | |
30 | evl1gsumadd.f | . . . . 5 β’ (π β (π₯ β π β¦ π) finSupp 0 ) | |
31 | 12 | eqcomd 2742 | . . . . . . 7 β’ (π β (Poly1β(π βΎs πΎ)) = π) |
32 | 31 | fveq2d 6804 | . . . . . 6 β’ (π β (0gβ(Poly1β(π βΎs πΎ))) = (0gβπ)) |
33 | evl1gsumadd.0 | . . . . . 6 β’ 0 = (0gβπ) | |
34 | 32, 33 | eqtr4di 2794 | . . . . 5 β’ (π β (0gβ(Poly1β(π βΎs πΎ))) = 0 ) |
35 | 30, 34 | breqtrrd 5109 | . . . 4 β’ (π β (π₯ β π β¦ π) finSupp (0gβ(Poly1β(π βΎs πΎ)))) |
36 | 14, 2, 15, 16, 17, 18, 19, 7, 22, 28, 29, 35 | evls1gsumadd 21531 | . . 3 β’ (π β ((π evalSub1 πΎ)β((Poly1β(π βΎs πΎ)) Ξ£g (π₯ β π β¦ π))) = (π Ξ£g (π₯ β π β¦ ((π evalSub1 πΎ)βπ)))) |
37 | 13, 36 | eqtrd 2776 | . 2 β’ (π β ((π evalSub1 πΎ)β(π Ξ£g (π₯ β π β¦ π))) = (π Ξ£g (π₯ β π β¦ ((π evalSub1 πΎ)βπ)))) |
38 | 4 | fveq1d 6802 | . . . . 5 β’ (π β (πβπ) = ((π evalSub1 πΎ)βπ)) |
39 | 38 | eqcomd 2742 | . . . 4 β’ (π β ((π evalSub1 πΎ)βπ) = (πβπ)) |
40 | 39 | mpteq2dv 5183 | . . 3 β’ (π β (π₯ β π β¦ ((π evalSub1 πΎ)βπ)) = (π₯ β π β¦ (πβπ))) |
41 | 40 | oveq2d 7319 | . 2 β’ (π β (π Ξ£g (π₯ β π β¦ ((π evalSub1 πΎ)βπ))) = (π Ξ£g (π₯ β π β¦ (πβπ)))) |
42 | 5, 37, 41 | 3eqtrd 2780 | 1 β’ (π β (πβ(π Ξ£g (π₯ β π β¦ π))) = (π Ξ£g (π₯ β π β¦ (πβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 β wss 3892 class class class wbr 5081 β¦ cmpt 5164 βcfv 6454 (class class class)co 7303 finSupp cfsupp 9168 β0cn0 12275 Basecbs 16953 βΎs cress 16982 0gc0g 17191 Ξ£g cgsu 17192 βs cpws 17198 Ringcrg 19824 CRingccrg 19825 SubRingcsubrg 20061 Poly1cpl1 21389 evalSub1 ces1 21520 eval1ce1 21521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-of 7561 df-ofr 7562 df-om 7741 df-1st 7859 df-2nd 7860 df-supp 8005 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-er 8525 df-map 8644 df-pm 8645 df-ixp 8713 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-fsupp 9169 df-sup 9241 df-oi 9309 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-7 12083 df-8 12084 df-9 12085 df-n0 12276 df-z 12362 df-dec 12480 df-uz 12625 df-fz 13282 df-fzo 13425 df-seq 13764 df-hash 14087 df-struct 16889 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-mulr 17017 df-sca 17019 df-vsca 17020 df-ip 17021 df-tset 17022 df-ple 17023 df-ds 17025 df-hom 17027 df-cco 17028 df-0g 17193 df-gsum 17194 df-prds 17199 df-pws 17201 df-mre 17336 df-mrc 17337 df-acs 17339 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-mhm 18471 df-submnd 18472 df-grp 18621 df-minusg 18622 df-sbg 18623 df-mulg 18742 df-subg 18793 df-ghm 18873 df-cntz 18964 df-cmn 19429 df-abl 19430 df-mgp 19762 df-ur 19779 df-srg 19783 df-ring 19826 df-cring 19827 df-rnghom 20000 df-subrg 20063 df-lmod 20166 df-lss 20235 df-lsp 20275 df-assa 21101 df-asp 21102 df-ascl 21103 df-psr 21153 df-mvr 21154 df-mpl 21155 df-opsr 21157 df-evls 21323 df-evl 21324 df-psr1 21392 df-ply1 21394 df-evls1 21522 df-evl1 21523 |
This theorem is referenced by: (None) |
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