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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 22064. (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 22071 | . . . 4 β’ π = (π evalSub1 πΎ) |
4 | 3 | a1i 11 | . . 3 β’ (π β π = (π evalSub1 πΎ)) |
5 | 4 | fveq1d 6894 | . 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 17194 | . . . . . . . 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 2737 | . . . . . 6 β’ (π β π = (π βΎs πΎ)) |
11 | 10 | fveq2d 6896 | . . . . 5 β’ (π β (Poly1βπ ) = (Poly1β(π βΎs πΎ))) |
12 | 6, 11 | eqtrid 2783 | . . . 4 β’ (π β π = (Poly1β(π βΎs πΎ))) |
13 | 12 | fvoveq1d 7434 | . . 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 2731 | . . . 4 β’ (π evalSub1 πΎ) = (π evalSub1 πΎ) | |
15 | eqid 2731 | . . . 4 β’ (Poly1β(π βΎs πΎ)) = (Poly1β(π βΎs πΎ)) | |
16 | eqid 2731 | . . . 4 β’ (0gβ(Poly1β(π βΎs πΎ))) = (0gβ(Poly1β(π βΎs πΎ))) | |
17 | eqid 2731 | . . . 4 β’ (π βΎs πΎ) = (π βΎs πΎ) | |
18 | evl1gsumadd.p | . . . 4 β’ π = (π βs πΎ) | |
19 | eqid 2731 | . . . 4 β’ (Baseβ(Poly1β(π βΎs πΎ))) = (Baseβ(Poly1β(π βΎs πΎ))) | |
20 | crngring 20140 | . . . . 5 β’ (π β CRing β π β Ring) | |
21 | 2 | subrgid 20464 | . . . . 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 480 | . . . . . . 7 β’ ((π β§ π₯ β π) β π = (Poly1β(π βΎs πΎ))) |
26 | 25 | fveq2d 6896 | . . . . . 6 β’ ((π β§ π₯ β π) β (Baseβπ) = (Baseβ(Poly1β(π βΎs πΎ)))) |
27 | 24, 26 | eqtrid 2783 | . . . . 5 β’ ((π β§ π₯ β π) β π΅ = (Baseβ(Poly1β(π βΎs πΎ)))) |
28 | 23, 27 | eleqtrd 2834 | . . . 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 2737 | . . . . . . 7 β’ (π β (Poly1β(π βΎs πΎ)) = π) |
32 | 31 | fveq2d 6896 | . . . . . 6 β’ (π β (0gβ(Poly1β(π βΎs πΎ))) = (0gβπ)) |
33 | evl1gsumadd.0 | . . . . . 6 β’ 0 = (0gβπ) | |
34 | 32, 33 | eqtr4di 2789 | . . . . 5 β’ (π β (0gβ(Poly1β(π βΎs πΎ))) = 0 ) |
35 | 30, 34 | breqtrrd 5177 | . . . 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 22064 | . . 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 2771 | . 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 6894 | . . . . 5 β’ (π β (πβπ) = ((π evalSub1 πΎ)βπ)) |
39 | 38 | eqcomd 2737 | . . . 4 β’ (π β ((π evalSub1 πΎ)βπ) = (πβπ)) |
40 | 39 | mpteq2dv 5251 | . . 3 β’ (π β (π₯ β π β¦ ((π evalSub1 πΎ)βπ)) = (π₯ β π β¦ (πβπ))) |
41 | 40 | oveq2d 7428 | . 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 2775 | 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 395 = wceq 1540 β wcel 2105 β wss 3949 class class class wbr 5149 β¦ cmpt 5232 βcfv 6544 (class class class)co 7412 finSupp cfsupp 9364 β0cn0 12477 Basecbs 17149 βΎs cress 17178 0gc0g 17390 Ξ£g cgsu 17391 βs cpws 17397 Ringcrg 20128 CRingccrg 20129 SubRingcsubrg 20458 Poly1cpl1 21921 evalSub1 ces1 22053 eval1ce1 22054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-ofr 7674 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-lsp 20728 df-assa 21628 df-asp 21629 df-ascl 21630 df-psr 21682 df-mvr 21683 df-mpl 21684 df-opsr 21686 df-evls 21855 df-evl 21856 df-psr1 21924 df-ply1 21926 df-evls1 22055 df-evl1 22056 |
This theorem is referenced by: (None) |
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