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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 | ⊢ 𝑄 = (eval1‘𝑅) |
evl1gsummon.k | ⊢ 𝐾 = (Base‘𝑅) |
evl1gsummon.w | ⊢ 𝑊 = (Poly1‘𝑅) |
evl1gsummon.b | ⊢ 𝐵 = (Base‘𝑊) |
evl1gsummon.x | ⊢ 𝑋 = (var1‘𝑅) |
evl1gsummon.h | ⊢ 𝐻 = (mulGrp‘𝑅) |
evl1gsummon.e | ⊢ 𝐸 = (.g‘𝐻) |
evl1gsummon.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evl1gsummon.p | ⊢ ↑ = (.g‘𝐺) |
evl1gsummon.t1 | ⊢ × = ( ·𝑠 ‘𝑊) |
evl1gsummon.t2 | ⊢ · = (.r‘𝑅) |
evl1gsummon.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evl1gsummon.a | ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾) |
evl1gsummon.m | ⊢ (𝜑 → 𝑀 ⊆ ℕ0) |
evl1gsummon.f | ⊢ (𝜑 → 𝑀 ∈ Fin) |
evl1gsummon.n | ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0) |
evl1gsummon.c | ⊢ (𝜑 → 𝐶 ∈ 𝐾) |
Ref | Expression |
---|---|
evl1gsummon | ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsummon.q | . . 3 ⊢ 𝑄 = (eval1‘𝑅) | |
2 | evl1gsummon.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
3 | evl1gsummon.w | . . 3 ⊢ 𝑊 = (Poly1‘𝑅) | |
4 | eqid 2778 | . . 3 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾) | |
5 | evl1gsummon.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
6 | evl1gsummon.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | crngring 19031 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | 3 | ply1lmod 20123 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
11 | 10 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑊 ∈ LMod) |
12 | evl1gsummon.a | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾) | |
13 | 12 | r19.21bi 3158 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ 𝐾) |
14 | 3 | ply1sca 20124 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑊)) |
16 | 15 | fveq2d 6503 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑊))) |
17 | 2, 16 | syl5eq 2826 | . . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊))) |
18 | 17 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐾 = (Base‘(Scalar‘𝑊))) |
19 | 13, 18 | eleqtrd 2868 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
20 | 3 | ply1ring 20119 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
21 | 8, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
22 | evl1gsummon.g | . . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊) | |
23 | 22 | ringmgp 19026 | . . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd) |
24 | 21, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
25 | 24 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐺 ∈ Mnd) |
26 | evl1gsummon.n | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0) | |
27 | 26 | r19.21bi 3158 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑁 ∈ ℕ0) |
28 | 8 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring) |
29 | evl1gsummon.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
30 | 29, 3, 5 | vr1cl 20088 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
31 | 28, 30 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
32 | 22, 5 | mgpbas 18968 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) |
33 | evl1gsummon.p | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
34 | 32, 33 | mulgnn0cl 18029 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
35 | 25, 27, 31, 34 | syl3anc 1351 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
36 | eqid 2778 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
37 | evl1gsummon.t1 | . . . . 5 ⊢ × = ( ·𝑠 ‘𝑊) | |
38 | eqid 2778 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
39 | 5, 36, 37, 38 | lmodvscl 19373 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ 𝐵) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
40 | 11, 19, 35, 39 | syl3anc 1351 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
41 | evl1gsummon.m | . . 3 ⊢ (𝜑 → 𝑀 ⊆ ℕ0) | |
42 | evl1gsummon.f | . . 3 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
43 | evl1gsummon.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
44 | 1, 2, 3, 4, 5, 6, 40, 41, 42, 43 | evl1gsumaddval 20224 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)))) |
45 | 6 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ CRing) |
46 | 43 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐶 ∈ 𝐾) |
47 | evl1gsummon.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑅) | |
48 | evl1gsummon.e | . . . . 5 ⊢ 𝐸 = (.g‘𝐻) | |
49 | evl1gsummon.t2 | . . . . 5 ⊢ · = (.r‘𝑅) | |
50 | 1, 3, 22, 29, 2, 33, 45, 27, 37, 13, 46, 47, 48, 49 | evl1scvarpwval 20229 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
51 | 50 | mpteq2dva 5022 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)) = (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))) |
52 | 51 | oveq2d 6992 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶))) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
53 | 44, 52 | eqtrd 2814 | 1 ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ⊆ wss 3829 ↦ cmpt 5008 ‘cfv 6188 (class class class)co 6976 Fincfn 8306 ℕ0cn0 11707 Basecbs 16339 .rcmulr 16422 Scalarcsca 16424 ·𝑠 cvsca 16425 Σg cgsu 16570 ↑s cpws 16576 Mndcmnd 17762 .gcmg 18011 mulGrpcmgp 18962 Ringcrg 19020 CRingccrg 19021 LModclmod 19356 var1cv1 20047 Poly1cpl1 20048 eval1ce1 20180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-ofr 7228 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-hom 16445 df-cco 16446 df-0g 16571 df-gsum 16572 df-prds 16577 df-pws 16579 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-ghm 18127 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-srg 18979 df-ring 19022 df-cring 19023 df-rnghom 19190 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-assa 19806 df-asp 19807 df-ascl 19808 df-psr 19850 df-mvr 19851 df-mpl 19852 df-opsr 19854 df-evls 19999 df-evl 20000 df-psr1 20051 df-vr1 20052 df-ply1 20053 df-evl1 20182 |
This theorem is referenced by: (None) |
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