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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 2737 | . . 3 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾) | |
5 | evl1gsummon.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
6 | evl1gsummon.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | crngring 19574 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | 3 | ply1lmod 21173 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
11 | 10 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑊 ∈ LMod) |
12 | evl1gsummon.a | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾) | |
13 | 12 | r19.21bi 3130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ 𝐾) |
14 | 3 | ply1sca 21174 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑊)) |
16 | 15 | fveq2d 6721 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑊))) |
17 | 2, 16 | syl5eq 2790 | . . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊))) |
18 | 17 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐾 = (Base‘(Scalar‘𝑊))) |
19 | 13, 18 | eleqtrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
20 | 3 | ply1ring 21169 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
21 | 8, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
22 | evl1gsummon.g | . . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊) | |
23 | 22 | ringmgp 19568 | . . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd) |
24 | 21, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
25 | 24 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐺 ∈ Mnd) |
26 | evl1gsummon.n | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0) | |
27 | 26 | r19.21bi 3130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑁 ∈ ℕ0) |
28 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring) |
29 | evl1gsummon.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
30 | 29, 3, 5 | vr1cl 21138 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
31 | 28, 30 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
32 | 22, 5 | mgpbas 19510 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) |
33 | evl1gsummon.p | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
34 | 32, 33 | mulgnn0cl 18508 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
35 | 25, 27, 31, 34 | syl3anc 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
36 | eqid 2737 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
37 | evl1gsummon.t1 | . . . . 5 ⊢ × = ( ·𝑠 ‘𝑊) | |
38 | eqid 2737 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
39 | 5, 36, 37, 38 | lmodvscl 19916 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ 𝐵) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
40 | 11, 19, 35, 39 | syl3anc 1373 | . . 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 21275 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)))) |
45 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ CRing) |
46 | 43 | adantr 484 | . . . . 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 21280 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
51 | 50 | mpteq2dva 5150 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)) = (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))) |
52 | 51 | oveq2d 7229 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶))) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
53 | 44, 52 | eqtrd 2777 | 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 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3866 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 ℕ0cn0 12090 Basecbs 16760 .rcmulr 16803 Scalarcsca 16805 ·𝑠 cvsca 16806 Σg cgsu 16945 ↑s cpws 16951 Mndcmnd 18173 .gcmg 18488 mulGrpcmgp 19504 Ringcrg 19562 CRingccrg 19563 LModclmod 19899 var1cv1 21097 Poly1cpl1 21098 eval1ce1 21230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-hom 16826 df-cco 16827 df-0g 16946 df-gsum 16947 df-prds 16952 df-pws 16954 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-srg 19521 df-ring 19564 df-cring 19565 df-rnghom 19735 df-subrg 19798 df-lmod 19901 df-lss 19969 df-lsp 20009 df-assa 20815 df-asp 20816 df-ascl 20817 df-psr 20868 df-mvr 20869 df-mpl 20870 df-opsr 20872 df-evls 21032 df-evl 21033 df-psr1 21101 df-vr1 21102 df-ply1 21103 df-evl1 21232 |
This theorem is referenced by: (None) |
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