![]() |
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 | ⊢ 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 2736 | . . 3 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾) | |
5 | evl1gsummon.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
6 | evl1gsummon.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | crngring 19976 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | 3 | ply1lmod 21623 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
11 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑊 ∈ LMod) |
12 | evl1gsummon.a | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾) | |
13 | 12 | r19.21bi 3234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ 𝐾) |
14 | 3 | ply1sca 21624 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑊)) |
16 | 15 | fveq2d 6846 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑊))) |
17 | 2, 16 | eqtrid 2788 | . . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊))) |
18 | 17 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐾 = (Base‘(Scalar‘𝑊))) |
19 | 13, 18 | eleqtrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
20 | evl1gsummon.g | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑊) | |
21 | 20, 5 | mgpbas 19902 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) |
22 | evl1gsummon.p | . . . . 5 ⊢ ↑ = (.g‘𝐺) | |
23 | 3 | ply1ring 21619 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
24 | 8, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
25 | 20 | ringmgp 19970 | . . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd) |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
27 | 26 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐺 ∈ Mnd) |
28 | evl1gsummon.n | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0) | |
29 | 28 | r19.21bi 3234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑁 ∈ ℕ0) |
30 | 8 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring) |
31 | evl1gsummon.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
32 | 31, 3, 5 | vr1cl 21588 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
33 | 30, 32 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
34 | 21, 22, 27, 29, 33 | mulgnn0cld 18897 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
35 | eqid 2736 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
36 | evl1gsummon.t1 | . . . . 5 ⊢ × = ( ·𝑠 ‘𝑊) | |
37 | eqid 2736 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
38 | 5, 35, 36, 37 | lmodvscl 20339 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ 𝐵) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
39 | 11, 19, 34, 38 | syl3anc 1371 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
40 | evl1gsummon.m | . . 3 ⊢ (𝜑 → 𝑀 ⊆ ℕ0) | |
41 | evl1gsummon.f | . . 3 ⊢ (𝜑 → 𝑀 ∈ Fin) | |
42 | evl1gsummon.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐾) | |
43 | 1, 2, 3, 4, 5, 6, 39, 40, 41, 42 | evl1gsumaddval 21725 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)))) |
44 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ CRing) |
45 | 42 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐶 ∈ 𝐾) |
46 | evl1gsummon.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑅) | |
47 | evl1gsummon.e | . . . . 5 ⊢ 𝐸 = (.g‘𝐻) | |
48 | evl1gsummon.t2 | . . . . 5 ⊢ · = (.r‘𝑅) | |
49 | 1, 3, 20, 31, 2, 22, 44, 29, 36, 13, 45, 46, 47, 48 | evl1scvarpwval 21730 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
50 | 49 | mpteq2dva 5205 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)) = (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))) |
51 | 50 | oveq2d 7373 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶))) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
52 | 43, 51 | eqtrd 2776 | 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3910 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℕ0cn0 12413 Basecbs 17083 .rcmulr 17134 Scalarcsca 17136 ·𝑠 cvsca 17137 Σg cgsu 17322 ↑s cpws 17328 Mndcmnd 18556 .gcmg 18872 mulGrpcmgp 19896 Ringcrg 19964 CRingccrg 19965 LModclmod 20322 var1cv1 21547 Poly1cpl1 21548 eval1ce1 21680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-opsr 21315 df-evls 21482 df-evl 21483 df-psr1 21551 df-vr1 21552 df-ply1 21553 df-evl1 21682 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |