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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 2821 | . . 3 ⊢ (𝑅 ↑s 𝐾) = (𝑅 ↑s 𝐾) | |
5 | evl1gsummon.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
6 | evl1gsummon.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | crngring 19308 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | 3 | ply1lmod 20420 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
11 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑊 ∈ LMod) |
12 | evl1gsummon.a | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾) | |
13 | 12 | r19.21bi 3208 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ 𝐾) |
14 | 3 | ply1sca 20421 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
15 | 6, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑊)) |
16 | 15 | fveq2d 6674 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑊))) |
17 | 2, 16 | syl5eq 2868 | . . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊))) |
18 | 17 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐾 = (Base‘(Scalar‘𝑊))) |
19 | 13, 18 | eleqtrd 2915 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
20 | 3 | ply1ring 20416 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
21 | 8, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
22 | evl1gsummon.g | . . . . . . . 8 ⊢ 𝐺 = (mulGrp‘𝑊) | |
23 | 22 | ringmgp 19303 | . . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd) |
24 | 21, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
25 | 24 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐺 ∈ Mnd) |
26 | evl1gsummon.n | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0) | |
27 | 26 | r19.21bi 3208 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑁 ∈ ℕ0) |
28 | 8 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring) |
29 | evl1gsummon.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
30 | 29, 3, 5 | vr1cl 20385 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
31 | 28, 30 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
32 | 22, 5 | mgpbas 19245 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) |
33 | evl1gsummon.p | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
34 | 32, 33 | mulgnn0cl 18244 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
35 | 25, 27, 31, 34 | syl3anc 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
36 | eqid 2821 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
37 | evl1gsummon.t1 | . . . . 5 ⊢ × = ( ·𝑠 ‘𝑊) | |
38 | eqid 2821 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
39 | 5, 36, 37, 38 | lmodvscl 19651 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ 𝐵) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
40 | 11, 19, 35, 39 | syl3anc 1367 | . . 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 20522 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)))) |
45 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ CRing) |
46 | 43 | adantr 483 | . . . . 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 20527 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
51 | 50 | mpteq2dva 5161 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)) = (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))) |
52 | 51 | oveq2d 7172 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶))) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
53 | 44, 52 | eqtrd 2856 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 ℕ0cn0 11898 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 Σg cgsu 16714 ↑s cpws 16720 Mndcmnd 17911 .gcmg 18224 mulGrpcmgp 19239 Ringcrg 19297 CRingccrg 19298 LModclmod 19634 var1cv1 20344 Poly1cpl1 20345 eval1ce1 20477 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-srg 19256 df-ring 19299 df-cring 19300 df-rnghom 19467 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-assa 20085 df-asp 20086 df-ascl 20087 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-evls 20286 df-evl 20287 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-evl1 20479 |
This theorem is referenced by: (None) |
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