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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 20242 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) |
| 9 | 3 | ply1lmod 22253 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ LMod) |
| 10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
| 11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑊 ∈ LMod) |
| 12 | evl1gsummon.a | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝐴 ∈ 𝐾) | |
| 13 | 12 | r19.21bi 3251 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ 𝐾) |
| 14 | 3 | ply1sca 22254 | . . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑊)) |
| 15 | 6, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑊)) |
| 16 | 15 | fveq2d 6910 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑊))) |
| 17 | 2, 16 | eqtrid 2789 | . . . . . 6 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑊))) |
| 18 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐾 = (Base‘(Scalar‘𝑊))) |
| 19 | 13, 18 | eleqtrd 2843 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 ∈ (Base‘(Scalar‘𝑊))) |
| 20 | evl1gsummon.g | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑊) | |
| 21 | 20, 5 | mgpbas 20142 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) |
| 22 | evl1gsummon.p | . . . . 5 ⊢ ↑ = (.g‘𝐺) | |
| 23 | 3 | ply1ring 22249 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑊 ∈ Ring) |
| 24 | 8, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring) |
| 25 | 20 | ringmgp 20236 | . . . . . . 7 ⊢ (𝑊 ∈ Ring → 𝐺 ∈ Mnd) |
| 26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 27 | 26 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐺 ∈ Mnd) |
| 28 | evl1gsummon.n | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑀 𝑁 ∈ ℕ0) | |
| 29 | 28 | r19.21bi 3251 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑁 ∈ ℕ0) |
| 30 | 8 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring) |
| 31 | evl1gsummon.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
| 32 | 31, 3, 5 | vr1cl 22219 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
| 33 | 30, 32 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑋 ∈ 𝐵) |
| 34 | 21, 22, 27, 29, 33 | mulgnn0cld 19113 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑁 ↑ 𝑋) ∈ 𝐵) |
| 35 | eqid 2737 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 36 | evl1gsummon.t1 | . . . . 5 ⊢ × = ( ·𝑠 ‘𝑊) | |
| 37 | eqid 2737 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 38 | 5, 35, 36, 37 | lmodvscl 20876 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑁 ↑ 𝑋) ∈ 𝐵) → (𝐴 × (𝑁 ↑ 𝑋)) ∈ 𝐵) |
| 39 | 11, 19, 34, 38 | syl3anc 1373 | . . 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 22363 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑊 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 × (𝑁 ↑ 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)))) |
| 44 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑅 ∈ CRing) |
| 45 | 42 | adantr 480 | . . . . 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 22368 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶))) |
| 50 | 49 | mpteq2dva 5242 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶)) = (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))) |
| 51 | 50 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝑅 Σg (𝑥 ∈ 𝑀 ↦ ((𝑄‘(𝐴 × (𝑁 ↑ 𝑋)))‘𝐶))) = (𝑅 Σg (𝑥 ∈ 𝑀 ↦ (𝐴 · (𝑁𝐸𝐶))))) |
| 52 | 43, 51 | 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 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 Σg cgsu 17485 ↑s cpws 17491 Mndcmnd 18747 .gcmg 19085 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 LModclmod 20858 var1cv1 22177 Poly1cpl1 22178 eval1ce1 22318 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-evl1 22320 |
| This theorem is referenced by: (None) |
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