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Mirrors > Home > MPE Home > Th. List > evl1gsummul | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 15-Sep-2019.) |
Ref | Expression |
---|---|
evl1gsumadd.q | ⊢ 𝑄 = (eval1‘𝑅) |
evl1gsumadd.k | ⊢ 𝐾 = (Base‘𝑅) |
evl1gsumadd.w | ⊢ 𝑊 = (Poly1‘𝑅) |
evl1gsumadd.p | ⊢ 𝑃 = (𝑅 ↑s 𝐾) |
evl1gsumadd.b | ⊢ 𝐵 = (Base‘𝑊) |
evl1gsumadd.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evl1gsumadd.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evl1gsumadd.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evl1gsummul.1 | ⊢ 1 = (1r‘𝑊) |
evl1gsummul.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evl1gsummul.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evl1gsummul.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) |
Ref | Expression |
---|---|
evl1gsummul | ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evl1gsummul.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
2 | evl1gsumadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | mgpbas 18495 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evl1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 18503 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evl1gsumadd.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | evl1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑅) | |
8 | 7 | ply1crng 19568 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑊 ∈ CRing) |
9 | 1 | crngmgp 18555 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
10 | 6, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
11 | crngring 18558 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | evl1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
14 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
15 | 13, 14 | eqeltri 2697 | . . . . 5 ⊢ 𝐾 ∈ V |
16 | 12, 15 | jctir 561 | . . . 4 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐾 ∈ V)) |
17 | evl1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
18 | 17 | pwsring 18615 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
19 | evl1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
20 | 19 | ringmgp 18553 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
21 | 16, 18, 20 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
22 | nn0ex 11298 | . . . . 5 ⊢ ℕ0 ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
24 | evl1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
25 | 23, 24 | ssexd 4805 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
26 | evl1gsumadd.q | . . . . 5 ⊢ 𝑄 = (eval1‘𝑅) | |
27 | 26, 7, 17, 13 | evl1rhm 19696 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
28 | 1, 19 | rhmmhm 18722 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
29 | 6, 27, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
30 | evl1gsumadd.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
31 | evl1gsummul.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) | |
32 | 3, 5, 10, 21, 25, 29, 30, 31 | gsummptmhm 18340 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
33 | 32 | eqcomd 2628 | 1 ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 finSupp cfsupp 8275 ℕ0cn0 11292 Basecbs 15857 Σg cgsu 16101 ↑s cpws 16107 Mndcmnd 17294 MndHom cmhm 17333 CMndccmn 18193 mulGrpcmgp 18489 1rcur 18501 Ringcrg 18547 CRingccrg 18548 RingHom crh 18712 Poly1cpl1 19547 eval1ce1 19679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-evls 19506 df-evl 19507 df-psr1 19550 df-ply1 19552 df-evl1 19681 |
This theorem is referenced by: (None) |
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