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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 18703 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evl1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 18711 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evl1gsumadd.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | evl1gsumadd.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑅) | |
8 | 7 | ply1crng 19783 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑊 ∈ CRing) |
9 | 1 | crngmgp 18763 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
10 | 6, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
11 | crngring 18766 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
13 | evl1gsumadd.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
14 | fvex 6342 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
15 | 13, 14 | eqeltri 2846 | . . . . 5 ⊢ 𝐾 ∈ V |
16 | 12, 15 | jctir 510 | . . . 4 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐾 ∈ V)) |
17 | evl1gsumadd.p | . . . . 5 ⊢ 𝑃 = (𝑅 ↑s 𝐾) | |
18 | 17 | pwsring 18823 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
19 | evl1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
20 | 19 | ringmgp 18761 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
21 | 16, 18, 20 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
22 | nn0ex 11500 | . . . . 5 ⊢ ℕ0 ∈ V | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
24 | evl1gsumadd.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
25 | 23, 24 | ssexd 4939 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
26 | evl1gsumadd.q | . . . . 5 ⊢ 𝑄 = (eval1‘𝑅) | |
27 | 26, 7, 17, 13 | evl1rhm 19911 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
28 | 1, 19 | rhmmhm 18932 | . . . 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 18547 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
33 | 32 | eqcomd 2777 | 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 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ⊆ wss 3723 class class class wbr 4786 ↦ cmpt 4863 ‘cfv 6031 (class class class)co 6793 finSupp cfsupp 8431 ℕ0cn0 11494 Basecbs 16064 Σg cgsu 16309 ↑s cpws 16315 Mndcmnd 17502 MndHom cmhm 17541 CMndccmn 18400 mulGrpcmgp 18697 1rcur 18709 Ringcrg 18755 CRingccrg 18756 RingHom crh 18922 Poly1cpl1 19762 eval1ce1 19894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-ofr 7045 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-fzo 12674 df-seq 13009 df-hash 13322 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-hom 16174 df-cco 16175 df-0g 16310 df-gsum 16311 df-prds 16316 df-pws 16318 df-mre 16454 df-mrc 16455 df-acs 16457 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-mulg 17749 df-subg 17799 df-ghm 17866 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-srg 18714 df-ring 18757 df-cring 18758 df-rnghom 18925 df-subrg 18988 df-lmod 19075 df-lss 19143 df-lsp 19185 df-assa 19527 df-asp 19528 df-ascl 19529 df-psr 19571 df-mvr 19572 df-mpl 19573 df-opsr 19575 df-evls 19721 df-evl 19722 df-psr1 19765 df-ply1 19767 df-evl1 19896 |
This theorem is referenced by: (None) |
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