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Mirrors > Home > MPE Home > Th. List > evls1gsummul | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 14-Sep-2019.) |
Ref | Expression |
---|---|
evls1gsummul.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1gsummul.k | ⊢ 𝐾 = (Base‘𝑆) |
evls1gsummul.w | ⊢ 𝑊 = (Poly1‘𝑈) |
evls1gsummul.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evls1gsummul.1 | ⊢ 1 = (1r‘𝑊) |
evls1gsummul.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1gsummul.p | ⊢ 𝑃 = (𝑆 ↑s 𝐾) |
evls1gsummul.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evls1gsummul.b | ⊢ 𝐵 = (Base‘𝑊) |
evls1gsummul.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1gsummul.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1gsummul.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evls1gsummul.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evls1gsummul.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) |
Ref | Expression |
---|---|
evls1gsummul | ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1gsummul.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
2 | evls1gsummul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | mgpbas 18968 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evls1gsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 18976 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evls1gsummul.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | evls1gsummul.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
8 | evls1gsummul.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
9 | 8 | subrgcrng 19262 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing) |
10 | 6, 7, 9 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ CRing) |
11 | evls1gsummul.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑈) | |
12 | 11 | ply1crng 20069 | . . . 4 ⊢ (𝑈 ∈ CRing → 𝑊 ∈ CRing) |
13 | 1 | crngmgp 19028 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
14 | 10, 12, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
15 | crngring 19031 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
16 | 6, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
17 | evls1gsummul.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
18 | 17 | fvexi 6513 | . . . . 5 ⊢ 𝐾 ∈ V |
19 | 16, 18 | jctir 513 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ 𝐾 ∈ V)) |
20 | evls1gsummul.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s 𝐾) | |
21 | 20 | pwsring 19088 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐾 ∈ V) → 𝑃 ∈ Ring) |
22 | evls1gsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
23 | 22 | ringmgp 19026 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
24 | 19, 21, 23 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
25 | nn0ex 11714 | . . . . 5 ⊢ ℕ0 ∈ V | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
27 | evls1gsummul.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
28 | 26, 27 | ssexd 5084 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
29 | evls1gsummul.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
30 | 29, 17, 20, 8, 11 | evls1rhm 20188 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
31 | 6, 7, 30 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
32 | 1, 22 | rhmmhm 19197 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
33 | 31, 32 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
34 | evls1gsummul.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
35 | evls1gsummul.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) | |
36 | 3, 5, 14, 24, 28, 33, 34, 35 | gsummptmhm 18813 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
37 | 36 | eqcomd 2784 | 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 387 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ⊆ wss 3829 class class class wbr 4929 ↦ cmpt 5008 ‘cfv 6188 (class class class)co 6976 finSupp cfsupp 8628 ℕ0cn0 11707 Basecbs 16339 ↾s cress 16340 Σg cgsu 16570 ↑s cpws 16576 Mndcmnd 17762 MndHom cmhm 17801 CMndccmn 18666 mulGrpcmgp 18962 1rcur 18974 Ringcrg 19020 CRingccrg 19021 RingHom crh 19187 SubRingcsubrg 19254 Poly1cpl1 20048 evalSub1 ces1 20179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-ofr 7228 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-sup 8701 df-oi 8769 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-hom 16445 df-cco 16446 df-0g 16571 df-gsum 16572 df-prds 16577 df-pws 16579 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-ghm 18127 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-srg 18979 df-ring 19022 df-cring 19023 df-rnghom 19190 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-assa 19806 df-asp 19807 df-ascl 19808 df-psr 19850 df-mvr 19851 df-mpl 19852 df-opsr 19854 df-evls 19999 df-psr1 20051 df-ply1 20053 df-evls1 20181 |
This theorem is referenced by: (None) |
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