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Mirrors > Home > MPE Home > Th. List > evlsgsummul | Structured version Visualization version GIF version |
Description: Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.) |
Ref | Expression |
---|---|
evlsgsummul.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsgsummul.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlsgsummul.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evlsgsummul.1 | ⊢ 1 = (1r‘𝑊) |
evlsgsummul.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsgsummul.p | ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) |
evlsgsummul.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evlsgsummul.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsgsummul.b | ⊢ 𝐵 = (Base‘𝑊) |
evlsgsummul.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsgsummul.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsgsummul.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsgsummul.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) |
evlsgsummul.n | ⊢ (𝜑 → 𝑁 ⊆ ℕ0) |
evlsgsummul.f | ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) |
Ref | Expression |
---|---|
evlsgsummul | ⊢ (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsgsummul.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
2 | evlsgsummul.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 1, 2 | mgpbas 19245 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
4 | evlsgsummul.1 | . . . 4 ⊢ 1 = (1r‘𝑊) | |
5 | 1, 4 | ringidval 19253 | . . 3 ⊢ 1 = (0g‘𝐺) |
6 | evlsgsummul.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | evlsgsummul.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
8 | evlsgsummul.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
9 | evlsgsummul.u | . . . . . . 7 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
10 | 9 | subrgcrng 19539 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing) |
11 | 7, 8, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ CRing) |
12 | evlsgsummul.w | . . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
13 | 12 | mplcrng 20234 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → 𝑊 ∈ CRing) |
14 | 6, 11, 13 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ CRing) |
15 | 1 | crngmgp 19305 | . . . 4 ⊢ (𝑊 ∈ CRing → 𝐺 ∈ CMnd) |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
17 | crngring 19308 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
18 | 7, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
19 | ovex 7189 | . . . . 5 ⊢ (𝐾 ↑m 𝐼) ∈ V | |
20 | 18, 19 | jctir 523 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ Ring ∧ (𝐾 ↑m 𝐼) ∈ V)) |
21 | evlsgsummul.p | . . . . 5 ⊢ 𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
22 | 21 | pwsring 19365 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑m 𝐼) ∈ V) → 𝑃 ∈ Ring) |
23 | evlsgsummul.h | . . . . 5 ⊢ 𝐻 = (mulGrp‘𝑃) | |
24 | 23 | ringmgp 19303 | . . . 4 ⊢ (𝑃 ∈ Ring → 𝐻 ∈ Mnd) |
25 | 20, 22, 24 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
26 | nn0ex 11904 | . . . . 5 ⊢ ℕ0 ∈ V | |
27 | 26 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ0 ∈ V) |
28 | evlsgsummul.n | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ0) | |
29 | 27, 28 | ssexd 5228 | . . 3 ⊢ (𝜑 → 𝑁 ∈ V) |
30 | evlsgsummul.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
31 | evlsgsummul.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
32 | 30, 12, 9, 21, 31 | evlsrhm 20301 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
33 | 6, 7, 8, 32 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑃)) |
34 | 1, 23 | rhmmhm 19474 | . . . 4 ⊢ (𝑄 ∈ (𝑊 RingHom 𝑃) → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
35 | 33, 34 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom 𝐻)) |
36 | evlsgsummul.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵) | |
37 | evlsgsummul.f | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 ) | |
38 | 3, 5, 16, 25, 29, 35, 36, 37 | gsummptmhm 19060 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))) = (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌)))) |
39 | 38 | eqcomd 2827 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 finSupp cfsupp 8833 ℕ0cn0 11898 Basecbs 16483 ↾s cress 16484 Σg cgsu 16714 ↑s cpws 16720 Mndcmnd 17911 MndHom cmhm 17954 CMndccmn 18906 mulGrpcmgp 19239 1rcur 19251 Ringcrg 19297 CRingccrg 19298 RingHom crh 19464 SubRingcsubrg 19531 mPoly cmpl 20133 evalSub ces 20284 |
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-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-evls 20286 |
This theorem is referenced by: (None) |
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