![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > evlsrhm | Structured version Visualization version GIF version |
Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.) |
Ref | Expression |
---|---|
evlsrhm.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsrhm.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlsrhm.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsrhm.t | ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼)) |
evlsrhm.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
evlsrhm | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsrhm.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | evlsrhm.w | . . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
3 | eqid 2733 | . . 3 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
4 | evlsrhm.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
5 | evlsrhm.t | . . 3 ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
6 | evlsrhm.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
7 | eqid 2733 | . . 3 ⊢ (algSc‘𝑊) = (algSc‘𝑊) | |
8 | eqid 2733 | . . 3 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
9 | eqid 2733 | . . 3 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | evlsval2 21631 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom 𝑇) ∧ ((𝑄 ∘ (algSc‘𝑊)) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) ∧ (𝑄 ∘ (𝐼 mVar 𝑈)) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑m 𝐼) ↦ (𝑦‘𝑥)))))) |
11 | 10 | simpld 496 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {csn 4626 ↦ cmpt 5229 × cxp 5672 ∘ ccom 5678 ‘cfv 6539 (class class class)co 7403 ↑m cmap 8815 Basecbs 17139 ↾s cress 17168 ↑s cpws 17387 CRingccrg 20047 RingHom crh 20236 SubRingcsubrg 20346 algSccascl 21390 mVar cmvr 21439 mPoly cmpl 21440 evalSub ces 21614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-fz 13480 df-fzo 13623 df-seq 13962 df-hash 14286 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-hom 17216 df-cco 17217 df-0g 17382 df-gsum 17383 df-prds 17388 df-pws 17390 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-mhm 18666 df-submnd 18667 df-grp 18817 df-minusg 18818 df-sbg 18819 df-mulg 18944 df-subg 18996 df-ghm 19083 df-cntz 19174 df-cmn 19642 df-abl 19643 df-mgp 19979 df-ur 19996 df-srg 20000 df-ring 20048 df-cring 20049 df-rnghom 20239 df-subrg 20348 df-lmod 20460 df-lss 20530 df-lsp 20570 df-assa 21391 df-asp 21392 df-ascl 21393 df-psr 21443 df-mvr 21444 df-mpl 21445 df-evls 21616 |
This theorem is referenced by: evlsgsumadd 21635 evlsgsummul 21636 evlspw 21637 evlrhm 21640 mpfconst 21645 mpfproj 21646 mpfsubrg 21647 mpfind 21651 evls1val 21820 evls1rhm 21822 evls1sca 21823 evlscl 41079 evlsexpval 41088 evlsaddval 41089 evlsmulval 41090 selvcllemh 41101 |
Copyright terms: Public domain | W3C validator |