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| Mirrors > Home > MPE Home > Th. List > evlrhm | Structured version Visualization version GIF version | ||
| Description: The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) | 
| Ref | Expression | 
|---|---|
| evlval.q | ⊢ 𝑄 = (𝐼 eval 𝑅) | 
| evlval.b | ⊢ 𝐵 = (Base‘𝑅) | 
| evlrhm.w | ⊢ 𝑊 = (𝐼 mPoly 𝑅) | 
| evlrhm.t | ⊢ 𝑇 = (𝑅 ↑s (𝐵 ↑m 𝐼)) | 
| Ref | Expression | 
|---|---|
| evlrhm | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | crngring 20243 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 2 | 1 | adantl 481 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring) | 
| 3 | evlval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 4 | 3 | subrgid 20574 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅)) | 
| 5 | 2, 4 | syl 17 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝐵 ∈ (SubRing‘𝑅)) | 
| 6 | evlval.q | . . . . 5 ⊢ 𝑄 = (𝐼 eval 𝑅) | |
| 7 | 6, 3 | evlval 22120 | . . . 4 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵) | 
| 8 | eqid 2736 | . . . 4 ⊢ (𝐼 mPoly (𝑅 ↾s 𝐵)) = (𝐼 mPoly (𝑅 ↾s 𝐵)) | |
| 9 | eqid 2736 | . . . 4 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵) | |
| 10 | evlrhm.t | . . . 4 ⊢ 𝑇 = (𝑅 ↑s (𝐵 ↑m 𝐼)) | |
| 11 | 7, 8, 9, 10, 3 | evlsrhm 22113 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing ∧ 𝐵 ∈ (SubRing‘𝑅)) → 𝑄 ∈ ((𝐼 mPoly (𝑅 ↾s 𝐵)) RingHom 𝑇)) | 
| 12 | 5, 11 | mpd3an3 1463 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑄 ∈ ((𝐼 mPoly (𝑅 ↾s 𝐵)) RingHom 𝑇)) | 
| 13 | 3 | ressid 17291 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅) | 
| 14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → (𝑅 ↾s 𝐵) = 𝑅) | 
| 15 | 14 | oveq2d 7448 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → (𝐼 mPoly (𝑅 ↾s 𝐵)) = (𝐼 mPoly 𝑅)) | 
| 16 | evlrhm.w | . . . 4 ⊢ 𝑊 = (𝐼 mPoly 𝑅) | |
| 17 | 15, 16 | eqtr4di 2794 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → (𝐼 mPoly (𝑅 ↾s 𝐵)) = 𝑊) | 
| 18 | 17 | oveq1d 7447 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → ((𝐼 mPoly (𝑅 ↾s 𝐵)) RingHom 𝑇) = (𝑊 RingHom 𝑇)) | 
| 19 | 12, 18 | eleqtrd 2842 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑄 ∈ (𝑊 RingHom 𝑇)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 Basecbs 17248 ↾s cress 17275 ↑s cpws 17492 Ringcrg 20231 CRingccrg 20232 RingHom crh 20470 SubRingcsubrg 20570 mPoly cmpl 21927 eval cevl 22098 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-srg 20185 df-ring 20233 df-cring 20234 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-lsp 20971 df-assa 21874 df-asp 21875 df-ascl 21876 df-psr 21930 df-mvr 21931 df-mpl 21932 df-evls 22099 df-evl 22100 | 
| This theorem is referenced by: evl1val 22334 evl1rhm 22337 mpfpf1 22356 pf1mpf 22357 evlcl 42587 evladdval 42590 evlmulval 42591 | 
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