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Mirrors > Home > MPE Home > Th. List > evls1val | Structured version Visualization version GIF version |
Description: Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.) |
Ref | Expression |
---|---|
evls1fval.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1fval.e | ⊢ 𝐸 = (1𝑜 evalSub 𝑆) |
evls1fval.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1val.m | ⊢ 𝑀 = (1𝑜 mPoly (𝑆 ↾s 𝑅)) |
evls1val.k | ⊢ 𝐾 = (Base‘𝑀) |
Ref | Expression |
---|---|
evls1val | ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1fval.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆) | |
2 | 1 | subrgss 19099 | . . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
3 | 2 | adantl 474 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ⊆ 𝐵) |
4 | elpwg 4357 | . . . . . . 7 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) | |
5 | 4 | adantl 474 | . . . . . 6 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵)) |
6 | 3, 5 | mpbird 249 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑅 ∈ 𝒫 𝐵) |
7 | evls1fval.q | . . . . . 6 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
8 | evls1fval.e | . . . . . 6 ⊢ 𝐸 = (1𝑜 evalSub 𝑆) | |
9 | 7, 8, 1 | evls1fval 20006 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))) |
10 | 6, 9 | syldan 586 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))) |
11 | 10 | fveq1d 6413 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄‘𝐴) = (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))‘𝐴)) |
12 | 11 | 3adant3 1163 | . 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) = (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))‘𝐴)) |
13 | 1on 7806 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
14 | 13 | a1i 11 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → 1𝑜 ∈ On) |
15 | simp1 1167 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → 𝑆 ∈ CRing) | |
16 | simp2 1168 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → 𝑅 ∈ (SubRing‘𝑆)) | |
17 | 8 | fveq1i 6412 | . . . . . 6 ⊢ (𝐸‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅) |
18 | evls1val.m | . . . . . 6 ⊢ 𝑀 = (1𝑜 mPoly (𝑆 ↾s 𝑅)) | |
19 | eqid 2799 | . . . . . 6 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅) | |
20 | eqid 2799 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) | |
21 | 17, 18, 19, 20, 1 | evlsrhm 19843 | . . . . 5 ⊢ ((1𝑜 ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝐸‘𝑅) ∈ (𝑀 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
22 | 14, 15, 16, 21 | syl3anc 1491 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝐸‘𝑅) ∈ (𝑀 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
23 | evls1val.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑀) | |
24 | eqid 2799 | . . . . 5 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) | |
25 | 23, 24 | rhmf 19044 | . . . 4 ⊢ ((𝐸‘𝑅) ∈ (𝑀 RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) → (𝐸‘𝑅):𝐾⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
26 | 22, 25 | syl 17 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝐸‘𝑅):𝐾⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
27 | simp3 1169 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → 𝐴 ∈ 𝐾) | |
28 | fvco3 6500 | . . 3 ⊢ (((𝐸‘𝑅):𝐾⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) ∧ 𝐴 ∈ 𝐾) → (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))‘𝐴) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐸‘𝑅)‘𝐴))) | |
29 | 26, 27, 28 | syl2anc 580 | . 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅))‘𝐴) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐸‘𝑅)‘𝐴))) |
30 | 26, 27 | ffvelrnd 6586 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → ((𝐸‘𝑅)‘𝐴) ∈ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
31 | ovex 6910 | . . . . 5 ⊢ (𝐵 ↑𝑚 1𝑜) ∈ V | |
32 | 20, 1 | pwsbas 16462 | . . . . 5 ⊢ ((𝑆 ∈ CRing ∧ (𝐵 ↑𝑚 1𝑜) ∈ V) → (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
33 | 15, 31, 32 | sylancl 581 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))) |
34 | 30, 33 | eleqtrrd 2881 | . . 3 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → ((𝐸‘𝑅)‘𝐴) ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜))) |
35 | coeq1 5483 | . . . 4 ⊢ (𝑥 = ((𝐸‘𝑅)‘𝐴) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) | |
36 | eqid 2799 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) | |
37 | fvex 6424 | . . . . 5 ⊢ ((𝐸‘𝑅)‘𝐴) ∈ V | |
38 | 1 | fvexi 6425 | . . . . . 6 ⊢ 𝐵 ∈ V |
39 | 38 | mptex 6715 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) ∈ V |
40 | 37, 39 | coex 7353 | . . . 4 ⊢ (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V |
41 | 35, 36, 40 | fvmpt 6507 | . . 3 ⊢ (((𝐸‘𝑅)‘𝐴) ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐸‘𝑅)‘𝐴)) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
42 | 34, 41 | syl 17 | . 2 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐸‘𝑅)‘𝐴)) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
43 | 12, 29, 42 | 3eqtrd 2837 | 1 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ 𝐴 ∈ 𝐾) → (𝑄‘𝐴) = (((𝐸‘𝑅)‘𝐴) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ⊆ wss 3769 𝒫 cpw 4349 {csn 4368 ↦ cmpt 4922 × cxp 5310 ∘ ccom 5316 Oncon0 5941 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 1𝑜c1o 7792 ↑𝑚 cmap 8095 Basecbs 16184 ↾s cress 16185 ↑s cpws 16422 CRingccrg 18864 RingHom crh 19030 SubRingcsubrg 19094 mPoly cmpl 19676 evalSub ces 19826 evalSub1 ces1 20000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-ofr 7132 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-sup 8590 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-hom 16291 df-cco 16292 df-0g 16417 df-gsum 16418 df-prds 16423 df-pws 16425 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mhm 17650 df-submnd 17651 df-grp 17741 df-minusg 17742 df-sbg 17743 df-mulg 17857 df-subg 17904 df-ghm 17971 df-cntz 18062 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-srg 18822 df-ring 18865 df-cring 18866 df-rnghom 19033 df-subrg 19096 df-lmod 19183 df-lss 19251 df-lsp 19293 df-assa 19635 df-asp 19636 df-ascl 19637 df-psr 19679 df-mvr 19680 df-mpl 19681 df-evls 19828 df-evls1 20002 |
This theorem is referenced by: evls1var 20024 |
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