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Mirrors > Home > MPE Home > Th. List > evlssca | Structured version Visualization version GIF version |
Description: Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.) |
Ref | Expression |
---|---|
evlssca.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlssca.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlssca.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlssca.b | ⊢ 𝐵 = (Base‘𝑆) |
evlssca.a | ⊢ 𝐴 = (algSc‘𝑊) |
evlssca.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlssca.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlssca.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlssca.x | ⊢ (𝜑 → 𝑋 ∈ 𝑅) |
Ref | Expression |
---|---|
evlssca | ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlssca.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
2 | evlssca.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
3 | evlssca.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evlssca.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
5 | evlssca.w | . . . . . 6 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
6 | eqid 2825 | . . . . . 6 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
7 | evlssca.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
8 | eqid 2825 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) = (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) | |
9 | evlssca.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
10 | evlssca.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑊) | |
11 | eqid 2825 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) | |
12 | eqid 2825 | . . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑦‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑦‘𝑥))) | |
13 | 4, 5, 6, 7, 8, 9, 10, 11, 12 | evlsval2 19887 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))) ∧ ((𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) ∧ (𝑄 ∘ (𝐼 mVar 𝑈)) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑦‘𝑥)))))) |
14 | 1, 2, 3, 13 | syl3anc 1494 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))) ∧ ((𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) ∧ (𝑄 ∘ (𝐼 mVar 𝑈)) = (𝑥 ∈ 𝐼 ↦ (𝑦 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑦‘𝑥)))))) |
15 | 14 | simprld 788 | . . 3 ⊢ (𝜑 → (𝑄 ∘ 𝐴) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥}))) |
16 | 15 | fveq1d 6439 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥}))‘𝑋)) |
17 | eqid 2825 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2825 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
19 | 7 | subrgring 19146 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
20 | 3, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
21 | 5, 17, 18, 10, 1, 20 | mplasclf 19864 | . . . 4 ⊢ (𝜑 → 𝐴:(Base‘𝑈)⟶(Base‘𝑊)) |
22 | 9 | subrgss 19144 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
23 | 7, 9 | ressbas2 16301 | . . . . . 6 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈)) |
24 | 3, 22, 23 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Base‘𝑈)) |
25 | 24 | feq2d 6268 | . . . 4 ⊢ (𝜑 → (𝐴:𝑅⟶(Base‘𝑊) ↔ 𝐴:(Base‘𝑈)⟶(Base‘𝑊))) |
26 | 21, 25 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝐴:𝑅⟶(Base‘𝑊)) |
27 | evlssca.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
28 | fvco3 6526 | . . 3 ⊢ ((𝐴:𝑅⟶(Base‘𝑊) ∧ 𝑋 ∈ 𝑅) → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) | |
29 | 26, 27, 28 | syl2anc 579 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝐴)‘𝑋) = (𝑄‘(𝐴‘𝑋))) |
30 | sneq 4409 | . . . . 5 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
31 | 30 | xpeq2d 5376 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐵 ↑𝑚 𝐼) × {𝑥}) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
32 | ovex 6942 | . . . . 5 ⊢ (𝐵 ↑𝑚 𝐼) ∈ V | |
33 | snex 5131 | . . . . 5 ⊢ {𝑋} ∈ V | |
34 | 32, 33 | xpex 7228 | . . . 4 ⊢ ((𝐵 ↑𝑚 𝐼) × {𝑋}) ∈ V |
35 | 31, 11, 34 | fvmpt 6533 | . . 3 ⊢ (𝑋 ∈ 𝑅 → ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥}))‘𝑋) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
36 | 27, 35 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥}))‘𝑋) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
37 | 16, 29, 36 | 3eqtr3d 2869 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 𝐼) × {𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 {csn 4399 ↦ cmpt 4954 × cxp 5344 ∘ ccom 5350 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 Basecbs 16229 ↾s cress 16230 ↑s cpws 16467 Ringcrg 18908 CRingccrg 18909 RingHom crh 19075 SubRingcsubrg 19139 algSccascl 19679 mVar cmvr 19720 mPoly cmpl 19721 evalSub ces 19871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-ofr 7163 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-srg 18867 df-ring 18910 df-cring 18911 df-rnghom 19078 df-subrg 19141 df-lmod 19228 df-lss 19296 df-lsp 19338 df-assa 19680 df-asp 19681 df-ascl 19682 df-psr 19724 df-mvr 19725 df-mpl 19726 df-evls 19873 |
This theorem is referenced by: evlsscasrng 19893 evlsca 19894 mpfconst 19897 mpfind 19903 evls1sca 20055 evl1sca 20065 pf1ind 20086 |
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