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Mirrors > Home > MPE Home > Th. List > evlsvar | Structured version Visualization version GIF version |
Description: Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.) (Proof shortened by AV, 18-Sep-2021.) |
Ref | Expression |
---|---|
evlsvar.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsvar.v | ⊢ 𝑉 = (𝐼 mVar 𝑈) |
evlsvar.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsvar.b | ⊢ 𝐵 = (Base‘𝑆) |
evlsvar.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
evlsvar.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsvar.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsvar.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
evlsvar | ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsvar.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
2 | evlsvar.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
3 | evlsvar.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evlsvar.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
5 | eqid 2622 | . . . . . 6 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly 𝑈) | |
6 | evlsvar.v | . . . . . 6 ⊢ 𝑉 = (𝐼 mVar 𝑈) | |
7 | evlsvar.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
8 | eqid 2622 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) = (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) | |
9 | evlsvar.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
10 | eqid 2622 | . . . . . 6 ⊢ (algSc‘(𝐼 mPoly 𝑈)) = (algSc‘(𝐼 mPoly 𝑈)) | |
11 | eqid 2622 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) | |
12 | eqid 2622 | . . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥))) | |
13 | 4, 5, 6, 7, 8, 9, 10, 11, 12 | evlsval2 19520 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → (𝑄 ∈ ((𝐼 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))) ∧ ((𝑄 ∘ (algSc‘(𝐼 mPoly 𝑈))) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) ∧ (𝑄 ∘ 𝑉) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))) |
14 | 1, 2, 3, 13 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ ((𝐼 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 𝐼))) ∧ ((𝑄 ∘ (algSc‘(𝐼 mPoly 𝑈))) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) ∧ (𝑄 ∘ 𝑉) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))) |
15 | 14 | simprrd 797 | . . 3 ⊢ (𝜑 → (𝑄 ∘ 𝑉) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))) |
16 | 15 | fveq1d 6193 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))‘𝑋)) |
17 | eqid 2622 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈)) | |
18 | 7 | subrgring 18783 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
19 | 3, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
20 | 5, 6, 17, 1, 19 | mvrf2 19492 | . . . 4 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPoly 𝑈))) |
21 | 20 | ffnd 6046 | . . 3 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
22 | evlsvar.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
23 | fvco2 6273 | . . 3 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑋 ∈ 𝐼) → ((𝑄 ∘ 𝑉)‘𝑋) = (𝑄‘(𝑉‘𝑋))) | |
24 | 21, 22, 23 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = (𝑄‘(𝑉‘𝑋))) |
25 | fveq2 6191 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑔‘𝑥) = (𝑔‘𝑋)) | |
26 | 25 | mpteq2dv 4745 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋))) |
27 | ovex 6678 | . . . . 5 ⊢ (𝐵 ↑𝑚 𝐼) ∈ V | |
28 | 27 | mptex 6486 | . . . 4 ⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋)) ∈ V |
29 | 26, 12, 28 | fvmpt 6282 | . . 3 ⊢ (𝑋 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))‘𝑋) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋))) |
30 | 22, 29 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))‘𝑋) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋))) |
31 | 16, 24, 30 | 3eqtr3d 2664 | 1 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 ↦ cmpt 4729 × cxp 5112 ∘ ccom 5118 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Basecbs 15857 ↾s cress 15858 ↑s cpws 16107 Ringcrg 18547 CRingccrg 18548 RingHom crh 18712 SubRingcsubrg 18776 algSccascl 19311 mVar cmvr 19352 mPoly cmpl 19353 evalSub ces 19504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-evls 19506 |
This theorem is referenced by: evlsvarsrng 19528 evlvar 19529 mpfproj 19531 mpfind 19536 evl1var 19700 |
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