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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 2736 | . . . . . 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 2736 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
9 | evlsvar.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
10 | eqid 2736 | . . . . . 6 ⊢ (algSc‘(𝐼 mPoly 𝑈)) = (algSc‘(𝐼 mPoly 𝑈)) | |
11 | eqid 2736 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
12 | eqid 2736 | . . . . . 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 21497 | . . . . 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 1371 | . . . 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 772 | . . 3 ⊢ (𝜑 → (𝑄 ∘ 𝑉) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))) |
16 | 15 | fveq1d 6844 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))‘𝑋)) |
17 | eqid 2736 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈)) | |
18 | 7 | subrgring 20225 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
19 | 3, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
20 | 5, 6, 17, 1, 19 | mvrf2 21468 | . . . 4 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPoly 𝑈))) |
21 | 20 | ffnd 6669 | . . 3 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
22 | evlsvar.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
23 | fvco2 6938 | . . 3 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑋 ∈ 𝐼) → ((𝑄 ∘ 𝑉)‘𝑋) = (𝑄‘(𝑉‘𝑋))) | |
24 | 21, 22, 23 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = (𝑄‘(𝑉‘𝑋))) |
25 | fveq2 6842 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑔‘𝑥) = (𝑔‘𝑋)) | |
26 | 25 | mpteq2dv 5207 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
27 | ovex 7390 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
28 | 27 | mptex 7173 | . . . 4 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋)) ∈ V |
29 | 26, 12, 28 | fvmpt 6948 | . . 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 2784 | 1 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4586 ↦ cmpt 5188 × cxp 5631 ∘ ccom 5637 Fn wfn 6491 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 Basecbs 17083 ↾s cress 17112 ↑s cpws 17328 Ringcrg 19964 CRingccrg 19965 RingHom crh 20143 SubRingcsubrg 20218 algSccascl 21258 mVar cmvr 21307 mPoly cmpl 21308 evalSub ces 21480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-evls 21482 |
This theorem is referenced by: evlsvarsrng 21509 evlvar 21510 mpfproj 21512 mpfind 21517 evl1var 21702 evlsvarval 40735 |
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