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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 2821 | . . . . . 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 2821 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
9 | evlsvar.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
10 | eqid 2821 | . . . . . 6 ⊢ (algSc‘(𝐼 mPoly 𝑈)) = (algSc‘(𝐼 mPoly 𝑈)) | |
11 | eqid 2821 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
12 | eqid 2821 | . . . . . 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 20294 | . . . . 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 1367 | . . . 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 6667 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))‘𝑋)) |
17 | eqid 2821 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈)) | |
18 | 7 | subrgring 19532 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
19 | 3, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
20 | 5, 6, 17, 1, 19 | mvrf2 20266 | . . . 4 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPoly 𝑈))) |
21 | 20 | ffnd 6510 | . . 3 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
22 | evlsvar.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
23 | fvco2 6753 | . . 3 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑋 ∈ 𝐼) → ((𝑄 ∘ 𝑉)‘𝑋) = (𝑄‘(𝑉‘𝑋))) | |
24 | 21, 22, 23 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = (𝑄‘(𝑉‘𝑋))) |
25 | fveq2 6665 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑔‘𝑥) = (𝑔‘𝑋)) | |
26 | 25 | mpteq2dv 5155 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
27 | ovex 7183 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
28 | 27 | mptex 6980 | . . . 4 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋)) ∈ V |
29 | 26, 12, 28 | fvmpt 6763 | . . 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 2864 | 1 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4561 ↦ cmpt 5139 × cxp 5548 ∘ ccom 5554 Fn wfn 6345 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Basecbs 16477 ↾s cress 16478 ↑s cpws 16714 Ringcrg 19291 CRingccrg 19292 RingHom crh 19458 SubRingcsubrg 19525 algSccascl 20078 mVar cmvr 20126 mPoly cmpl 20127 evalSub ces 20278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-srg 19250 df-ring 19293 df-cring 19294 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-assa 20079 df-asp 20080 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-evls 20280 |
This theorem is referenced by: evlsvarsrng 20306 evlvar 20307 mpfproj 20309 mpfind 20314 evl1var 20493 |
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