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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 2823 | . . . . . 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 2823 | . . . . . 6 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
9 | evlsvar.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
10 | eqid 2823 | . . . . . 6 ⊢ (algSc‘(𝐼 mPoly 𝑈)) = (algSc‘(𝐼 mPoly 𝑈)) | |
11 | eqid 2823 | . . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑m 𝐼) × {𝑥})) | |
12 | eqid 2823 | . . . . . 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 20302 | . . . . 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 6674 | . 2 ⊢ (𝜑 → ((𝑄 ∘ 𝑉)‘𝑋) = ((𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)))‘𝑋)) |
17 | eqid 2823 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈)) | |
18 | 7 | subrgring 19540 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
19 | 3, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ Ring) |
20 | 5, 6, 17, 1, 19 | mvrf2 20274 | . . . 4 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘(𝐼 mPoly 𝑈))) |
21 | 20 | ffnd 6517 | . . 3 ⊢ (𝜑 → 𝑉 Fn 𝐼) |
22 | evlsvar.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
23 | fvco2 6760 | . . 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 6672 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑔‘𝑥) = (𝑔‘𝑋)) | |
26 | 25 | mpteq2dv 5164 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑥)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
27 | ovex 7191 | . . . . 5 ⊢ (𝐵 ↑m 𝐼) ∈ V | |
28 | 27 | mptex 6988 | . . . 4 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋)) ∈ V |
29 | 26, 12, 28 | fvmpt 6770 | . . 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 2866 | 1 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 ↦ cmpt 5148 × cxp 5555 ∘ ccom 5561 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Basecbs 16485 ↾s cress 16486 ↑s cpws 16722 Ringcrg 19299 CRingccrg 19300 RingHom crh 19466 SubRingcsubrg 19533 algSccascl 20086 mVar cmvr 20134 mPoly cmpl 20135 evalSub ces 20286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-assa 20087 df-asp 20088 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-evls 20288 |
This theorem is referenced by: evlsvarsrng 20314 evlvar 20315 mpfproj 20317 mpfind 20322 evl1var 20501 |
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