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Mirrors > Home > MPE Home > Th. List > mpfproj | Structured version Visualization version GIF version |
Description: Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
mpfconst.b | ⊢ 𝐵 = (Base‘𝑆) |
mpfconst.q | ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) |
mpfconst.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mpfconst.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
mpfconst.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
mpfproj.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
Ref | Expression |
---|---|
mpfproj | ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽)) ∈ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | eqid 2726 | . . 3 ⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅)) | |
3 | eqid 2726 | . . 3 ⊢ (𝑆 ↾s 𝑅) = (𝑆 ↾s 𝑅) | |
4 | mpfconst.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
5 | mpfconst.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
6 | mpfconst.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | mpfconst.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
8 | mpfproj.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | evlsvar 22101 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽)) = (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽))) |
10 | eqid 2726 | . . . . . . 7 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅)) | |
11 | eqid 2726 | . . . . . . 7 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
12 | 1, 10, 3, 11, 4 | evlsrhm 22099 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
13 | 5, 6, 7, 12 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
14 | eqid 2726 | . . . . . 6 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) | |
15 | eqid 2726 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) | |
16 | 14, 15 | rhmf 20463 | . . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼)))) |
17 | ffn 6720 | . . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅):(Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))⟶(Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) | |
18 | 13, 16, 17 | 3syl 18 | . . . 4 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
19 | 3 | subrgring 20554 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝑅) ∈ Ring) |
20 | 7, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑆 ↾s 𝑅) ∈ Ring) |
21 | 10, 2, 14, 5, 20, 8 | mvrcl 21997 | . . . 4 ⊢ (𝜑 → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
22 | fnfvelrn 7086 | . . . 4 ⊢ ((((𝐼 evalSub 𝑆)‘𝑅) Fn (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) ∧ ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽)) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) | |
23 | 18, 21, 22 | syl2anc 582 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽)) ∈ ran ((𝐼 evalSub 𝑆)‘𝑅)) |
24 | mpfconst.q | . . 3 ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅) | |
25 | 23, 24 | eleqtrrdi 2837 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽)) ∈ 𝑄) |
26 | 9, 25 | eqeltrrd 2827 | 1 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽)) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5228 ran crn 5675 Fn wfn 6541 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ↑m cmap 8847 Basecbs 17208 ↾s cress 17237 ↑s cpws 17456 Ringcrg 20212 CRingccrg 20213 RingHom crh 20447 SubRingcsubrg 20547 mVar cmvr 21898 mPoly cmpl 21899 evalSub ces 22081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-fz 13533 df-fzo 13676 df-seq 14016 df-hash 14343 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-hom 17285 df-cco 17286 df-0g 17451 df-gsum 17452 df-prds 17457 df-pws 17459 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-cntz 19307 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-srg 20166 df-ring 20214 df-cring 20215 df-rhm 20450 df-subrng 20524 df-subrg 20549 df-lmod 20834 df-lss 20905 df-lsp 20945 df-assa 21847 df-asp 21848 df-ascl 21849 df-psr 21902 df-mvr 21903 df-mpl 21904 df-evls 22083 |
This theorem is referenced by: mzpmfp 42441 |
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