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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 2819 | . . 3 ⊢ ((𝐼 evalSub 𝑆)‘𝑅) = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | eqid 2819 | . . 3 ⊢ (𝐼 mVar (𝑆 ↾s 𝑅)) = (𝐼 mVar (𝑆 ↾s 𝑅)) | |
3 | eqid 2819 | . . 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 20295 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽)) = (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽))) |
10 | eqid 2819 | . . . . . . 7 ⊢ (𝐼 mPoly (𝑆 ↾s 𝑅)) = (𝐼 mPoly (𝑆 ↾s 𝑅)) | |
11 | eqid 2819 | . . . . . . 7 ⊢ (𝑆 ↑s (𝐵 ↑m 𝐼)) = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
12 | 1, 10, 3, 11, 4 | evlsrhm 20293 | . . . . . 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 1366 | . . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑆)‘𝑅) ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s (𝐵 ↑m 𝐼)))) |
14 | eqid 2819 | . . . . . 6 ⊢ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) = (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅))) | |
15 | eqid 2819 | . . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐵 ↑m 𝐼))) | |
16 | 14, 15 | rhmf 19470 | . . . . 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 6507 | . . . . 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 19530 | . . . . . 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 20221 | . . . 4 ⊢ (𝜑 → ((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽) ∈ (Base‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) |
22 | fnfvelrn 6841 | . . . 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 586 | . . 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 2922 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝑅)‘((𝐼 mVar (𝑆 ↾s 𝑅))‘𝐽)) ∈ 𝑄) |
26 | 9, 25 | eqeltrrd 2912 | 1 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑m 𝐼) ↦ (𝑓‘𝐽)) ∈ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ↦ cmpt 5137 ran crn 5549 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ↑m cmap 8398 Basecbs 16475 ↾s cress 16476 ↑s cpws 16712 Ringcrg 19289 CRingccrg 19290 RingHom crh 19456 SubRingcsubrg 19523 mVar cmvr 20124 mPoly cmpl 20125 evalSub ces 20276 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-srg 19248 df-ring 19291 df-cring 19292 df-rnghom 19459 df-subrg 19525 df-lmod 19628 df-lss 19696 df-lsp 19736 df-assa 20077 df-asp 20078 df-ascl 20079 df-psr 20128 df-mvr 20129 df-mpl 20130 df-evls 20278 |
This theorem is referenced by: mzpmfp 39335 |
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