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Mirrors > Home > MPE Home > Th. List > Mathboxes > selvcl | Structured version Visualization version GIF version |
Description: Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.) |
Ref | Expression |
---|---|
selvcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
selvcl.b | ⊢ 𝐵 = (Base‘𝑃) |
selvcl.u | ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) |
selvcl.t | ⊢ 𝑇 = (𝐽 mPoly 𝑈) |
selvcl.e | ⊢ 𝐸 = (Base‘𝑇) |
selvcl.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
selvcl.j | ⊢ (𝜑 → 𝐽 ⊆ 𝐼) |
selvcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
selvcl | ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | selvcl.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | selvcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
3 | selvcl.u | . . 3 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) | |
4 | selvcl.t | . . 3 ⊢ 𝑇 = (𝐽 mPoly 𝑈) | |
5 | eqid 2736 | . . 3 ⊢ (algSc‘𝑇) = (algSc‘𝑇) | |
6 | eqid 2736 | . . 3 ⊢ ((algSc‘𝑇) ∘ (algSc‘𝑈)) = ((algSc‘𝑇) ∘ (algSc‘𝑈)) | |
7 | selvcl.j | . . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼) | |
8 | selvcl.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | selvval 22131 | . 2 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))‘(((algSc‘𝑇) ∘ (algSc‘𝑈)) ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), ((algSc‘𝑇)‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))) |
10 | eqid 2736 | . . . 4 ⊢ (𝑇 ↑s (𝐸 ↑m 𝐼)) = (𝑇 ↑s (𝐸 ↑m 𝐼)) | |
11 | selvcl.e | . . . 4 ⊢ 𝐸 = (Base‘𝑇) | |
12 | eqid 2736 | . . . 4 ⊢ (Base‘(𝑇 ↑s (𝐸 ↑m 𝐼))) = (Base‘(𝑇 ↑s (𝐸 ↑m 𝐼))) | |
13 | 1, 2 | mplrcl 22006 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) |
14 | 8, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
15 | 14, 7 | ssexd 5322 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) |
16 | 14 | difexd 5329 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V) |
17 | selvcl.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
18 | 3, 16, 17 | mplcrngd 42535 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ CRing) |
19 | 4, 15, 18 | mplcrngd 42535 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ CRing) |
20 | ovexd 7464 | . . . 4 ⊢ (𝜑 → (𝐸 ↑m 𝐼) ∈ V) | |
21 | eqid 2736 | . . . . . . 7 ⊢ ((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈))) = ((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈))) | |
22 | eqid 2736 | . . . . . . 7 ⊢ (𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))) = (𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))) | |
23 | eqid 2736 | . . . . . . 7 ⊢ (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈))) = (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈))) | |
24 | 3, 4, 5, 6, 21, 22, 23, 10, 11, 14, 17, 7 | selvcllemh 42568 | . . . . . 6 ⊢ (𝜑 → ((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈))) ∈ ((𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))) RingHom (𝑇 ↑s (𝐸 ↑m 𝐼)))) |
25 | eqid 2736 | . . . . . . 7 ⊢ (Base‘(𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈))))) = (Base‘(𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈))))) | |
26 | 25, 12 | rhmf 20477 | . . . . . 6 ⊢ (((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈))) ∈ ((𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))) RingHom (𝑇 ↑s (𝐸 ↑m 𝐼))) → ((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈))):(Base‘(𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))))⟶(Base‘(𝑇 ↑s (𝐸 ↑m 𝐼)))) |
27 | 24, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈))):(Base‘(𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))))⟶(Base‘(𝑇 ↑s (𝐸 ↑m 𝐼)))) |
28 | 1, 2, 3, 4, 5, 6, 23, 22, 25, 17, 7, 8 | selvcllem4 42569 | . . . . 5 ⊢ (𝜑 → (((algSc‘𝑇) ∘ (algSc‘𝑈)) ∘ 𝐹) ∈ (Base‘(𝐼 mPoly (𝑇 ↾s ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))))) |
29 | 27, 28 | ffvelcdmd 7103 | . . . 4 ⊢ (𝜑 → (((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))‘(((algSc‘𝑇) ∘ (algSc‘𝑈)) ∘ 𝐹)) ∈ (Base‘(𝑇 ↑s (𝐸 ↑m 𝐼)))) |
30 | 10, 11, 12, 19, 20, 29 | pwselbas 17530 | . . 3 ⊢ (𝜑 → (((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))‘(((algSc‘𝑇) ∘ (algSc‘𝑈)) ∘ 𝐹)):(𝐸 ↑m 𝐼)⟶𝐸) |
31 | eqid 2736 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), ((algSc‘𝑇)‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), ((algSc‘𝑇)‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))) | |
32 | 3, 4, 5, 11, 31, 14, 17, 7 | selvcllem5 42570 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), ((algSc‘𝑇)‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))) ∈ (𝐸 ↑m 𝐼)) |
33 | 30, 32 | ffvelcdmd 7103 | . 2 ⊢ (𝜑 → ((((𝐼 evalSub 𝑇)‘ran ((algSc‘𝑇) ∘ (algSc‘𝑈)))‘(((algSc‘𝑇) ∘ (algSc‘𝑈)) ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), ((algSc‘𝑇)‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))) ∈ 𝐸) |
34 | 9, 33 | eqeltrd 2840 | 1 ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ifcif 4524 ↦ cmpt 5223 ran crn 5684 ∘ ccom 5687 ⟶wf 6555 ‘cfv 6559 (class class class)co 7429 ↑m cmap 8862 Basecbs 17243 ↾s cress 17270 ↑s cpws 17487 CRingccrg 20227 RingHom crh 20461 algSccascl 21864 mVar cmvr 21917 mPoly cmpl 21918 evalSub ces 22088 selectVars cslv 22124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-ofr 7695 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-hom 17317 df-cco 17318 df-0g 17482 df-gsum 17483 df-prds 17488 df-pws 17490 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-mhm 18792 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-ghm 19227 df-cntz 19331 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-srg 20180 df-ring 20228 df-cring 20229 df-rhm 20464 df-subrng 20538 df-subrg 20562 df-lmod 20852 df-lss 20922 df-lsp 20962 df-assa 21865 df-asp 21866 df-ascl 21867 df-psr 21921 df-mvr 21922 df-mpl 21923 df-evls 22090 df-selv 22128 |
This theorem is referenced by: evlselv 42575 |
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