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Mirrors > Home > MPE Home > Th. List > Mathboxes > scmfsupp | Structured version Visualization version GIF version |
Description: A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
scmsuppfi.s | ⊢ 𝑆 = (Scalar‘𝑀) |
scmsuppfi.r | ⊢ 𝑅 = (Base‘𝑆) |
Ref | Expression |
---|---|
scmfsupp | ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6270 | . . 3 ⊢ Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) | |
2 | 1 | a1i 11 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))) |
3 | id 22 | . . . 4 ⊢ (𝐴 finSupp (0g‘𝑆) → 𝐴 finSupp (0g‘𝑆)) | |
4 | 3 | fsuppimpd 8693 | . . 3 ⊢ (𝐴 finSupp (0g‘𝑆) → (𝐴 supp (0g‘𝑆)) ∈ Fin) |
5 | scmsuppfi.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑀) | |
6 | scmsuppfi.r | . . . 4 ⊢ 𝑅 = (Base‘𝑆) | |
7 | 5, 6 | scmsuppfi 43927 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin) |
8 | 4, 7 | syl3an3 1158 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin) |
9 | mptexg 6857 | . . . . 5 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V) | |
10 | 9 | adantl 482 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V) |
11 | 10 | 3ad2ant1 1126 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V) |
12 | fvex 6558 | . . 3 ⊢ (0g‘𝑀) ∈ V | |
13 | isfsupp 8690 | . . 3 ⊢ (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V ∧ (0g‘𝑀) ∈ V) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin))) | |
14 | 11, 12, 13 | sylancl 586 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin))) |
15 | 2, 8, 14 | mpbir2and 709 | 1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 Vcvv 3440 𝒫 cpw 4459 class class class wbr 4968 ↦ cmpt 5047 Fun wfun 6226 ‘cfv 6232 (class class class)co 7023 supp csupp 7688 ↑𝑚 cmap 8263 Fincfn 8364 finSupp cfsupp 8686 Basecbs 16316 Scalarcsca 16401 ·𝑠 cvsca 16402 0gc0g 16546 LModclmod 19328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-er 8146 df-map 8265 df-en 8365 df-fin 8368 df-fsupp 8687 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-grp 17868 df-ring 18993 df-lmod 19330 |
This theorem is referenced by: gsumlsscl 43933 lincfsuppcl 43970 linccl 43971 lincdifsn 43981 lincsum 43986 lincscm 43987 lincresunit3lem2 44037 lincresunit3 44038 |
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