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Theorem mndpfsupp 45385
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.)
Hypothesis
Ref Expression
mndpsuppfi.r 𝑅 = (Base‘𝑀)
Assertion
Ref Expression
mndpfsupp (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (𝐴f (+g𝑀)𝐵) finSupp (0g𝑀))

Proof of Theorem mndpfsupp
StepHypRef Expression
1 elmapfn 8546 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
21adantr 484 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴 Fn 𝑉)
323ad2ant2 1136 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐴 Fn 𝑉)
4 elmapfn 8546 . . . . 5 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
54adantl 485 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵 Fn 𝑉)
653ad2ant2 1136 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐵 Fn 𝑉)
7 simp1r 1200 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝑉𝑋)
83, 6, 7, 7offun 7482 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → Fun (𝐴f (+g𝑀)𝐵))
9 id 22 . . . . 5 (𝐴 finSupp (0g𝑀) → 𝐴 finSupp (0g𝑀))
109fsuppimpd 8992 . . . 4 (𝐴 finSupp (0g𝑀) → (𝐴 supp (0g𝑀)) ∈ Fin)
11 id 22 . . . . 5 (𝐵 finSupp (0g𝑀) → 𝐵 finSupp (0g𝑀))
1211fsuppimpd 8992 . . . 4 (𝐵 finSupp (0g𝑀) → (𝐵 supp (0g𝑀)) ∈ Fin)
1310, 12anim12i 616 . . 3 ((𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀)) → ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin))
14 mndpsuppfi.r . . . 4 𝑅 = (Base‘𝑀)
1514mndpsuppfi 45384 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin)) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)
1613, 15syl3an3 1167 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)
17 ovex 7246 . . 3 (𝐴f (+g𝑀)𝐵) ∈ V
18 fvexd 6732 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (0g𝑀) ∈ V)
19 isfsupp 8989 . . 3 (((𝐴f (+g𝑀)𝐵) ∈ V ∧ (0g𝑀) ∈ V) → ((𝐴f (+g𝑀)𝐵) finSupp (0g𝑀) ↔ (Fun (𝐴f (+g𝑀)𝐵) ∧ ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)))
2017, 18, 19sylancr 590 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → ((𝐴f (+g𝑀)𝐵) finSupp (0g𝑀) ↔ (Fun (𝐴f (+g𝑀)𝐵) ∧ ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)))
218, 16, 20mpbir2and 713 1 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (𝐴f (+g𝑀)𝐵) finSupp (0g𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  Vcvv 3408   class class class wbr 5053  Fun wfun 6374   Fn wfn 6375  cfv 6380  (class class class)co 7213  f cof 7467   supp csupp 7903  m cmap 8508  Fincfn 8626   finSupp cfsupp 8985  Basecbs 16760  +gcplusg 16802  0gc0g 16944  Mndcmnd 18173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-1o 8202  df-map 8510  df-en 8627  df-fin 8630  df-fsupp 8986  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174
This theorem is referenced by:  lincsumcl  45445
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