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Theorem mndpfsupp 44793
 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 8414 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
21adantr 484 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴 Fn 𝑉)
323ad2ant2 1131 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐴 Fn 𝑉)
4 elmapfn 8414 . . . . 5 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
54adantl 485 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵 Fn 𝑉)
653ad2ant2 1131 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐵 Fn 𝑉)
7 simp1r 1195 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝑉𝑋)
83, 6, 7, 7offun 7402 . 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 8826 . . . 4 (𝐴 finSupp (0g𝑀) → (𝐴 supp (0g𝑀)) ∈ Fin)
11 id 22 . . . . 5 (𝐵 finSupp (0g𝑀) → 𝐵 finSupp (0g𝑀))
1211fsuppimpd 8826 . . . 4 (𝐵 finSupp (0g𝑀) → (𝐵 supp (0g𝑀)) ∈ Fin)
1310, 12anim12i 615 . . 3 ((𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀)) → ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin))
14 mndpsuppfi.r . . . 4 𝑅 = (Base‘𝑀)
1514mndpsuppfi 44792 . . 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 1162 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)
17 ovex 7168 . . 3 (𝐴f (+g𝑀)𝐵) ∈ V
18 fvexd 6660 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (0g𝑀) ∈ V)
19 isfsupp 8823 . . 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 712 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 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   class class class wbr 5030  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   ∘f cof 7388   supp csupp 7815   ↑m cmap 8391  Fincfn 8494   finSupp cfsupp 8819  Basecbs 16477  +gcplusg 16559  0gc0g 16707  Mndcmnd 17905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7390  df-om 7563  df-1st 7673  df-2nd 7674  df-supp 7816  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-fin 8498  df-fsupp 8820  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906 This theorem is referenced by:  lincsumcl  44854
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