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Theorem mndpfsupp 48101
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 8923 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
21adantr 480 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴 Fn 𝑉)
323ad2ant2 1134 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐴 Fn 𝑉)
4 elmapfn 8923 . . . . 5 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
54adantl 481 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵 Fn 𝑉)
653ad2ant2 1134 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐵 Fn 𝑉)
7 simp1r 1198 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝑉𝑋)
83, 6, 7, 7offun 7728 . 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 9439 . . . 4 (𝐴 finSupp (0g𝑀) → (𝐴 supp (0g𝑀)) ∈ Fin)
11 id 22 . . . . 5 (𝐵 finSupp (0g𝑀) → 𝐵 finSupp (0g𝑀))
1211fsuppimpd 9439 . . . 4 (𝐵 finSupp (0g𝑀) → (𝐵 supp (0g𝑀)) ∈ Fin)
1310, 12anim12i 612 . . 3 ((𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀)) → ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin))
14 mndpsuppfi.r . . . 4 𝑅 = (Base‘𝑀)
1514mndpsuppfi 48100 . . 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 1165 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)
17 ovex 7481 . . 3 (𝐴f (+g𝑀)𝐵) ∈ V
18 fvexd 6935 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (0g𝑀) ∈ V)
19 isfsupp 9435 . . 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 586 . 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  Vcvv 3488   class class class wbr 5166  Fun wfun 6567   Fn wfn 6568  cfv 6573  (class class class)co 7448  f cof 7712   supp csupp 8201  m cmap 8884  Fincfn 9003   finSupp cfsupp 9431  Basecbs 17258  +gcplusg 17311  0gc0g 17499  Mndcmnd 18772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-1o 8522  df-map 8886  df-en 9004  df-fin 9007  df-fsupp 9432  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773
This theorem is referenced by:  lincsumcl  48160
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