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Theorem mndpfsupp 46442
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 8803 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
21adantr 481 . . . 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 8803 . . . . 5 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
54adantl 482 . . . 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 7631 . 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 9312 . . . 4 (𝐴 finSupp (0g𝑀) → (𝐴 supp (0g𝑀)) ∈ Fin)
11 id 22 . . . . 5 (𝐵 finSupp (0g𝑀) → 𝐵 finSupp (0g𝑀))
1211fsuppimpd 9312 . . . 4 (𝐵 finSupp (0g𝑀) → (𝐵 supp (0g𝑀)) ∈ Fin)
1310, 12anim12i 613 . . 3 ((𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀)) → ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin))
14 mndpsuppfi.r . . . 4 𝑅 = (Base‘𝑀)
1514mndpsuppfi 46441 . . 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 7390 . . 3 (𝐴f (+g𝑀)𝐵) ∈ V
18 fvexd 6857 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (0g𝑀) ∈ V)
19 isfsupp 9309 . . 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 587 . 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 711 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445   class class class wbr 5105  Fun wfun 6490   Fn wfn 6491  cfv 6496  (class class class)co 7357  f cof 7615   supp csupp 8092  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Mndcmnd 18556
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-1o 8412  df-map 8767  df-en 8884  df-fin 8887  df-fsupp 9306  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557
This theorem is referenced by:  lincsumcl  46502
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