Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mndpfsupp Structured version   Visualization version   GIF version

Theorem mndpfsupp 46542
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 8809 . . . . 5 (𝐴 ∈ (𝑅m 𝑉) → 𝐴 Fn 𝑉)
21adantr 482 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐴 Fn 𝑉)
323ad2ant2 1135 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐴 Fn 𝑉)
4 elmapfn 8809 . . . . 5 (𝐵 ∈ (𝑅m 𝑉) → 𝐵 Fn 𝑉)
54adantl 483 . . . 4 ((𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) → 𝐵 Fn 𝑉)
653ad2ant2 1135 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝐵 Fn 𝑉)
7 simp1r 1199 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → 𝑉𝑋)
83, 6, 7, 7offun 7635 . 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 9319 . . . 4 (𝐴 finSupp (0g𝑀) → (𝐴 supp (0g𝑀)) ∈ Fin)
11 id 22 . . . . 5 (𝐵 finSupp (0g𝑀) → 𝐵 finSupp (0g𝑀))
1211fsuppimpd 9319 . . . 4 (𝐵 finSupp (0g𝑀) → (𝐵 supp (0g𝑀)) ∈ Fin)
1310, 12anim12i 614 . . 3 ((𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀)) → ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin))
14 mndpsuppfi.r . . . 4 𝑅 = (Base‘𝑀)
1514mndpsuppfi 46541 . . 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 1166 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → ((𝐴f (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)
17 ovex 7394 . . 3 (𝐴f (+g𝑀)𝐵) ∈ V
18 fvexd 6861 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (0g𝑀) ∈ V)
19 isfsupp 9315 . . 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 588 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3447   class class class wbr 5109  Fun wfun 6494   Fn wfn 6495  cfv 6500  (class class class)co 7361  f cof 7619   supp csupp 8096  m cmap 8771  Fincfn 8889   finSupp cfsupp 9311  Basecbs 17091  +gcplusg 17141  0gc0g 17329  Mndcmnd 18564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-1o 8416  df-map 8773  df-en 8890  df-fin 8893  df-fsupp 9312  df-0g 17331  df-mgm 18505  df-sgrp 18554  df-mnd 18565
This theorem is referenced by:  lincsumcl  46602
  Copyright terms: Public domain W3C validator