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Theorem mptscmfsuppd 19691
Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 20923. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.)
Hypotheses
Ref Expression
mptscmfsuppd.b 𝐵 = (Base‘𝑃)
mptscmfsuppd.s 𝑆 = (Scalar‘𝑃)
mptscmfsuppd.n · = ( ·𝑠𝑃)
mptscmfsuppd.p (𝜑𝑃 ∈ LMod)
mptscmfsuppd.x (𝜑𝑋𝑉)
mptscmfsuppd.z ((𝜑𝑘𝑋) → 𝑍𝐵)
mptscmfsuppd.a (𝜑𝐴:𝑋𝑌)
mptscmfsuppd.f (𝜑𝐴 finSupp (0g𝑆))
Assertion
Ref Expression
mptscmfsuppd (𝜑 → (𝑘𝑋 ↦ ((𝐴𝑘) · 𝑍)) finSupp (0g𝑃))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑃,𝑘   𝑆,𝑘   𝑘,𝑋   · ,𝑘   𝜑,𝑘
Allowed substitution hints:   𝑉(𝑘)   𝑌(𝑘)   𝑍(𝑘)

Proof of Theorem mptscmfsuppd
StepHypRef Expression
1 mptscmfsuppd.x . 2 (𝜑𝑋𝑉)
2 mptscmfsuppd.p . 2 (𝜑𝑃 ∈ LMod)
3 mptscmfsuppd.s . . 3 𝑆 = (Scalar‘𝑃)
43a1i 11 . 2 (𝜑𝑆 = (Scalar‘𝑃))
5 mptscmfsuppd.b . 2 𝐵 = (Base‘𝑃)
6 fvexd 6667 . 2 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ V)
7 mptscmfsuppd.z . 2 ((𝜑𝑘𝑋) → 𝑍𝐵)
8 eqid 2822 . 2 (0g𝑃) = (0g𝑃)
9 eqid 2822 . 2 (0g𝑆) = (0g𝑆)
10 mptscmfsuppd.n . 2 · = ( ·𝑠𝑃)
11 mptscmfsuppd.a . . . 4 (𝜑𝐴:𝑋𝑌)
1211feqmptd 6715 . . 3 (𝜑𝐴 = (𝑘𝑋 ↦ (𝐴𝑘)))
13 mptscmfsuppd.f . . 3 (𝜑𝐴 finSupp (0g𝑆))
1412, 13eqbrtrrd 5066 . 2 (𝜑 → (𝑘𝑋 ↦ (𝐴𝑘)) finSupp (0g𝑆))
151, 2, 4, 5, 6, 7, 8, 9, 10, 14mptscmfsupp0 19690 1 (𝜑 → (𝑘𝑋 ↦ ((𝐴𝑘) · 𝑍)) finSupp (0g𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469   class class class wbr 5042  cmpt 5122  wf 6330  cfv 6334  (class class class)co 7140   finSupp cfsupp 8821  Basecbs 16474  Scalarcsca 16559   ·𝑠 cvsca 16560  0gc0g 16704  LModclmod 19625
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-supp 7818  df-er 8276  df-en 8497  df-fin 8500  df-fsupp 8822  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-ring 19290  df-lmod 19627
This theorem is referenced by:  ply1coefsupp  20922
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