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Theorem mptscmfsuppd 18752
 Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 19487. (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 fvex 6113 . . 3 (𝐴𝑘) ∈ V
76a1i 11 . 2 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ V)
8 mptscmfsuppd.z . 2 ((𝜑𝑘𝑋) → 𝑍𝐵)
9 eqid 2610 . 2 (0g𝑃) = (0g𝑃)
10 eqid 2610 . 2 (0g𝑆) = (0g𝑆)
11 mptscmfsuppd.n . 2 · = ( ·𝑠𝑃)
12 mptscmfsuppd.a . . . 4 (𝜑𝐴:𝑋𝑌)
1312feqmptd 6159 . . 3 (𝜑𝐴 = (𝑘𝑋 ↦ (𝐴𝑘)))
14 mptscmfsuppd.f . . 3 (𝜑𝐴 finSupp (0g𝑆))
1513, 14eqbrtrrd 4607 . 2 (𝜑 → (𝑘𝑋 ↦ (𝐴𝑘)) finSupp (0g𝑆))
161, 2, 4, 5, 7, 8, 9, 10, 11, 15mptscmfsupp0 18751 1 (𝜑 → (𝑘𝑋 ↦ ((𝐴𝑘) · 𝑍)) finSupp (0g𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LModclmod 18686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-supp 7183  df-er 7629  df-en 7842  df-fin 7845  df-fsupp 8159  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-ring 18372  df-lmod 18688 This theorem is referenced by:  ply1coefsupp  19486
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