Theorem mptscmfsuppd 20863

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 22214. (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	⊢ · = ( ·𝑠 ‘𝑃)
mptscmfsuppd.p	⊢ (𝜑 → 𝑃 ∈ LMod)
mptscmfsuppd.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mptscmfsuppd.z	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)
mptscmfsuppd.a	⊢ (𝜑 → 𝐴:𝑋⟶𝑌)
mptscmfsuppd.f	⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))

Assertion

Ref	Expression
mptscmfsuppd	⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))

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

Step	Hyp	Ref	Expression
1		mptscmfsuppd.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		mptscmfsuppd.p	. 2 ⊢ (𝜑 → 𝑃 ∈ LMod)
3		mptscmfsuppd.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑃)
4	3	a1i 11	. 2 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃))
5		mptscmfsuppd.b	. 2 ⊢ 𝐵 = (Base‘𝑃)
6		fvexd 6843	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V)
7		mptscmfsuppd.z	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)
8		eqid 2733	. 2 ⊢ (0g‘𝑃) = (0g‘𝑃)
9		eqid 2733	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
10		mptscmfsuppd.n	. 2 ⊢ · = ( ·𝑠 ‘𝑃)
11		mptscmfsuppd.a	. . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌)
12	11	feqmptd 6896	. . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)))
13		mptscmfsuppd.f	. . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))
14	12, 13	eqbrtrrd 5117	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆))
15	1, 2, 4, 5, 6, 7, 8, 9, 10, 14	mptscmfsupp0 20862	1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   class class class wbr 5093   ↦ cmpt 5174  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   finSupp cfsupp 9252  Basecbs 17122  Scalarcsca 17166   ·_𝑠 cvsca 17167  0_gc0g 17345  LModclmod 20795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-supp 8097  df-1o 8391  df-en 8876  df-fin 8879  df-fsupp 9253  df-0g 17347  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-grp 18851  df-ring 20155  df-lmod 20797

This theorem is referenced by:  ply1coefsupp  22213