Theorem mptscmfsuppd 20859

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 22211. (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 6837	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V)
7		mptscmfsuppd.z	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)
8		eqid 2731	. 2 ⊢ (0g‘𝑃) = (0g‘𝑃)
9		eqid 2731	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
10		mptscmfsuppd.n	. 2 ⊢ · = ( ·𝑠 ‘𝑃)
11		mptscmfsuppd.a	. . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌)
12	11	feqmptd 6890	. . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)))
13		mptscmfsuppd.f	. . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))
14	12, 13	eqbrtrrd 5115	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆))
15	1, 2, 4, 5, 6, 7, 8, 9, 10, 14	mptscmfsupp0 20858	1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5091   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   finSupp cfsupp 9245  Basecbs 17117  Scalarcsca 17161   ·_𝑠 cvsca 17162  0_gc0g 17340  LModclmod 20791

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-supp 8091  df-1o 8385  df-en 8870  df-fin 8873  df-fsupp 9246  df-0g 17342  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-grp 18846  df-ring 20151  df-lmod 20793

This theorem is referenced by:  ply1coefsupp  22210