Theorem mptscmfsuppd 20849

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 22201. (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 6841	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V)
7		mptscmfsuppd.z	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)
8		eqid 2729	. 2 ⊢ (0g‘𝑃) = (0g‘𝑃)
9		eqid 2729	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
10		mptscmfsuppd.n	. 2 ⊢ · = ( ·𝑠 ‘𝑃)
11		mptscmfsuppd.a	. . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌)
12	11	feqmptd 6895	. . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)))
13		mptscmfsuppd.f	. . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))
14	12, 13	eqbrtrrd 5119	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆))
15	1, 2, 4, 5, 6, 7, 8, 9, 10, 14	mptscmfsupp0 20848	1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   class class class wbr 5095   ↦ cmpt 5176  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353   finSupp cfsupp 9270  Basecbs 17138  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361  LModclmod 20781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-supp 8101  df-1o 8395  df-en 8880  df-fin 8883  df-fsupp 9271  df-0g 17363  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-ring 20138  df-lmod 20783

This theorem is referenced by:  ply1coefsupp  22200