Theorem mptscmfsuppd 20879

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 22242. (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 6849	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V)
7		mptscmfsuppd.z	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)
8		eqid 2736	. 2 ⊢ (0g‘𝑃) = (0g‘𝑃)
9		eqid 2736	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
10		mptscmfsuppd.n	. 2 ⊢ · = ( ·𝑠 ‘𝑃)
11		mptscmfsuppd.a	. . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌)
12	11	feqmptd 6902	. . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)))
13		mptscmfsuppd.f	. . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))
14	12, 13	eqbrtrrd 5122	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆))
15	1, 2, 4, 5, 6, 7, 8, 9, 10, 14	mptscmfsupp0 20878	1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   class class class wbr 5098   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   finSupp cfsupp 9264  Basecbs 17136  Scalarcsca 17180   ·_𝑠 cvsca 17181  0_gc0g 17359  LModclmod 20811

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-supp 8103  df-1o 8397  df-en 8884  df-fin 8887  df-fsupp 9265  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-ring 20170  df-lmod 20813

This theorem is referenced by:  ply1coefsupp  22241