Theorem mptscmfsuppd 21209

Description: A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 22622. (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 6901	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ V)
7		mptscmfsuppd.z	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)
8		eqid 2760	. 2 ⊢ (0g‘𝑃) = (0g‘𝑃)
9		eqid 2760	. 2 ⊢ (0g‘𝑆) = (0g‘𝑆)
10		mptscmfsuppd.n	. 2 ⊢ · = ( ·𝑠 ‘𝑃)
11		mptscmfsuppd.a	. . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶𝑌)
12	11	feqmptd 6954	. . 3 ⊢ (𝜑 → 𝐴 = (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)))
13		mptscmfsuppd.f	. . 3 ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))
14	12, 13	eqbrtrrd 5128	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (𝐴‘𝑘)) finSupp (0g‘𝑆))
15	1, 2, 4, 5, 6, 7, 8, 9, 10, 14	mptscmfsupp0 21208	1 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   finSupp cfsupp 9357  Basecbs 17393  Scalarcsca 17437   ·_𝑠 cvsca 17438  0_gc0g 17616  LModclmod 21141

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-supp 8181  df-1o 8480  df-en 8978  df-fin 8981  df-fsupp 9358  df-0g 17618  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-grp 19153  df-ring 20467  df-lmod 21143

This theorem is used by:  ply1coefsupp  22621