Theorem scmsuppfi 45713

Description: The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)

Hypotheses

Ref	Expression
scmsuppfi.s	⊢ 𝑆 = (Scalar‘𝑀)
scmsuppfi.r	⊢ 𝑅 = (Base‘𝑆)

Assertion

Ref	Expression
scmsuppfi	⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)

Distinct variable groups:   𝑣,𝐴   𝑣,𝑀   𝑣,𝑅   𝑣,𝑉

Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmsuppfi

Step	Hyp	Ref	Expression
1		simp3 1137	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → (𝐴 supp (0g‘𝑆)) ∈ Fin)
2		simpll 764	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝑀 ∈ LMod)
3		simplr 766	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
4		simpr 485	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉))
5	2, 3, 4	3jca 1127	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)))
6	5	3adant3 1131	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)))
7		scmsuppfi.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑀)
8		scmsuppfi.r	. . . 4 ⊢ 𝑅 = (Base‘𝑆)
9	7, 8	scmsuppss 45708	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑆)))
10	6, 9	syl 17	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑆)))
11		ssfi 8956	. 2 ⊢ (((𝐴 supp (0g‘𝑆)) ∈ Fin ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑆))) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)
12	1, 10, 11	syl2anc 584	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  𝒫 cpw 4533   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   supp csupp 7977   ↑_m cmap 8615  Fincfn 8733  Basecbs 16912  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150  LModclmod 20123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-1o 8297  df-map 8617  df-en 8734  df-fin 8737  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-ring 19785  df-lmod 20125

This theorem is referenced by:  scmfsupp  45714