Theorem scmfsupp 49409

Description: A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmfsupp

Step	Hyp	Ref	Expression
1		funmpt 6566	. . 3 ⊢ Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))
2	1	a1i 11	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
3		id 23	. . . 4 ⊢ (𝐴 finSupp (0g‘𝑆) → 𝐴 finSupp (0g‘𝑆))
4	3	fsuppimpd 9339	. . 3 ⊢ (𝐴 finSupp (0g‘𝑆) → (𝐴 supp (0g‘𝑆)) ∈ Fin)
5		scmsuppfi.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑀)
6		scmsuppfi.r	. . . 4 ⊢ 𝑅 = (Base‘𝑆)
7	5, 6	scmsuppfi 49408	. . 3 ⊢ (((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)
8	4, 7	syl3an3 1183	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)
9		mptexg 7215	. . . . 5 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
10	9	adantl 487	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
11	10	3ad2ant1 1151	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
12		fvex 6886	. . 3 ⊢ (0g‘𝑀) ∈ V
13		isfsupp 9335	. . 3 ⊢ (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V ∧ (0g‘𝑀) ∈ V) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)))
14	11, 12, 13	sylancl 598	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀) ↔ (Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)))
15	2, 8, 14	mpbir2and 726	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185  Fun wfun 6521  ‘cfv 6527  (class class class)co 7408   supp csupp 8155   ↑_m cmap 8825  Fincfn 8951   finSupp cfsupp 9331  Basecbs 17349  Scalarcsca 17393   ·_𝑠 cvsca 17394  0_gc0g 17572  LModclmod 21097

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-1o 8454  df-map 8827  df-en 8952  df-fin 8955  df-fsupp 9332  df-0g 17574  df-mgm 18778  df-sgrp 18870  df-mnd 18886  df-grp 19109  df-ring 20423  df-lmod 21099

This theorem is used by:  gsumlsscl  49414  lincfsuppcl  49447  linccl  49448  lincdifsn  49458  lincsum  49463  lincscm  49464  lincresunit3lem2  49514  lincresunit3  49515