scmsuppfi - Mathbox for Alexander van der Vekens

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Theorem scmsuppfi 47553

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 (0_g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) supp (0_g‘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 1135	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ (𝐴 supp (0_g‘𝑆)) ∈ Fin) → (𝐴 supp (0_g‘𝑆)) ∈ Fin)
2		simpll 765	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉)) → 𝑀 ∈ LMod)
3		simplr 767	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
4		simpr 483	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉)) → 𝐴 ∈ (𝑅 ↑_m 𝑉))
5	2, 3, 4	3jca 1125	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉)))
6	5	3adant3 1129	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ (𝐴 supp (0_g‘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 47548	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) supp (0_g‘𝑀)) ⊆ (𝐴 supp (0_g‘𝑆)))
10	6, 9	syl 17	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ (𝐴 supp (0_g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) supp (0_g‘𝑀)) ⊆ (𝐴 supp (0_g‘𝑆)))
11		ssfi 9196	. 2 ⊢ (((𝐴 supp (0_g‘𝑆)) ∈ Fin ∧ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) supp (0_g‘𝑀)) ⊆ (𝐴 supp (0_g‘𝑆))) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) supp (0_g‘𝑀)) ∈ Fin)
12	1, 10, 11	syl2anc 582	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ (𝐴 supp (0_g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) supp (0_g‘𝑀)) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3939 𝒫 cpw 4598 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 supp csupp 8163 ↑_m cmap 8843 Fincfn 8962 Basecbs 17179 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 LModclmod 20747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-1o 8485 df-map 8845 df-en 8963 df-fin 8966 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-ring 20179 df-lmod 20749

This theorem is referenced by: scmfsupp 47554

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