scmsuppfi - Mathbox for Alexander van der Vekens

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Theorem scmsuppfi 47329

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 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 484	. . . . 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 47324	. . 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 9175	. 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 583	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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 𝒫 cpw 4597 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 supp csupp 8146 ↑_m cmap 8822 Fincfn 8941 Basecbs 17153 Scalarcsca 17209 ·_𝑠 cvsca 17210 0_gc0g 17394 LModclmod 20706

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-1o 8467 df-map 8824 df-en 8942 df-fin 8945 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-ring 20140 df-lmod 20708

This theorem is referenced by: scmfsupp 47330

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