Theorem scmsuppss 45335

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

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   𝑆(𝑣)

Proof of Theorem scmsuppss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8519	. . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
2		fdm 6543	. . . . . 6 ⊢ (𝐴:𝑉⟶𝑅 → dom 𝐴 = 𝑉)
3		eqidd 2735	. . . . . . . . . . . 12 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
4		fveq2 6706	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → (𝐴‘𝑣) = (𝐴‘𝑥))
5		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
6	4, 5	oveq12d 7220	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))
7	6	adantl 485	. . . . . . . . . . . 12 ⊢ (((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) ∧ 𝑣 = 𝑥) → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))
8		simpr 488	. . . . . . . . . . . 12 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
9		ovex 7235	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ V
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ V)
11	3, 7, 8, 10	fvmptd 6814	. . . . . . . . . . 11 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) = ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))
12	11	neeq1d 2994	. . . . . . . . . 10 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀) ↔ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ≠ (0g‘𝑀)))
13		oveq1 7209	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑥) = (0g‘𝑆) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑥))
14		simplrr 778	. . . . . . . . . . . . . 14 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
15		elelpwi 4515	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
16	15	expcom 417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
17	16	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
18	17	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
19	18	imp 410	. . . . . . . . . . . . . 14 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑀))
20		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑀) = (Base‘𝑀)
21		scmsuppss.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Scalar‘𝑀)
22		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
23		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑆) = (0g‘𝑆)
24		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	20, 21, 22, 23, 24	lmod0vs 19904	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑥) = (0g‘𝑀))
26	14, 19, 25	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑥) = (0g‘𝑀))
27	13, 26	sylan9eqr 2796	. . . . . . . . . . . 12 ⊢ (((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) ∧ (𝐴‘𝑥) = (0g‘𝑆)) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) = (0g‘𝑀))
28	27	ex 416	. . . . . . . . . . 11 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → ((𝐴‘𝑥) = (0g‘𝑆) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) = (0g‘𝑀)))
29	28	necon3d 2956	. . . . . . . . . 10 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ≠ (0g‘𝑀) → (𝐴‘𝑥) ≠ (0g‘𝑆)))
30	12, 29	sylbid 243	. . . . . . . . 9 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → (((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀) → (𝐴‘𝑥) ≠ (0g‘𝑆)))
31	30	ss2rabdv 3979	. . . . . . . 8 ⊢ (((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥 ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
32		ovex 7235	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ V
33		eqid 2734	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))
34	32, 33	dmmpti 6511	. . . . . . . . . . . 12 ⊢ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = 𝑉
35		rabeq 3387	. . . . . . . . . . . 12 ⊢ (dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = 𝑉 → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} = {𝑥 ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)})
36	34, 35	mp1i 13	. . . . . . . . . . 11 ⊢ (dom 𝐴 = 𝑉 → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} = {𝑥 ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)})
37		rabeq 3387	. . . . . . . . . . 11 ⊢ (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)} = {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
38	36, 37	sseq12d 3924	. . . . . . . . . 10 ⊢ (dom 𝐴 = 𝑉 → ({𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)} ↔ {𝑥 ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)}))
39	38	adantr 484	. . . . . . . . 9 ⊢ ((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) → ({𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)} ↔ {𝑥 ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)}))
40	39	adantr 484	. . . . . . . 8 ⊢ (((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → ({𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)} ↔ {𝑥 ∈ 𝑉 ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)}))
41	31, 40	mpbird 260	. . . . . . 7 ⊢ (((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
42	41	exp43 440	. . . . . 6 ⊢ (dom 𝐴 = 𝑉 → (𝐴:𝑉⟶𝑅 → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑀 ∈ LMod → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)}))))
43	2, 42	mpcom 38	. . . . 5 ⊢ (𝐴:𝑉⟶𝑅 → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑀 ∈ LMod → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})))
44	1, 43	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑀 ∈ LMod → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})))
45	44	com13 88	. . 3 ⊢ (𝑀 ∈ LMod → (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝐴 ∈ (𝑅 ↑m 𝑉) → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})))
46	45	3imp 1113	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
47		funmpt 6407	. . . 4 ⊢ Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))
48	47	a1i 11	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
49		mptexg 7026	. . . 4 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
50	49	3ad2ant2 1136	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
51		fvexd 6721	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑀) ∈ V)
52		suppval1 7898	. . 3 ⊢ ((Fun (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∧ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V ∧ (0g‘𝑀) ∈ V) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) = {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)})
53	48, 50, 51, 52	syl3anc 1373	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) = {𝑥 ∈ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∣ ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))‘𝑥) ≠ (0g‘𝑀)})
54		elmapfun 8536	. . . 4 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → Fun 𝐴)
55	54	3ad2ant3 1137	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → Fun 𝐴)
56		simp3 1140	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉))
57		fvexd 6721	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑆) ∈ V)
58		suppval1 7898	. . 3 ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (0g‘𝑆) ∈ V) → (𝐴 supp (0g‘𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
59	55, 56, 57, 58	syl3anc 1373	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (𝐴 supp (0g‘𝑆)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
60	46, 53, 59	3sstr4d 3938	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2935  {crab 3058  Vcvv 3401   ⊆ wss 3857  𝒫 cpw 4503   ↦ cmpt 5124  dom cdm 5540  Fun wfun 6363  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202   supp csupp 7892   ↑_m cmap 8497  Basecbs 16684  Scalarcsca 16770   ·_𝑠 cvsca 16771  0_gc0g 16916  LModclmod 19871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-1st 7750  df-2nd 7751  df-supp 7893  df-map 8499  df-0g 16918  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-grp 18340  df-ring 19536  df-lmod 19873

This theorem is referenced by:  scmsuppfi  45340