Theorem scmsuppss 48097

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 8907	. . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
2		fdm 6756	. . . . . 6 ⊢ (𝐴:𝑉⟶𝑅 → dom 𝐴 = 𝑉)
3		eqidd 2741	. . . . . . . . . . . 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 6920	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → (𝐴‘𝑣) = (𝐴‘𝑥))
5		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
6	4, 5	oveq12d 7466	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))
7	6	adantl 481	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
9		ovex 7481	. . . . . . . . . . . . 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 7036	. . . . . . . . . . 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 3006	. . . . . . . . . 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 7455	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑥) = (0g‘𝑆) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑥))
14		simplrr 777	. . . . . . . . . . . . . 14 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
15		elelpwi 4632	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
16	15	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
17	16	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
18	17	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
19	18	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((dom 𝐴 = 𝑉 ∧ 𝐴:𝑉⟶𝑅) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑀))
20		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑀) = (Base‘𝑀)
21		scmsuppss.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (Scalar‘𝑀)
22		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
23		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑆) = (0g‘𝑆)
24		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	20, 21, 22, 23, 24	lmod0vs 20915	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑀)) → ((0g‘𝑆)( ·𝑠 ‘𝑀)𝑥) = (0g‘𝑀))
26	14, 19, 25	syl2anc 583	. . . . . . . . . . . . 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 2802	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 2967	. . . . . . . . . 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 240	. . . . . . . . 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 4099	. . . . . . . 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 7481	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ V
33		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))
34	32, 33	dmmpti 6724	. . . . . . . . . . . 12 ⊢ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = 𝑉
35		rabeq 3458	. . . . . . . . . . . 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 3458	. . . . . . . . . . 11 ⊢ (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)} = {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑆)})
38	36, 37	sseq12d 4042	. . . . . . . . . 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 480	. . . . . . . . 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 480	. . . . . . . 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 257	. . . . . . 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 436	. . . . . 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 1111	. 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 6616	. . . 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 7258	. . . 4 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
50	49	3ad2ant2 1134	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) ∈ V)
51		fvexd 6935	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑀) ∈ V)
52		suppval1 8207	. . 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 1371	. 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 8924	. . . 4 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → Fun 𝐴)
55	54	3ad2ant3 1135	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → Fun 𝐴)
56		simp3 1138	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉))
57		fvexd 6935	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → (0g‘𝑆) ∈ V)
58		suppval1 8207	. . 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 1371	. 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 4056	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  {crab 3443  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   ↦ cmpt 5249  dom cdm 5700  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   supp csupp 8201   ↑_m cmap 8884  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  LModclmod 20880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-supp 8202  df-map 8886  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-ring 20262  df-lmod 20882

This theorem is referenced by:  scmsuppfi  48102