Theorem mndpsuppss 18818

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.)

Hypothesis

Ref	Expression
mndpsuppss.r	⊢ 𝑅 = (Base‘𝑀)

Assertion

Ref	Expression
mndpsuppss	⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ⊆ ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))))

Proof of Theorem mndpsuppss

Dummy variables 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioran 999	. . . . . 6 ⊢ (¬ ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀)) ↔ (¬ (𝐴‘𝑥) ≠ (0g‘𝑀) ∧ ¬ (𝐵‘𝑥) ≠ (0g‘𝑀)))
2		nne 2962	. . . . . . 7 ⊢ (¬ (𝐴‘𝑥) ≠ (0g‘𝑀) ↔ (𝐴‘𝑥) = (0g‘𝑀))
3		nne 2962	. . . . . . 7 ⊢ (¬ (𝐵‘𝑥) ≠ (0g‘𝑀) ↔ (𝐵‘𝑥) = (0g‘𝑀))
4	2, 3	anbi12i 639	. . . . . 6 ⊢ ((¬ (𝐴‘𝑥) ≠ (0g‘𝑀) ∧ ¬ (𝐵‘𝑥) ≠ (0g‘𝑀)) ↔ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀)))
5	1, 4	bitri 278	. . . . 5 ⊢ (¬ ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀)) ↔ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀)))
6		elmapfn 8858	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 Fn 𝑉)
7	6	ad2antrl 740	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴 Fn 𝑉)
8	7	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → 𝐴 Fn 𝑉)
9		elmapfn 8858	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 Fn 𝑉)
10	9	ad2antll 741	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵 Fn 𝑉)
11	10	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → 𝐵 Fn 𝑉)
12		simplr 780	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑉 ∈ 𝑋)
13	12	adantr 485	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → 𝑉 ∈ 𝑋)
14		inidm 4179	. . . . . . . . . 10 ⊢ (𝑉 ∩ 𝑉) = 𝑉
15		simplrl 788	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) = (0g‘𝑀))
16		simplrr 789	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) = (0g‘𝑀))
17	8, 11, 13, 13, 14, 15, 16	ofval 7685	. . . . . . . . 9 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) ∧ 𝑥 ∈ 𝑉) → ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) = ((0g‘𝑀)(+g‘𝑀)(0g‘𝑀)))
18	17	an32s 664	. . . . . . . 8 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) = ((0g‘𝑀)(+g‘𝑀)(0g‘𝑀)))
19		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
20		eqid 2763	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
21	19, 20	mndidcl 18802	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
22	21	ancli 557	. . . . . . . . . 10 ⊢ (𝑀 ∈ Mnd → (𝑀 ∈ Mnd ∧ (0g‘𝑀) ∈ (Base‘𝑀)))
23	22	ad4antr 744	. . . . . . . . 9 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → (𝑀 ∈ Mnd ∧ (0g‘𝑀) ∈ (Base‘𝑀)))
24		eqid 2763	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
25	19, 24, 20	mndlid 18807	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ (0g‘𝑀) ∈ (Base‘𝑀)) → ((0g‘𝑀)(+g‘𝑀)(0g‘𝑀)) = (0g‘𝑀))
26	23, 25	syl 18	. . . . . . . 8 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → ((0g‘𝑀)(+g‘𝑀)(0g‘𝑀)) = (0g‘𝑀))
27	18, 26	eqtrd 2798	. . . . . . 7 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) = (0g‘𝑀))
28		nne 2962	. . . . . . 7 ⊢ (¬ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀) ↔ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) = (0g‘𝑀))
29	27, 28	sylibr 237	. . . . . 6 ⊢ (((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) ∧ ((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀))) → ¬ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀))
30	29	ex 417	. . . . 5 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (((𝐴‘𝑥) = (0g‘𝑀) ∧ (𝐵‘𝑥) = (0g‘𝑀)) → ¬ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)))
31	5, 30	biimtrid 245	. . . 4 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (¬ ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀)) → ¬ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)))
32	31	con4d 116	. . 3 ⊢ ((((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀) → ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀))))
33	32	ss2rabdv 4029	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → {𝑥 ∈ 𝑉 ∣ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)} ⊆ {𝑥 ∈ 𝑉 ∣ ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀))})
34	7, 10, 12, 12	offun 7688	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → Fun (𝐴 ∘f (+g‘𝑀)𝐵))
35		ovexd 7445	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f (+g‘𝑀)𝐵) ∈ V)
36		fvexd 6896	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (0g‘𝑀) ∈ V)
37		suppval1 8158	. . . 4 ⊢ ((Fun (𝐴 ∘f (+g‘𝑀)𝐵) ∧ (𝐴 ∘f (+g‘𝑀)𝐵) ∈ V ∧ (0g‘𝑀) ∈ V) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) = {𝑥 ∈ dom (𝐴 ∘f (+g‘𝑀)𝐵) ∣ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)})
38	34, 35, 36, 37	syl3anc 1398	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) = {𝑥 ∈ dom (𝐴 ∘f (+g‘𝑀)𝐵) ∣ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)})
39	12, 7, 10	offvalfv 7696	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f (+g‘𝑀)𝐵) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(+g‘𝑀)(𝐵‘𝑣))))
40	39	dmeqd 5895	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → dom (𝐴 ∘f (+g‘𝑀)𝐵) = dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(+g‘𝑀)(𝐵‘𝑣))))
41		ovex 7443	. . . . . 6 ⊢ ((𝐴‘𝑣)(+g‘𝑀)(𝐵‘𝑣)) ∈ V
42		eqid 2763	. . . . . 6 ⊢ (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(+g‘𝑀)(𝐵‘𝑣))) = (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(+g‘𝑀)(𝐵‘𝑣)))
43	41, 42	dmmpti 6679	. . . . 5 ⊢ dom (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(+g‘𝑀)(𝐵‘𝑣))) = 𝑉
44	40, 43	eqtrdi 2814	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → dom (𝐴 ∘f (+g‘𝑀)𝐵) = 𝑉)
45	44	rabeqdv 3431	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → {𝑥 ∈ dom (𝐴 ∘f (+g‘𝑀)𝐵) ∣ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)} = {𝑥 ∈ 𝑉 ∣ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)})
46	38, 45	eqtrd 2798	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) = {𝑥 ∈ 𝑉 ∣ ((𝐴 ∘f (+g‘𝑀)𝐵)‘𝑥) ≠ (0g‘𝑀)})
47		elmapfun 8859	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → Fun 𝐴)
48		id 23	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 ∈ (𝑅 ↑m 𝑉))
49		fvexd 6896	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (0g‘𝑀) ∈ V)
50		suppval1 8158	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (0g‘𝑀) ∈ V) → (𝐴 supp (0g‘𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)})
51	47, 48, 49, 50	syl3anc 1398	. . . . . 6 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝐴 supp (0g‘𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)})
52		elmapi 8842	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
53		fdm 6715	. . . . . . 7 ⊢ (𝐴:𝑉⟶𝑅 → dom 𝐴 = 𝑉)
54		rabeq 3430	. . . . . . 7 ⊢ (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)} = {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)})
55	52, 53, 54	3syl 19	. . . . . 6 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → {𝑥 ∈ dom 𝐴 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)} = {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)})
56	51, 55	eqtrd 2798	. . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝐴 supp (0g‘𝑀)) = {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)})
57	56	ad2antrl 740	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 supp (0g‘𝑀)) = {𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)})
58		elmapfun 8859	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → Fun 𝐵)
59	58	ad2antll 741	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → Fun 𝐵)
60		simprr 784	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵 ∈ (𝑅 ↑m 𝑉))
61		suppval1 8158	. . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐵 ∈ (𝑅 ↑m 𝑉) ∧ (0g‘𝑀) ∈ V) → (𝐵 supp (0g‘𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)})
62	59, 60, 36, 61	syl3anc 1398	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐵 supp (0g‘𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)})
63		elmapi 8842	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵:𝑉⟶𝑅)
64	63	fdmd 6716	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → dom 𝐵 = 𝑉)
65	64	ad2antll 741	. . . . . 6 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → dom 𝐵 = 𝑉)
66	65	rabeqdv 3431	. . . . 5 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → {𝑥 ∈ dom 𝐵 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)} = {𝑥 ∈ 𝑉 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)})
67	62, 66	eqtrd 2798	. . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐵 supp (0g‘𝑀)) = {𝑥 ∈ 𝑉 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)})
68	57, 67	uneq12d 4123	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))) = ({𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)} ∪ {𝑥 ∈ 𝑉 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)}))
69		unrab 4268	. . 3 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐴‘𝑥) ≠ (0g‘𝑀)} ∪ {𝑥 ∈ 𝑉 ∣ (𝐵‘𝑥) ≠ (0g‘𝑀)}) = {𝑥 ∈ 𝑉 ∣ ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀))}
70	68, 69	eqtrdi 2814	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))) = {𝑥 ∈ 𝑉 ∣ ((𝐴‘𝑥) ≠ (0g‘𝑀) ∨ (𝐵‘𝑥) ≠ (0g‘𝑀))})
71	33, 46, 70	3sstr4d 3992	1 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f (+g‘𝑀)𝐵) supp (0g‘𝑀)) ⊆ ((𝐴 supp (0g‘𝑀)) ∪ (𝐵 supp (0g‘𝑀))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  {crab 3416  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905   ↦ cmpt 5192  dom cdm 5661  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672   supp csupp 8152   ↑_m cmap 8820  Basecbs 17264  +_gcplusg 17305  0_gc0g 17487  Mndcmnd 18787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-1st 7982  df-2nd 7983  df-supp 8153  df-map 8822  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788

This theorem is referenced by:  mndpsuppfi  18819