Theorem fsuppssind 40205

Description: Induction on functions 𝐹:𝐴⟶𝐵 with finite support (see fsuppind 40202) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)

Hypotheses

Ref	Expression
fsuppssind.b	⊢ 𝐵 = (Base‘𝐺)
fsuppssind.z	⊢ 0 = (0g‘𝐺)
fsuppssind.p	⊢ + = (+g‘𝐺)
fsuppssind.g	⊢ (𝜑 → 𝐺 ∈ Grp)
fsuppssind.v	⊢ (𝜑 → 𝐼 ∈ 𝑉)
fsuppssind.s	⊢ (𝜑 → 𝑆 ⊆ 𝐼)
fsuppssind.0	⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
fsuppssind.1	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
fsuppssind.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘f + 𝑦) ∈ 𝐻)
fsuppssind.3	⊢ (𝜑 → 𝑋:𝐼⟶𝐵)
fsuppssind.4	⊢ (𝜑 → 𝑋 finSupp 0 )
fsuppssind.5	⊢ (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)

Assertion

Ref	Expression
fsuppssind	⊢ (𝜑 → 𝑋 ∈ 𝐻)

Distinct variable groups:   𝐵,𝑎,𝑏,𝑠   0 ,𝑎,𝑏,𝑠   𝑥, 0 ,𝑦   + ,𝑠,𝑥,𝑦   𝜑,𝑎,𝑏,𝑠   𝜑,𝑥,𝑦   𝐼,𝑎,𝑏,𝑠   𝑥,𝐼,𝑦,𝑠   𝑆,𝑎,𝑏   𝑥,𝑆,𝑦,𝑠   𝐻,𝑎,𝑏,𝑠   𝑥,𝐻,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   + (𝑎,𝑏)   𝐺(𝑥,𝑦,𝑠,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑠,𝑎,𝑏)   𝑋(𝑥,𝑦,𝑠,𝑎,𝑏)

Proof of Theorem fsuppssind

Dummy variables 𝑓 𝑡 𝑢 𝑣 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppssind.3	. . . . 5 ⊢ (𝜑 → 𝑋:𝐼⟶𝐵)
2		fsuppssind.s	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐼)
3	1, 2	fssresd 6625	. . . 4 ⊢ (𝜑 → (𝑋 ↾ 𝑆):𝑆⟶𝐵)
4		fsuppssind.4	. . . . 5 ⊢ (𝜑 → 𝑋 finSupp 0 )
5		fsuppssind.z	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
6	5	fvexi 6770	. . . . . 6 ⊢ 0 ∈ V
7	6	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ V)
8	4, 7	fsuppres 9083	. . . 4 ⊢ (𝜑 → (𝑋 ↾ 𝑆) finSupp 0 )
9	3, 8	jca 511	. . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑆):𝑆⟶𝐵 ∧ (𝑋 ↾ 𝑆) finSupp 0 ))
10		fsuppssind.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
11		fsuppssind.p	. . . 4 ⊢ + = (+g‘𝐺)
12		fsuppssind.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
13		fsuppssind.v	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14	13, 2	ssexd 5243	. . . 4 ⊢ (𝜑 → 𝑆 ∈ V)
15	10, 5	grpidcl 18522	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
16	12, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐵)
17		fconst6g 6647	. . . . . 6 ⊢ ( 0 ∈ 𝐵 → (𝑆 × { 0 }):𝑆⟶𝐵)
18	16, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝑆 × { 0 }):𝑆⟶𝐵)
19		xpundir 5647	. . . . . . 7 ⊢ ((𝑆 ∪ (𝐼 ∖ 𝑆)) × { 0 }) = ((𝑆 × { 0 }) ∪ ((𝐼 ∖ 𝑆) × { 0 }))
20		undif 4412	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐼 ↔ (𝑆 ∪ (𝐼 ∖ 𝑆)) = 𝐼)
21	2, 20	sylib 217	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∪ (𝐼 ∖ 𝑆)) = 𝐼)
22	21	xpeq1d 5609	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∪ (𝐼 ∖ 𝑆)) × { 0 }) = (𝐼 × { 0 }))
23	19, 22	eqtr3id 2793	. . . . . 6 ⊢ (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼 ∖ 𝑆) × { 0 })) = (𝐼 × { 0 }))
24		fsuppssind.0	. . . . . 6 ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
25	23, 24	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)
26	10	fvexi 6770	. . . . . . 7 ⊢ 𝐵 ∈ V
27	26	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
28	27, 13, 2	fsuppssindlem2 40204	. . . . 5 ⊢ (𝜑 → ((𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ ((𝑆 × { 0 }):𝑆⟶𝐵 ∧ ((𝑆 × { 0 }) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
29	18, 25, 28	mpbir2and 709	. . . 4 ⊢ (𝜑 → (𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})
30		simplrr 774	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝑏 ∈ 𝐵)
31	16	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑠 ∈ 𝑆) → 0 ∈ 𝐵)
32	30, 31	ifcld 4502	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑠 ∈ 𝑆) → if(𝑠 = 𝑎, 𝑏, 0 ) ∈ 𝐵)
33	32	fmpttd 6971	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆⟶𝐵)
34		fconstmpt 5640	. . . . . . . 8 ⊢ ((𝐼 ∖ 𝑆) × { 0 }) = (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ 0 )
35	34	uneq2i 4090	. . . . . . 7 ⊢ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼 ∖ 𝑆) × { 0 })) = ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ 0 ))
36		eldifn 4058	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (𝐼 ∖ 𝑆) → ¬ 𝑠 ∈ 𝑆)
37		eleq1a 2834	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ 𝑆 → (𝑠 = 𝑎 → 𝑠 ∈ 𝑆))
38	37	con3dimp 408	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑆 ∧ ¬ 𝑠 ∈ 𝑆) → ¬ 𝑠 = 𝑎)
39	38	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵) ∧ ¬ 𝑠 ∈ 𝑆) → ¬ 𝑠 = 𝑎)
40	39	adantll 710	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) ∧ ¬ 𝑠 ∈ 𝑆) → ¬ 𝑠 = 𝑎)
41	36, 40	sylan2 592	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑠 ∈ (𝐼 ∖ 𝑆)) → ¬ 𝑠 = 𝑎)
42	41	iffalsed 4467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑠 ∈ (𝐼 ∖ 𝑆)) → if(𝑠 = 𝑎, 𝑏, 0 ) = 0 )
43	42	mpteq2dva 5170	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ 0 ))
44	43	uneq2d 4093	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ 0 )))
45		mptun 6563	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝑆 ∪ (𝐼 ∖ 𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
46	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → 𝑆 ⊆ 𝐼)
47	46, 20	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑆 ∪ (𝐼 ∖ 𝑆)) = 𝐼)
48	47	mpteq1d 5165	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ (𝑆 ∪ (𝐼 ∖ 𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
49	45, 48	eqtr3id 2793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
50	44, 49	eqtr3d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼 ∖ 𝑆) ↦ 0 )) = (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
51	35, 50	syl5eq 2791	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼 ∖ 𝑆) × { 0 })) = (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
52		fsuppssind.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
53	51, 52	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)
54	26	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → 𝐵 ∈ V)
55	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → 𝐼 ∈ 𝑉)
56	54, 55, 46	fsuppssindlem2 40204	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆⟶𝐵 ∧ ((𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
57	33, 53, 56	mpbir2and 709	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})
58	27, 13, 2	fsuppssindlem2 40204	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ (𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
59	27, 13, 2	fsuppssindlem2 40204	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
60	58, 59	anbi12d 630	. . . . 5 ⊢ (𝜑 → ((𝑠 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ∧ 𝑡 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻}) ↔ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))))
61	10, 11	grpcl 18500	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
62	12, 61	syl3an1 1161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
63	62	3expb 1118	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
64	63	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
65		simprll 775	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → 𝑠:𝑆⟶𝐵)
66		simprrl 777	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → 𝑡:𝑆⟶𝐵)
67	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → 𝑆 ∈ V)
68		inidm 4149	. . . . . . 7 ⊢ (𝑆 ∩ 𝑆) = 𝑆
69	64, 65, 66, 67, 67, 68	off 7529	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → (𝑠 ∘f + 𝑡):𝑆⟶𝐵)
70	65	ffnd 6585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → 𝑠 Fn 𝑆)
71	66	ffnd 6585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → 𝑡 Fn 𝑆)
72		fnconstg 6646	. . . . . . . . . 10 ⊢ ( 0 ∈ V → ((𝐼 ∖ 𝑆) × { 0 }) Fn (𝐼 ∖ 𝑆))
73	6, 72	mp1i 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝐼 ∖ 𝑆) × { 0 }) Fn (𝐼 ∖ 𝑆))
74	13	difexd 5248	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∖ 𝑆) ∈ V)
75	74	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → (𝐼 ∖ 𝑆) ∈ V)
76		disjdif 4402	. . . . . . . . . 10 ⊢ (𝑆 ∩ (𝐼 ∖ 𝑆)) = ∅
77	76	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → (𝑆 ∩ (𝐼 ∖ 𝑆)) = ∅)
78	70, 71, 73, 73, 67, 75, 77	ofun 40137	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 }))) = ((𝑠 ∘f + 𝑡) ∪ (((𝐼 ∖ 𝑆) × { 0 }) ∘f + ((𝐼 ∖ 𝑆) × { 0 }))))
79	6, 72	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼 ∖ 𝑆) × { 0 }) Fn (𝐼 ∖ 𝑆))
80		fvconst2g 7059	. . . . . . . . . . . 12 ⊢ (( 0 ∈ V ∧ 𝑗 ∈ (𝐼 ∖ 𝑆)) → (((𝐼 ∖ 𝑆) × { 0 })‘𝑗) = 0 )
81	7, 80	sylan 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐼 ∖ 𝑆)) → (((𝐼 ∖ 𝑆) × { 0 })‘𝑗) = 0 )
82	10, 11, 5	grplid 18524	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
83	12, 16, 82	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( 0 + 0 ) = 0 )
84	83	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐼 ∖ 𝑆)) → ( 0 + 0 ) = 0 )
85	6	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐼 ∖ 𝑆)) → 0 ∈ V)
86	85, 80	sylancom 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐼 ∖ 𝑆)) → (((𝐼 ∖ 𝑆) × { 0 })‘𝑗) = 0 )
87	84, 86	eqtr4d 2781	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝐼 ∖ 𝑆)) → ( 0 + 0 ) = (((𝐼 ∖ 𝑆) × { 0 })‘𝑗))
88	74, 79, 79, 79, 81, 81, 87	offveq 7535	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐼 ∖ 𝑆) × { 0 }) ∘f + ((𝐼 ∖ 𝑆) × { 0 })) = ((𝐼 ∖ 𝑆) × { 0 }))
89	88	uneq2d 4093	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 ∘f + 𝑡) ∪ (((𝐼 ∖ 𝑆) × { 0 }) ∘f + ((𝐼 ∖ 𝑆) × { 0 }))) = ((𝑠 ∘f + 𝑡) ∪ ((𝐼 ∖ 𝑆) × { 0 })))
90	89	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∘f + 𝑡) ∪ (((𝐼 ∖ 𝑆) × { 0 }) ∘f + ((𝐼 ∖ 𝑆) × { 0 }))) = ((𝑠 ∘f + 𝑡) ∪ ((𝐼 ∖ 𝑆) × { 0 })))
91	78, 90	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 }))) = ((𝑠 ∘f + 𝑡) ∪ ((𝐼 ∖ 𝑆) × { 0 })))
92		fsuppssind.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘f + 𝑦) ∈ 𝐻)
93	92	caovclg 7442	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)) → ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 }))) ∈ 𝐻)
94	93	adantrrl 720	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 }))) ∈ 𝐻)
95	94	adantrll 718	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 }))) ∈ 𝐻)
96	91, 95	eqeltrrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∘f + 𝑡) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)
97	27, 13, 2	fsuppssindlem2 40204	. . . . . . 7 ⊢ (𝜑 → ((𝑠 ∘f + 𝑡) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠 ∘f + 𝑡):𝑆⟶𝐵 ∧ ((𝑠 ∘f + 𝑡) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
98	97	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∘f + 𝑡) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠 ∘f + 𝑡):𝑆⟶𝐵 ∧ ((𝑠 ∘f + 𝑡) ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
99	69, 96, 98	mpbir2and 709	. . . . 5 ⊢ ((𝜑 ∧ ((𝑠:𝑆⟶𝐵 ∧ (𝑠 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆⟶𝐵 ∧ (𝑡 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻))) → (𝑠 ∘f + 𝑡) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})
100	60, 99	sylbida 591	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ∧ 𝑡 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})) → (𝑠 ∘f + 𝑡) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})
101	10, 5, 11, 12, 14, 29, 57, 100	fsuppind 40202	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ↾ 𝑆):𝑆⟶𝐵 ∧ (𝑋 ↾ 𝑆) finSupp 0 )) → (𝑋 ↾ 𝑆) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})
102	9, 101	mpdan 683	. 2 ⊢ (𝜑 → (𝑋 ↾ 𝑆) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻})
103	27, 14	elmapd 8587	. . . . 5 ⊢ (𝜑 → ((𝑋 ↾ 𝑆) ∈ (𝐵 ↑m 𝑆) ↔ (𝑋 ↾ 𝑆):𝑆⟶𝐵))
104	3, 103	mpbird 256	. . . 4 ⊢ (𝜑 → (𝑋 ↾ 𝑆) ∈ (𝐵 ↑m 𝑆))
105		fveq1 6755	. . . . . . . 8 ⊢ (𝑓 = (𝑋 ↾ 𝑆) → (𝑓‘𝑖) = ((𝑋 ↾ 𝑆)‘𝑖))
106	105	ifeq1d 4475	. . . . . . 7 ⊢ (𝑓 = (𝑋 ↾ 𝑆) → if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 ) = if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 ))
107	106	mpteq2dv 5172	. . . . . 6 ⊢ (𝑓 = (𝑋 ↾ 𝑆) → (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 )))
108	107	eleq1d 2823	. . . . 5 ⊢ (𝑓 = (𝑋 ↾ 𝑆) → ((𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻 ↔ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 )) ∈ 𝐻))
109	108	elrab3 3618	. . . 4 ⊢ ((𝑋 ↾ 𝑆) ∈ (𝐵 ↑m 𝑆) → ((𝑋 ↾ 𝑆) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 )) ∈ 𝐻))
110	104, 109	syl 17	. . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑆) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 )) ∈ 𝐻))
111		fsuppssind.5	. . . . 5 ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)
112	7, 13, 1, 111	fsuppssindlem1 40203	. . . 4 ⊢ (𝜑 → 𝑋 = (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 )))
113	112	eleq1d 2823	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, ((𝑋 ↾ 𝑆)‘𝑖), 0 )) ∈ 𝐻))
114	110, 113	bitr4d 281	. 2 ⊢ (𝜑 → ((𝑋 ↾ 𝑆) ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑖 ∈ 𝐼 ↦ if(𝑖 ∈ 𝑆, (𝑓‘𝑖), 0 )) ∈ 𝐻} ↔ 𝑋 ∈ 𝐻))
115	102, 114	mpbid 231	1 ⊢ (𝜑 → 𝑋 ∈ 𝐻)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578   ↾ cres 5582   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509   supp csupp 7948   ↑_m cmap 8573   finSupp cfsupp 9058  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Grpcgrp 18492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495

This theorem is referenced by:  mhpind  40206