Theorem gsumzsplit 18327

Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)

Hypotheses

Ref	Expression
gsumzsplit.b	⊢ 𝐵 = (Base‘𝐺)
gsumzsplit.0	⊢ 0 = (0g‘𝐺)
gsumzsplit.p	⊢ + = (+g‘𝐺)
gsumzsplit.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzsplit.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzsplit.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzsplit.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzsplit.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzsplit.w	⊢ (𝜑 → 𝐹 finSupp 0 )
gsumzsplit.i	⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
gsumzsplit.u	⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))

Assertion

Ref	Expression
gsumzsplit	⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))

Proof of Theorem gsumzsplit

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzsplit.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsumzsplit.0	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsumzsplit.p	. . 3 ⊢ + = (+g‘𝐺)
4		gsumzsplit.z	. . 3 ⊢ 𝑍 = (Cntz‘𝐺)
5		gsumzsplit.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
6		gsumzsplit.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		gsumzsplit.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		fvex 6201	. . . . . 6 ⊢ (0g‘𝐺) ∈ V
9	2, 8	eqeltri 2697	. . . . 5 ⊢ 0 ∈ V
10	9	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
11		gsumzsplit.w	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 )
12	7, 6, 10, 11	fsuppmptif 8305	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) finSupp 0 )
13	7, 6, 10, 11	fsuppmptif 8305	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) finSupp 0 )
14	1	submacs 17365	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
15		acsmre 16313	. . . . 5 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
16	5, 14, 15	3syl 18	. . . 4 ⊢ (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
17		frn 6053	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
18	7, 17	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
19		eqid 2622	. . . . 5 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
20	19	mrccl 16271	. . . 4 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
21	16, 18, 20	syl2anc 693	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
22		gsumzsplit.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
23		eqid 2622	. . . . . 6 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
24	4, 19, 23	cntzspan 18247	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
25	5, 22, 24	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
26	23, 4	submcmn2 18244	. . . . 5 ⊢ (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
27	21, 26	syl 17	. . . 4 ⊢ (𝜑 → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
28	25, 27	mpbid 222	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
29	16, 19, 18	mrcssidd 16285	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
30	29	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
31		ffn 6045	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
32	7, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
33		fnfvelrn 6356	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
34	32, 33	sylan 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
35	30, 34	sseldd 3604	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
36	2	subm0cl 17352	. . . . . . 7 ⊢ (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
37	21, 36	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
39	35, 38	ifcld 4131	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
40		eqid 2622	. . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))
41	39, 40	fmptd 6385	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
42	35, 38	ifcld 4131	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
43		eqid 2622	. . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))
44	42, 43	fmptd 6385	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
45	1, 2, 3, 4, 5, 6, 12, 13, 21, 28, 41, 44	gsumzadd 18322	. 2 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))) + (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))))
46	7	feqmptd 6249	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
47		iftrue 4092	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐶 → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
48	47	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
49		gsumzsplit.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
50		noel 3919	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑘 ∈ ∅
51		eleq2 2690	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∩ 𝐷) = ∅ → (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ 𝑘 ∈ ∅))
52	50, 51	mtbiri 317	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∩ 𝐷) = ∅ → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
53	49, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
54	53	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
55		elin 3796	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
56	54, 55	sylnib 318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
57		imnan 438	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷) ↔ ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
58	56, 57	sylibr 224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷))
59	58	imp 445	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ¬ 𝑘 ∈ 𝐷)
60	59	iffalsed 4097	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
61	48, 60	oveq12d 6668	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ((𝐹‘𝑘) + 0 ))
62	7	ffvelrnda 6359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
63	1, 3, 2	mndrid 17312	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
64	5, 63	sylan 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
65	62, 64	syldan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
66	65	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
67	61, 66	eqtrd 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
68	58	con2d 129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐷 → ¬ 𝑘 ∈ 𝐶))
69	68	imp 445	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ¬ 𝑘 ∈ 𝐶)
70	69	iffalsed 4097	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
71		iftrue 4092	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
72	71	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
73	70, 72	oveq12d 6668	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ( 0 + (𝐹‘𝑘)))
74	1, 3, 2	mndlid 17311	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
75	5, 74	sylan 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
76	62, 75	syldan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
77	76	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
78	73, 77	eqtrd 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
79		gsumzsplit.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))
80	79	eleq2d 2687	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ 𝑘 ∈ (𝐶 ∪ 𝐷)))
81		elun 3753	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐶 ∪ 𝐷) ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
82	80, 81	syl6bb 276	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷)))
83	82	biimpa 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
84	67, 78, 83	mpjaodan 827	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
85	84	mpteq2dva 4744	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
86	46, 85	eqtr4d 2659	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
87	1, 2	mndidcl 17308	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
88	5, 87	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝐵)
89	88	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
90	62, 89	ifcld 4131	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ 𝐵)
91	62, 89	ifcld 4131	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ 𝐵)
92		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
93		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
94	6, 90, 91, 92, 93	offval2 6914	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
95	86, 94	eqtr4d 2659	. . 3 ⊢ (𝜑 → 𝐹 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
96	95	oveq2d 6666	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))))
97	46	reseq1d 5395	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
98		ssun1 3776	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
99	98, 79	syl5sseqr 3654	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
100	47	mpteq2ia 4740	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘))
101		resmpt 5449	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
102		resmpt 5449	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘)))
103	100, 101, 102	3eqtr4a 2682	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
104	99, 103	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
105	97, 104	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶))
106	105	oveq2d 6666	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)))
107	90, 40	fmptd 6385	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
108		frn 6053	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
109	41, 108	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
110	4	cntzidss 17770	. . . . . 6 ⊢ ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
111	28, 109, 110	syl2anc 693	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
112		eldifn 3733	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → ¬ 𝑘 ∈ 𝐶)
113	112	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → ¬ 𝑘 ∈ 𝐶)
114	113	iffalsed 4097	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
115	114, 6	suppss2 7329	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐶)
116	1, 2, 4, 5, 6, 107, 111, 115, 12	gsumzres 18310	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
117	106, 116	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
118	46	reseq1d 5395	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
119		ssun2 3777	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
120	119, 79	syl5sseqr 3654	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
121	71	mpteq2ia 4740	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘))
122		resmpt 5449	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
123		resmpt 5449	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘)))
124	121, 122, 123	3eqtr4a 2682	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
125	120, 124	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
126	118, 125	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷))
127	126	oveq2d 6666	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)))
128	91, 43	fmptd 6385	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
129		frn 6053	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
130	44, 129	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
131	4	cntzidss 17770	. . . . . 6 ⊢ ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
132	28, 130, 131	syl2anc 693	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
133		eldifn 3733	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐷) → ¬ 𝑘 ∈ 𝐷)
134	133	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → ¬ 𝑘 ∈ 𝐷)
135	134	iffalsed 4097	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
136	135, 6	suppss2 7329	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐷)
137	1, 2, 4, 5, 6, 128, 132, 136, 13	gsumzres 18310	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
138	127, 137	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
139	117, 138	oveq12d 6668	. 2 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))) + (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))))
140	45, 96, 139	3eqtr4d 2666	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115   ↾ cres 5116   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   finSupp cfsupp 8275  Basecbs 15857   ↾_s cress 15858  +_gcplusg 15941  0_gc0g 16100   Σ_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245  Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsumsplit  18328  gsumzunsnd  18355  dpjidcl  18457