Theorem gsumzsplit 18641

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	2	fvexi 6426	. . . . 5 ⊢ 0 ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
10		gsumzsplit.w	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 )
11	7, 6, 9, 10	fsuppmptif 8548	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) finSupp 0 )
12	7, 6, 9, 10	fsuppmptif 8548	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) finSupp 0 )
13	1	submacs 17679	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
14		acsmre 16626	. . . . 5 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
15	5, 13, 14	3syl 18	. . . 4 ⊢ (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
16	7	frnd 6264	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
17		eqid 2800	. . . . 5 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
18	17	mrccl 16585	. . . 4 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
19	15, 16, 18	syl2anc 580	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
20		gsumzsplit.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
21		eqid 2800	. . . . . 6 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
22	4, 17, 21	cntzspan 18561	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
23	5, 20, 22	syl2anc 580	. . . 4 ⊢ (𝜑 → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
24	21, 4	submcmn2 18558	. . . . 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))))
25	19, 24	syl 17	. . . 4 ⊢ (𝜑 → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
26	23, 25	mpbid 224	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
27	15, 17, 16	mrcssidd 16599	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
28	27	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
29	7	ffnd 6258	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
30		fnfvelrn 6583	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
31	29, 30	sylan 576	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
32	28, 31	sseldd 3800	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
33	2	subm0cl 17666	. . . . . . 7 ⊢ (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
34	19, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
35	34	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
36	32, 35	ifcld 4323	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
37	36	fmpttd 6612	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
38	32, 35	ifcld 4323	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
39	38	fmpttd 6612	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
40	1, 2, 3, 4, 5, 6, 11, 12, 19, 26, 37, 39	gsumzadd 18636	. 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 )))))
41	7	feqmptd 6475	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
42		iftrue 4284	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐶 → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
43	42	adantl 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
44		gsumzsplit.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
45		noel 4120	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑘 ∈ ∅
46		eleq2 2868	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∩ 𝐷) = ∅ → (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ 𝑘 ∈ ∅))
47	45, 46	mtbiri 319	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∩ 𝐷) = ∅ → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
48	44, 47	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
49	48	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
50		elin 3995	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
51	49, 50	sylnib 320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
52		imnan 389	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷) ↔ ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
53	51, 52	sylibr 226	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷))
54	53	imp 396	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ¬ 𝑘 ∈ 𝐷)
55	54	iffalsed 4289	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
56	43, 55	oveq12d 6897	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ((𝐹‘𝑘) + 0 ))
57	7	ffvelrnda 6586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
58	1, 3, 2	mndrid 17626	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
59	5, 58	sylan 576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
60	57, 59	syldan 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
61	60	adantr 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
62	56, 61	eqtrd 2834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
63	53	con2d 132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐷 → ¬ 𝑘 ∈ 𝐶))
64	63	imp 396	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ¬ 𝑘 ∈ 𝐶)
65	64	iffalsed 4289	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
66		iftrue 4284	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
67	66	adantl 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
68	65, 67	oveq12d 6897	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ( 0 + (𝐹‘𝑘)))
69	1, 3, 2	mndlid 17625	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
70	5, 69	sylan 576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
71	57, 70	syldan 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
72	71	adantr 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
73	68, 72	eqtrd 2834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
74		gsumzsplit.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))
75	74	eleq2d 2865	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ 𝑘 ∈ (𝐶 ∪ 𝐷)))
76		elun 3952	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐶 ∪ 𝐷) ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
77	75, 76	syl6bb 279	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷)))
78	77	biimpa 469	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
79	62, 73, 78	mpjaodan 982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
80	79	mpteq2dva 4938	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
81	41, 80	eqtr4d 2837	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
82	1, 2	mndidcl 17622	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
83	5, 82	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝐵)
84	83	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
85	57, 84	ifcld 4323	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ 𝐵)
86	57, 84	ifcld 4323	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ 𝐵)
87		eqidd 2801	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
88		eqidd 2801	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
89	6, 85, 86, 87, 88	offval2 7149	. . . 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 ))))
90	81, 89	eqtr4d 2837	. . 3 ⊢ (𝜑 → 𝐹 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
91	90	oveq2d 6895	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))))
92	41	reseq1d 5600	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
93		ssun1 3975	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
94	93, 74	syl5sseqr 3851	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
95	42	mpteq2ia 4934	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘))
96		resmpt 5662	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
97		resmpt 5662	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘)))
98	95, 96, 97	3eqtr4a 2860	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
99	94, 98	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
100	92, 99	eqtr4d 2837	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶))
101	100	oveq2d 6895	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)))
102	85	fmpttd 6612	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
103	37	frnd 6264	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
104	4	cntzidss 18081	. . . . . 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 ))))
105	26, 103, 104	syl2anc 580	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
106		eldifn 3932	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → ¬ 𝑘 ∈ 𝐶)
107	106	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → ¬ 𝑘 ∈ 𝐶)
108	107	iffalsed 4289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
109	108, 6	suppss2 7568	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐶)
110	1, 2, 4, 5, 6, 102, 105, 109, 11	gsumzres 18624	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
111	101, 110	eqtrd 2834	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
112	41	reseq1d 5600	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
113		ssun2 3976	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
114	113, 74	syl5sseqr 3851	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
115	66	mpteq2ia 4934	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘))
116		resmpt 5662	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
117		resmpt 5662	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘)))
118	115, 116, 117	3eqtr4a 2860	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
119	114, 118	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
120	112, 119	eqtr4d 2837	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷))
121	120	oveq2d 6895	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)))
122	86	fmpttd 6612	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
123	39	frnd 6264	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
124	4	cntzidss 18081	. . . . . 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 ))))
125	26, 123, 124	syl2anc 580	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
126		eldifn 3932	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐷) → ¬ 𝑘 ∈ 𝐷)
127	126	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → ¬ 𝑘 ∈ 𝐷)
128	127	iffalsed 4289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
129	128, 6	suppss2 7568	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐷)
130	1, 2, 4, 5, 6, 122, 125, 129, 12	gsumzres 18624	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
131	121, 130	eqtrd 2834	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
132	111, 131	oveq12d 6897	. 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 )))))
133	40, 91, 132	3eqtr4d 2844	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 385   ∨ wo 874   = wceq 1653   ∈ wcel 2157  Vcvv 3386   ∖ cdif 3767   ∪ cun 3768   ∩ cin 3769   ⊆ wss 3770  ∅c0 4116  ifcif 4278   class class class wbr 4844   ↦ cmpt 4923  ran crn 5314   ↾ cres 5315   Fn wfn 6097  ⟶wf 6098  ‘cfv 6102  (class class class)co 6879   ∘_𝑓 cof 7130   finSupp cfsupp 8518  Basecbs 16183   ↾_s cress 16184  +_gcplusg 16266  0_gc0g 16414   Σ_g cgsu 16415  Moorecmre 16556  mrClscmrc 16557  ACScacs 16559  Mndcmnd 17608  SubMndcsubmnd 17648  Cntzccntz 18059  CMndccmn 18507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-cnex 10281  ax-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301  ax-pre-mulgt0 10302

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-iin 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-se 5273  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-isom 6111  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-of 7132  df-om 7301  df-1st 7402  df-2nd 7403  df-supp 7534  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-oadd 7804  df-er 7983  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-fsupp 8519  df-oi 8658  df-card 9052  df-pnf 10366  df-mnf 10367  df-xr 10368  df-ltxr 10369  df-le 10370  df-sub 10559  df-neg 10560  df-nn 11314  df-2 11375  df-n0 11580  df-z 11666  df-uz 11930  df-fz 12580  df-fzo 12720  df-seq 13055  df-hash 13370  df-ndx 16186  df-slot 16187  df-base 16189  df-sets 16190  df-ress 16191  df-plusg 16279  df-0g 16416  df-gsum 16417  df-mre 16560  df-mrc 16561  df-acs 16563  df-mgm 17556  df-sgrp 17598  df-mnd 17609  df-submnd 17650  df-cntz 18061  df-cmn 18509

This theorem is referenced by:  gsumsplit  18642  gsumzunsnd  18669  dpjidcl  18772