Theorem gsumzsplit 20121

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 6891	. . . . 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 9375	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) finSupp 0 )
12	7, 6, 9, 10	fsuppmptif 9375	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) finSupp 0 )
13	1	submacs 19003	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
14		acsmre 17806	. . . . 5 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
15	5, 13, 14	3syl 19	. . . 4 ⊢ (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
16	7	frnd 6710	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
17		eqid 2761	. . . . 5 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
18	17	mrccl 17765	. . . 4 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
19	15, 16, 18	syl2anc 596	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
20		gsumzsplit.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
21		eqid 2761	. . . . . 6 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
22	4, 17, 21	cntzspan 20038	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
23	5, 20, 22	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
24	21, 4	submcmn2 20033	. . . . 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 18	. . . 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 235	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
27	15, 17, 16	mrcssidd 17779	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
28	27	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
29	7	ffnd 6702	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
30		fnfvelrn 7072	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
31	29, 30	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
32	28, 31	sseldd 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
33	2	subm0cl 18986	. . . . . . 7 ⊢ (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
34	19, 33	syl 18	. . . . . 6 ⊢ (𝜑 → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
35	34	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
36	32, 35	ifcld 4529	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
37	36	fmpttd 7107	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
38	32, 35	ifcld 4529	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
39	38	fmpttd 7107	. . 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 20116	. 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 6945	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
42		iftrue 4488	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐶 → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
43	42	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
44		gsumzsplit.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
45		noel 4284	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑘 ∈ ∅
46		eleq2 2850	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∩ 𝐷) = ∅ → (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ 𝑘 ∈ ∅))
47	45, 46	mtbiri 330	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∩ 𝐷) = ∅ → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
48	44, 47	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
49	48	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
50		elin 3915	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
51	49, 50	sylnib 331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
52		imnan 405	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷) ↔ ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
53	51, 52	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷))
54	53	imp 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ¬ 𝑘 ∈ 𝐷)
55	54	iffalsed 4493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
56	43, 55	oveq12d 7430	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ((𝐹‘𝑘) + 0 ))
57	7	ffvelcdmda 7076	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
58	1, 3, 2	mndrid 18925	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
59	5, 57, 58	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
60	59	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
61	56, 60	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
62	53	con2d 135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐷 → ¬ 𝑘 ∈ 𝐶))
63	62	imp 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ¬ 𝑘 ∈ 𝐶)
64	63	iffalsed 4493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
65		iftrue 4488	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
66	65	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
67	64, 66	oveq12d 7430	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ( 0 + (𝐹‘𝑘)))
68	1, 3, 2	mndlid 18924	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
69	5, 57, 68	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
70	69	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
71	67, 70	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
72		gsumzsplit.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))
73	72	eleq2d 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ 𝑘 ∈ (𝐶 ∪ 𝐷)))
74		elun 4100	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐶 ∪ 𝐷) ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
75	73, 74	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷)))
76	75	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
77	61, 71, 76	mpjaodan 973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘))
78	77	mpteq2dva 5198	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
79	41, 78	eqtr4d 2799	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
80	1, 2	mndidcl 18919	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
81	5, 80	syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝐵)
82	81	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
83	57, 82	ifcld 4529	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ 𝐵)
84	57, 82	ifcld 4529	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ 𝐵)
85		eqidd 2762	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
86		eqidd 2762	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
87	6, 83, 84, 85, 86	offval2 7702	. . . 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 ))))
88	79, 87	eqtr4d 2799	. . 3 ⊢ (𝜑 → 𝐹 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘f + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
89	88	oveq2d 7428	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘f + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))))
90	41	reseq1d 5969	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
91		ssun1 4124	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
92	91, 72	sseqtrrid 3974	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
93	42	mpteq2ia 5200	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘))
94		resmpt 6031	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
95		resmpt 6031	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘)))
96	93, 94, 95	3eqtr4a 2822	. . . . . . 7 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
97	92, 96	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
98	90, 97	eqtr4d 2799	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶))
99	98	oveq2d 7428	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)))
100	83	fmpttd 7107	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
101	37	frnd 6710	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
102	4	cntzidss 19534	. . . . . 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 ))))
103	26, 101, 102	syl2anc 596	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
104		eldifn 4079	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → ¬ 𝑘 ∈ 𝐶)
105	104	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → ¬ 𝑘 ∈ 𝐶)
106	105	iffalsed 4493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
107	106, 6	suppss2 8201	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐶)
108	1, 2, 4, 5, 6, 100, 103, 107, 11	gsumzres 20103	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
109	99, 108	eqtrd 2796	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
110	41	reseq1d 5969	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
111		ssun2 4125	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
112	111, 72	sseqtrrid 3974	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
113	65	mpteq2ia 5200	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘))
114		resmpt 6031	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
115		resmpt 6031	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘)))
116	113, 114, 115	3eqtr4a 2822	. . . . . . 7 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
117	112, 116	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
118	110, 117	eqtr4d 2799	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷))
119	118	oveq2d 7428	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)))
120	84	fmpttd 7107	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
121	39	frnd 6710	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
122	4	cntzidss 19534	. . . . . 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 ))))
123	26, 121, 122	syl2anc 596	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
124		eldifn 4079	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐷) → ¬ 𝑘 ∈ 𝐷)
125	124	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → ¬ 𝑘 ∈ 𝐷)
126	125	iffalsed 4493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
127	126, 6	suppss2 8201	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐷)
128	1, 2, 4, 5, 6, 120, 123, 127, 12	gsumzres 20103	. . . 4 ⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
129	119, 128	eqtrd 2796	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
130	109, 129	oveq12d 7430	. 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 )))))
131	40, 89, 130	3eqtr4d 2806	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652   ↾ cres 5653   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ∘_f cof 7680   finSupp cfsupp 9337  Basecbs 17367   ↾_s cress 17388  +_gcplusg 17408  0_gc0g 17590   Σ_g cgsu 17591  Moorecmre 17732  mrClscmrc 17733  ACScacs 17735  Mndcmnd 18903  SubMndcsubmnd 18957  Cntzccntz 19509  CMndccmn 19974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-seq 14125  df-hash 14455  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-0g 17592  df-gsum 17593  df-mre 17736  df-mrc 17737  df-acs 17739  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-cntz 19511  df-cmn 19976

This theorem is used by:  gsumsplit  20122  gsumzunsnd  20150  dpjidcl  20254