Theorem gsumzsplit 19997

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 6896	. . . . 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 9359	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) finSupp 0 )
12	7, 6, 9, 10	fsuppmptif 9359	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) finSupp 0 )
13	1	submacs 18886	. . . . 5 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
14		acsmre 17708	. . . . 5 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
15	5, 13, 14	3syl 19	. . . 4 ⊢ (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
16	7	frnd 6715	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
17		eqid 2769	. . . . 5 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
18	17	mrccl 17667	. . . 4 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
19	15, 16, 18	syl2anc 595	. . 3 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
20		gsumzsplit.c	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
21		eqid 2769	. . . . . 6 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
22	4, 17, 21	cntzspan 19914	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
23	5, 20, 22	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
24	21, 4	submcmn2 19909	. . . . 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 17681	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
28	27	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
29	7	ffnd 6707	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
30		fnfvelrn 7076	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
31	29, 30	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹)
32	28, 31	sseldd 3946	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
33	2	subm0cl 18869	. . . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
36	32, 35	ifcld 4539	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
37	36	fmpttd 7111	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
38	32, 35	ifcld 4539	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
39	38	fmpttd 7111	. . 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 19992	. 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 6950	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
42		iftrue 4498	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐶 → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
43	42	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
44		gsumzsplit.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)
45		noel 4299	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑘 ∈ ∅
46		eleq2 2858	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷))
50		elin 3929	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
51	49, 50	sylnib 331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷))
52		imnan 404	. . . . . . . . . . . 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 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ¬ 𝑘 ∈ 𝐷)
55	54	iffalsed 4503	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
56	43, 55	oveq12d 7429	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ((𝐹‘𝑘) + 0 ))
57	7	ffvelcdmda 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵)
58	1, 3, 2	mndrid 18813	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
59	5, 57, 58	syl2an2r 697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
60	59	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘))
61	56, 60	eqtrd 2804	. . . . . . 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 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ¬ 𝑘 ∈ 𝐶)
64	63	iffalsed 4503	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
65		iftrue 4498	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐷 → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
66	65	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘))
67	64, 66	oveq12d 7429	. . . . . . . 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 18812	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
69	5, 57, 68	syl2an2r 697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
70	69	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘))
71	67, 70	eqtrd 2804	. . . . . . 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 2855	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ 𝑘 ∈ (𝐶 ∪ 𝐷)))
74		elun 4115	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐶 ∪ 𝐷) ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))
75	73, 74	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷)))
76	75	biimpa 481	. . . . . . 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 5208	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
79	41, 78	eqtr4d 2807	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
80	1, 2	mndidcl 18807	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
81	5, 80	syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝐵)
82	81	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵)
83	57, 82	ifcld 4539	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ 𝐵)
84	57, 82	ifcld 4539	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ 𝐵)
85		eqidd 2770	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
86		eqidd 2770	. . . . 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 7695	. . . 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 2807	. . 3 ⊢ (𝜑 → 𝐹 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ∘f + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
89	88	oveq2d 7427	. 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 5978	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶))
91		ssun1 4139	. . . . . . . 8 ⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷)
92	91, 72	sseqtrrid 3988	. . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
93	42	mpteq2ia 5210	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘))
94		resmpt 6040	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))
95		resmpt 6040	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘)))
96	93, 94, 95	3eqtr4a 2830	. . . . . . 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 2807	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶))
99	98	oveq2d 7427	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)))
100	83	fmpttd 7111	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
101	37	frnd 6715	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
102	4	cntzidss 19410	. . . . . 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 595	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
104		eldifn 4094	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → ¬ 𝑘 ∈ 𝐶)
105	104	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → ¬ 𝑘 ∈ 𝐶)
106	105	iffalsed 4503	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 )
107	106, 6	suppss2 8196	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐶)
108	1, 2, 4, 5, 6, 100, 103, 107, 11	gsumzres 19979	. . . 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 2804	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))))
110	41	reseq1d 5978	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷))
111		ssun2 4140	. . . . . . . 8 ⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷)
112	111, 72	sseqtrrid 3988	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
113	65	mpteq2ia 5210	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘))
114		resmpt 6040	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))
115		resmpt 6040	. . . . . . . 8 ⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘)))
116	113, 114, 115	3eqtr4a 2830	. . . . . . 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 2807	. . . . 5 ⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷))
119	118	oveq2d 7427	. . . 4 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)))
120	84	fmpttd 7111	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶𝐵)
121	39	frnd 6715	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
122	4	cntzidss 19410	. . . . . 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 595	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
124		eldifn 4094	. . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐷) → ¬ 𝑘 ∈ 𝐷)
125	124	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → ¬ 𝑘 ∈ 𝐷)
126	125	iffalsed 4503	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 )
127	126, 6	suppss2 8196	. . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐷)
128	1, 2, 4, 5, 6, 120, 123, 127, 12	gsumzres 19979	. . . 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 2804	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))
130	109, 129	oveq12d 7429	. 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 2814	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ifcif 4492   class class class wbr 5113   ↦ cmpt 5196  ran crn 5663   ↾ cres 5664   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∘_f cof 7673   finSupp cfsupp 9321  Basecbs 17269   ↾_s cress 17290  +_gcplusg 17310  0_gc0g 17492   Σ_g cgsu 17493  Moorecmre 17634  mrClscmrc 17635  ACScacs 17637  Mndcmnd 18792  SubMndcsubmnd 18840  Cntzccntz 19385  CMndccmn 19850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-0g 17494  df-gsum 17495  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-submnd 18842  df-cntz 19387  df-cmn 19852

This theorem is referenced by:  gsumsplit  19998  gsumzunsnd  20026  dpjidcl  20130