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Theorem gsumzres 19510

Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)

**Hypotheses**
Ref	Expression
gsumzcl.b	⊢ 𝐵 = (Base‘𝐺)
gsumzcl.0	⊢ 0 = (0_g‘𝐺)
gsumzcl.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzcl.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzcl.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzcl.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzcl.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzres.s	⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
gsumzres.w	⊢ (𝜑 → 𝐹 finSupp 0 )

**Assertion**
Ref	Expression
gsumzres	⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))

Proof of Theorem gsumzres Dummy variables 𝑓 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzcl.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		gsumzcl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		inex1g 5243	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝑊) ∈ V)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝑊) ∈ V)
5		gsumzcl.0	. . . . . . . 8 ⊢ 0 = (0_g‘𝐺)
6	5	gsumz 18474	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∩ 𝑊) ∈ V) → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = 0 )
7	1, 4, 6	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = 0 )
8	5	gsumz 18474	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	1, 2, 8	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	7, 9	eqtr4d 2781	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
11	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
12		resres 5904	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴 ∩ 𝑊))
13		gsumzcl.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
14		ffn 6600	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
15		fnresdm 6551	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
16	13, 14, 15	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	reseq1d 5890	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ 𝑊))
18	12, 17	eqtr3id 2792	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
19	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
20	5	fvexi 6788	. . . . . . . . . 10 ⊢ 0 ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
22		ssid 3943	. . . . . . . . . 10 ⊢ (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
23	22	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
24	13, 2, 21, 23	gsumcllem 19509	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
25	24	reseq1d 5890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)))
26		inss1 4162	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑊) ⊆ 𝐴
27		resmpt 5945	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝑊) ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )
29	25, 28	eqtrdi 2794	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
30	19, 29	eqtr3d 2780	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ 𝑊) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
31	30	oveq2d 7291	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )))
32	24	oveq2d 7291	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
33	11, 31, 32	3eqtr4d 2788	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))
34	33	ex 413	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹)))
35		f1ofo 6723	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
36		forn 6691	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
38	37	ad2antll 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 ))
39		gsumzres.s	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
40	39	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝑊)
41	38, 40	eqsstrd 3959	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 ⊆ 𝑊)
42		cores 6153	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ 𝑊 → ((𝐹 ↾ 𝑊) ∘ 𝑓) = (𝐹 ∘ 𝑓))
43	41, 42	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ↾ 𝑊) ∘ 𝑓) = (𝐹 ∘ 𝑓))
44	43	seqeq3d 13729	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → seq1((+_g‘𝐺), ((𝐹 ↾ 𝑊) ∘ 𝑓)) = seq1((+_g‘𝐺), (𝐹 ∘ 𝑓)))
45	44	fveq1d 6776	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (seq1((+_g‘𝐺), ((𝐹 ↾ 𝑊) ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))))
46		gsumzcl.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
47		eqid 2738	. . . . . . 7 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
49	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
50	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴 ∩ 𝑊) ∈ V)
51	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵)
52		fssres 6640	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑊) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵)
53	51, 26, 52	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵)
54	18	feq1d 6585	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵 ↔ (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵))
55	54	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵) → (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵)
56	53, 55	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵)
57		gsumzcl.c	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
58		resss 5916	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝑊) ⊆ 𝐹
59	58	rnssi 5849	. . . . . . . . 9 ⊢ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹
60	48	cntzidss 18944	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹) → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊)))
61	57, 59, 60	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊)))
62	61	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊)))
63		simprl 768	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
64		f1of1 6715	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
65	64	ad2antll 726	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
66		suppssdm 7993	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
67	66, 13	fssdm 6620	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
68	67, 39	ssind 4166	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊))
69	68	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊))
70		f1ss 6676	. . . . . . . 8 ⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊)) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐴 ∩ 𝑊))
71	65, 69, 70	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐴 ∩ 𝑊))
72	13, 2	fexd 7103	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ V)
73		ressuppss 7999	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
74	72, 20, 73	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
75		sseq2 3947	. . . . . . . . . . 11 ⊢ (ran 𝑓 = (𝐹 supp 0 ) → (((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓 ↔ ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 )))
76	74, 75	syl5ibr 245	. . . . . . . . . 10 ⊢ (ran 𝑓 = (𝐹 supp 0 ) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓))
77	35, 36, 76	3syl 18	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓))
78	77	adantl 482	. . . . . . . 8 ⊢ (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓))
79	78	impcom 408	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓)
80		eqid 2738	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑊) ∘ 𝑓) supp 0 ) = (((𝐹 ↾ 𝑊) ∘ 𝑓) supp 0 )
81	46, 5, 47, 48, 49, 50, 56, 62, 63, 71, 79, 80	gsumval3 19508	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (seq1((+_g‘𝐺), ((𝐹 ↾ 𝑊) ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))))
82	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
83	57	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
84	67	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
85		f1ss 6676	. . . . . . . 8 ⊢ ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴)
86	65, 84, 85	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→𝐴)
87	22, 38	sseqtrrid 3974	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
88		eqid 2738	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
89	46, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88	gsumval3 19508	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σ_g 𝐹) = (seq1((+_g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))))
90	45, 81, 89	3eqtr4d 2788	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))
91	90	expr 457	. . . 4 ⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹)))
92	91	exlimdv 1936	. . 3 ⊢ ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹)))
93	92	expimpd 454	. 2 ⊢ (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹)))
94		gsumzres.w	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
95		fsuppimp 9134	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
96	95	simprd 496	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
97		fz1f1o 15422	. . 3 ⊢ ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) = ∅ ∨ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
98	94, 96, 97	3syl 18	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ ∨ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
99	34, 93, 98	mpjaod 857	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 ↾ cres 5591 ∘ ccom 5593 Fun wfun 6427 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 supp csupp 7977 Fincfn 8733 finSupp cfsupp 9128 1c1 10872 ℕcn 11973 ...cfz 13239 seqcseq 13721 ♯chash 14044 Basecbs 16912 +_gcplusg 16962 0_gc0g 17150 Σ_g cgsu 17151 Mndcmnd 18385 Cntzccntz 18921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-cntz 18923

This theorem is referenced by: gsumres 19514 gsumzsplit 19528 gsumpt 19563 dmdprdsplitlem 19640 dpjidcl 19661 mplcoe5 21241

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