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Theorem gsumzres 20025

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 5290	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝑊) ∈ V)
4	2, 3	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝑊) ∈ V)
5		gsumzcl.0	. . . . . . . 8 ⊢ 0 = (0_g‘𝐺)
6	5	gsumz 18934	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∩ 𝑊) ∈ V) → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = 0 )
7	1, 4, 6	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = 0 )
8	5	gsumz 18934	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	1, 2, 8	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	7, 9	eqtr4d 2803	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
11	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
12		resres 5993	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴 ∩ 𝑊))
13		gsumzcl.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
14		ffn 6709	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
15		fnresdm 6658	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
16	13, 14, 15	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	reseq1d 5979	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ 𝑊))
18	12, 17	eqtr3id 2814	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
20	5	fvexi 6899	. . . . . . . . . 10 ⊢ 0 ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
22		ssid 3960	. . . . . . . . . 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 20024	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
25	24	reseq1d 5979	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)))
26		inss1 4189	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑊) ⊆ 𝐴
27		resmpt 6041	. . . . . . . 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 2816	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
30	19, 29	eqtr3d 2802	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ 𝑊) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
31	30	oveq2d 7435	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )))
32	24	oveq2d 7435	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
33	11, 31, 32	3eqtr4d 2810	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))
34	33	ex 418	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹)))
35		f1ofo 6832	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
36		forn 6799	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
38	37	ad2antll 742	. . . . . . . . . 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 486	. . . . . . . . . 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 3972	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 ⊆ 𝑊)
42		cores 6252	. . . . . . . . 9 ⊢ (ran 𝑓 ⊆ 𝑊 → ((𝐹 ↾ 𝑊) ∘ 𝑓) = (𝐹 ∘ 𝑓))
43	41, 42	syl 18	. . . . . . . 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 14065	. . . . . . 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 6887	. . . . . 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 2765	. . . . . . 7 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
49	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
50	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴 ∩ 𝑊) ∈ V)
51	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵)
52		fssres 6748	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ∩ 𝑊) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵)
53	51, 26, 52	sylancl 598	. . . . . . . 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 6691	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵 ↔ (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵))
55	54	biimpa 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵) → (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵)
56	53, 55	syldan 603	. . . . . . 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 6002	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝑊) ⊆ 𝐹
59	58	rnssi 5932	. . . . . . . . 9 ⊢ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹
60	48	cntzidss 19456	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹) → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊)))
61	57, 59, 60	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → ran (𝐹 ↾ 𝑊) ⊆ (𝑍‘ran (𝐹 ↾ 𝑊)))
62	61	adantr 486	. . . . . . 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 783	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
64		f1of1 6823	. . . . . . . . 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 742	. . . . . . . 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 8179	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
67	66, 13	fssdm 6729	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
68	67, 39	ssind 4193	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊))
69	68	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊))
70		f1ss 6785	. . . . . . . 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 596	. . . . . . 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 7232	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ V)
73		ressuppss 8185	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
74	72, 20, 73	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
75		sseq2 3964	. . . . . . . . . . 11 ⊢ (ran 𝑓 = (𝐹 supp 0 ) → (((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓 ↔ ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 )))
76	74, 75	imbitrrid 249	. . . . . . . . . 10 ⊢ (ran 𝑓 = (𝐹 supp 0 ) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓))
77	35, 36, 76	3syl 19	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝜑 → ((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓))
78	77	adantl 487	. . . . . . . 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 413	. . . . . . 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 2765	. . . . . . 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 20023	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
83	57	adantr 486	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
85		f1ss 6785	. . . . . . . 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 596	. . . . . . 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 3981	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
88		eqid 2765	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
89	46, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88	gsumval3 20023	. . . . . 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 2810	. . . . 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 462	. . . 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 1966	. . 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 459	. 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 9335	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
96	95	simprd 501	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
97		fz1f1o 15786	. . 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 19	. 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 874	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∃wex 1812 ∈ wcel 2146 Vcvv 3457 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 class class class wbr 5111 ↦ cmpt 5194 ran crn 5664 ↾ cres 5665 ∘ ccom 5667 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 supp csupp 8162 Fincfn 8949 finSupp cfsupp 9328 1c1 11118 ℕcn 12250 ...cfz 13553 seqcseq 14057 ♯chash 14386 Basecbs 17293 +_gcplusg 17334 0_gc0g 17516 Σ_g cgsu 17517 Mndcmnd 18826 Cntzccntz 19431

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-n0 12522 df-z 12609 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-0g 17518 df-gsum 17519 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-cntz 19433

This theorem is used by: gsumres 20029 gsumzsplit 20043 gsumpt 20078 dmdprdsplitlem 20155 dpjidcl 20176 mplcoe5 22243

W3C validator