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Theorem gsumzres 20039

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 5282	. . . . . . . 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 18948	. . . . . . 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 18948	. . . . . . 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 2798	. . . . 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 5985	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴 ∩ 𝑊))
13		gsumzcl.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
14		ffn 6703	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
15		fnresdm 6652	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
16	13, 14, 15	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	reseq1d 5971	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ 𝑊))
18	12, 17	eqtr3id 2809	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
20	5	fvexi 6893	. . . . . . . . . 10 ⊢ 0 ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
22		ssid 3953	. . . . . . . . . 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 20038	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
25	24	reseq1d 5971	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)))
26		inss1 4182	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑊) ⊆ 𝐴
27		resmpt 6033	. . . . . . . 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 2811	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
30	19, 29	eqtr3d 2797	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ 𝑊) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
31	30	oveq2d 7430	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )))
32	24	oveq2d 7430	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
33	11, 31, 32	3eqtr4d 2805	. . 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 6826	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
36		forn 6793	. . . . . . . . . . . 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 3965	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 ⊆ 𝑊)
42		cores 6245	. . . . . . . . 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 14076	. . . . . . 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 6881	. . . . . 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 2760	. . . . . . 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 6742	. . . . . . . . 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 6685	. . . . . . . . 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 5994	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝑊) ⊆ 𝐹
59	58	rnssi 5924	. . . . . . . . 9 ⊢ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹
60	48	cntzidss 19470	. . . . . . . . 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 6817	. . . . . . . . 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 8176	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
67	66, 13	fssdm 6723	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
68	67, 39	ssind 4186	. . . . . . . . 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 6779	. . . . . . . 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 7227	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ V)
73		ressuppss 8182	. . . . . . . . . . . 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 3957	. . . . . . . . . . 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 2760	. . . . . . 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 20037	. . . . . 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 6779	. . . . . . . 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 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 2760	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
89	46, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88	gsumval3 20037	. . . . . 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 2805	. . . . 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 9341	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
96	95	simprd 501	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
97		fz1f1o 15799	. . 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 2145 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7414 supp csupp 8159 Fincfn 8955 finSupp cfsupp 9334 1c1 11128 ℕcn 12260 ...cfz 13564 seqcseq 14068 ♯chash 14397 Basecbs 17304 +_gcplusg 17345 0_gc0g 17527 Σ_g cgsu 17528 Mndcmnd 18839 Cntzccntz 19445

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13565 df-fzo 13713 df-seq 14069 df-hash 14398 df-0g 17529 df-gsum 17530 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-cntz 19447

This theorem is used by: gsumres 20043 gsumzsplit 20057 gsumpt 20092 dmdprdsplitlem 20169 dpjidcl 20190 mplcoe5 22259

W3C validator