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Theorem gsumzres 19821

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 5255	. . . . . . . 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 18744	. . . . . . 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 18744	. . . . . . 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 2769	. . . . 5 ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
12		resres 5940	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴 ∩ 𝑊))
13		gsumzcl.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
14		ffn 6651	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
15		fnresdm 6600	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
16	13, 14, 15	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
17	16	reseq1d 5926	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) ↾ 𝑊) = (𝐹 ↾ 𝑊))
18	12, 17	eqtr3id 2780	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝐹 ↾ 𝑊))
20	5	fvexi 6836	. . . . . . . . . 10 ⊢ 0 ∈ V
21	20	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
22		ssid 3952	. . . . . . . . . 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 19820	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
25	24	reseq1d 5926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = ((𝑘 ∈ 𝐴 ↦ 0 ) ↾ (𝐴 ∩ 𝑊)))
26		inss1 4184	. . . . . . . 8 ⊢ (𝐴 ∩ 𝑊) ⊆ 𝐴
27		resmpt 5985	. . . . . . . 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 2782	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴 ∩ 𝑊)) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
30	19, 29	eqtr3d 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ 𝑊) = (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 ))
31	30	oveq2d 7362	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g (𝑘 ∈ (𝐴 ∩ 𝑊) ↦ 0 )))
32	24	oveq2d 7362	. . . 4 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑘 ∈ 𝐴 ↦ 0 )))
33	11, 31, 32	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))
34	33	ex 412	. 2 ⊢ (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹)))
35		f1ofo 6770	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
36		forn 6738	. . . . . . . . . . . 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 729	. . . . . . . . . 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 480	. . . . . . . . . 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 3964	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 ⊆ 𝑊)
42		cores 6196	. . . . . . . . 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 13916	. . . . . . 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 6824	. . . . . 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 2731	. . . . . . 7 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
48		gsumzcl.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
49	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
50	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴 ∩ 𝑊) ∈ V)
51	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴⟶𝐵)
52		fssres 6689	. . . . . . . . 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 6633	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∩ 𝑊)):(𝐴 ∩ 𝑊)⟶𝐵 ↔ (𝐹 ↾ 𝑊):(𝐴 ∩ 𝑊)⟶𝐵))
55	54	biimpa 476	. . . . . . . 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 5949	. . . . . . . . . 10 ⊢ (𝐹 ↾ 𝑊) ⊆ 𝐹
59	58	rnssi 5879	. . . . . . . . 9 ⊢ ran (𝐹 ↾ 𝑊) ⊆ ran 𝐹
60	48	cntzidss 19252	. . . . . . . . 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 480	. . . . . . 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 770	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
64		f1of1 6762	. . . . . . . . 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 729	. . . . . . . 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 8107	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
67	66, 13	fssdm 6670	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
68	67, 39	ssind 4188	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊))
69	68	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴 ∩ 𝑊))
70		f1ss 6724	. . . . . . . 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 7161	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ V)
73		ressuppss 8113	. . . . . . . . . . . 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 3956	. . . . . . . . . . 11 ⊢ (ran 𝑓 = (𝐹 supp 0 ) → (((𝐹 ↾ 𝑊) supp 0 ) ⊆ ran 𝑓 ↔ ((𝐹 ↾ 𝑊) supp 0 ) ⊆ (𝐹 supp 0 )))
76	74, 75	imbitrrid 246	. . . . . . . . . 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 481	. . . . . . . 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 407	. . . . . . 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 2731	. . . . . . 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 19819	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴 ∈ 𝑉)
83	57	adantr 480	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
85		f1ss 6724	. . . . . . . 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 3973	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
88		eqid 2731	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
89	46, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88	gsumval3 19819	. . . . . 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 2776	. . . . 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 456	. . . 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 1934	. . 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 453	. 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 9252	. . . 4 ⊢ (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
96	95	simprd 495	. . 3 ⊢ (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
97		fz1f1o 15617	. . 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 860	1 ⊢ (𝜑 → (𝐺 Σ_g (𝐹 ↾ 𝑊)) = (𝐺 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∃wex 1780 ∈ wcel 2111 Vcvv 3436 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 class class class wbr 5089 ↦ cmpt 5170 ran crn 5615 ↾ cres 5616 ∘ ccom 5618 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 supp csupp 8090 Fincfn 8869 finSupp cfsupp 9245 1c1 11007 ℕcn 12125 ...cfz 13407 seqcseq 13908 ♯chash 14237 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Σ_g cgsu 17344 Mndcmnd 18642 Cntzccntz 19227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-0g 17345 df-gsum 17346 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-cntz 19229

This theorem is referenced by: gsumres 19825 gsumzsplit 19839 gsumpt 19874 dmdprdsplitlem 19951 dpjidcl 19972 mplcoe5 21975

W3C validator