Theorem gfsump1 16689

Description: Splitting off one element from a finite group sum. This would typically used in a proof by induction. (Contributed by Jim Kingdon, 3-Apr-2026.)

Hypotheses

Ref	Expression
gfsump1.b	⊢ 𝐵 = (Base‘𝐺)
gfsump1.p	⊢ + = (+g‘𝐺)
gfsump1.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gfsump1.f	⊢ (𝜑 → 𝐹:(𝑌 ∪ {𝑍})⟶𝐵)
gfsump1.fi	⊢ (𝜑 → 𝑌 ∈ Fin)
gfsump1.zv	⊢ (𝜑 → 𝑍 ∈ 𝑉)
gfsump1.z	⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)

Assertion

Ref	Expression
gfsump1	⊢ (𝜑 → (𝐺 Σgf 𝐹) = ((𝐺 Σgf (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))

Proof of Theorem gfsump1

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gfsump1.fi	. . 3 ⊢ (𝜑 → 𝑌 ∈ Fin)
2		fzf1o 11937	. . 3 ⊢ (𝑌 ∈ Fin → ∃ℎ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌)
3	1, 2	syl 14	. 2 ⊢ (𝜑 → ∃ℎ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌)
4		gfsump1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5		gfsump1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
6	5	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝐺 ∈ CMnd)
7		gfsump1.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝑌 ∪ {𝑍})⟶𝐵)
8	7	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝐹:(𝑌 ∪ {𝑍})⟶𝐵)
9		gfsump1.zv	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
10		gfsump1.z	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
11		unsnfi 7111	. . . . . . 7 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝑌) → (𝑌 ∪ {𝑍}) ∈ Fin)
12	1, 9, 10, 11	syl3anc 1273	. . . . . 6 ⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin)
13	12	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝑌 ∪ {𝑍}) ∈ Fin)
14		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌)
15		hashcl 11043	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ0)
16	1, 15	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ0)
17		peano2nn0 9442	. . . . . . . . . 10 ⊢ ((♯‘𝑌) ∈ ℕ0 → ((♯‘𝑌) + 1) ∈ ℕ0)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ ℕ0)
19		f1osng 5626	. . . . . . . . 9 ⊢ ((((♯‘𝑌) + 1) ∈ ℕ0 ∧ 𝑍 ∈ 𝑉) → {⟨((♯‘𝑌) + 1), 𝑍⟩}:{((♯‘𝑌) + 1)}–1-1-onto→{𝑍})
20	18, 9, 19	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → {⟨((♯‘𝑌) + 1), 𝑍⟩}:{((♯‘𝑌) + 1)}–1-1-onto→{𝑍})
21	20	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → {⟨((♯‘𝑌) + 1), 𝑍⟩}:{((♯‘𝑌) + 1)}–1-1-onto→{𝑍})
22		fzp1disj 10315	. . . . . . . 8 ⊢ ((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅
23	22	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅)
24		disjsn 3731	. . . . . . . . 9 ⊢ ((𝑌 ∩ {𝑍}) = ∅ ↔ ¬ 𝑍 ∈ 𝑌)
25	10, 24	sylibr 134	. . . . . . . 8 ⊢ (𝜑 → (𝑌 ∩ {𝑍}) = ∅)
26	25	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝑌 ∩ {𝑍}) = ∅)
27		f1oun 5603	. . . . . . 7 ⊢ (((ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 ∧ {⟨((♯‘𝑌) + 1), 𝑍⟩}:{((♯‘𝑌) + 1)}–1-1-onto→{𝑍}) ∧ (((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅ ∧ (𝑌 ∩ {𝑍}) = ∅)) → (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)})–1-1-onto→(𝑌 ∪ {𝑍}))
28	14, 21, 23, 26, 27	syl22anc 1274	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)})–1-1-onto→(𝑌 ∪ {𝑍}))
29	1, 10	jca 306	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ Fin ∧ ¬ 𝑍 ∈ 𝑌))
30		hashunsng 11071	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝑉 → ((𝑌 ∈ Fin ∧ ¬ 𝑍 ∈ 𝑌) → (♯‘(𝑌 ∪ {𝑍})) = ((♯‘𝑌) + 1)))
31	9, 29, 30	sylc 62	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑌 ∪ {𝑍})) = ((♯‘𝑌) + 1))
32	31	oveq2d 6034	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘(𝑌 ∪ {𝑍}))) = (1...((♯‘𝑌) + 1)))
33		1z 9505	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
34		nn0uz 9791	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
35		1m1e0 9212	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
36	35	fveq2i 5642	. . . . . . . . . . . 12 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
37	34, 36	eqtr4i 2255	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
38	16, 37	eleqtrdi 2324	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑌) ∈ (ℤ≥‘(1 − 1)))
39		fzsuc2 10314	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑌) ∈ (ℤ≥‘(1 − 1))) → (1...((♯‘𝑌) + 1)) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
40	33, 38, 39	sylancr 414	. . . . . . . . 9 ⊢ (𝜑 → (1...((♯‘𝑌) + 1)) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
41	32, 40	eqtrd 2264	. . . . . . . 8 ⊢ (𝜑 → (1...(♯‘(𝑌 ∪ {𝑍}))) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
42	41	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (1...(♯‘(𝑌 ∪ {𝑍}))) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
43	42	f1oeq2d 5579	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...(♯‘(𝑌 ∪ {𝑍})))–1-1-onto→(𝑌 ∪ {𝑍}) ↔ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)})–1-1-onto→(𝑌 ∪ {𝑍})))
44	28, 43	mpbird 167	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...(♯‘(𝑌 ∪ {𝑍})))–1-1-onto→(𝑌 ∪ {𝑍}))
45	4, 6, 8, 13, 44	gfsumval 16683	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgf 𝐹) = (𝐺 Σg (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))))
46		gfsump1.p	. . . . 5 ⊢ + = (+g‘𝐺)
47	5	cmnmndd 13896	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
48	47	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝐺 ∈ Mnd)
49		1zzd 9506	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 1 ∈ ℤ)
50	38	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (♯‘𝑌) ∈ (ℤ≥‘(1 − 1)))
51	40	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (1...((♯‘𝑌) + 1)) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
52	51	f1oeq2d 5579	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))–1-1-onto→(𝑌 ∪ {𝑍}) ↔ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)})–1-1-onto→(𝑌 ∪ {𝑍})))
53	28, 52	mpbird 167	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))–1-1-onto→(𝑌 ∪ {𝑍}))
54		f1of 5583	. . . . . . 7 ⊢ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))–1-1-onto→(𝑌 ∪ {𝑍}) → (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))⟶(𝑌 ∪ {𝑍}))
55	53, 54	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))⟶(𝑌 ∪ {𝑍}))
56		fco 5500	. . . . . 6 ⊢ ((𝐹:(𝑌 ∪ {𝑍})⟶𝐵 ∧ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))⟶(𝑌 ∪ {𝑍})) → (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})):(1...((♯‘𝑌) + 1))⟶𝐵)
57	8, 55, 56	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})):(1...((♯‘𝑌) + 1))⟶𝐵)
58	4, 46, 48, 49, 50, 57	gsumsplit0 13934	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))) = ((𝐺 Σg ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) + ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1))))
59	45, 58	eqtrd 2264	. . 3 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgf 𝐹) = ((𝐺 Σg ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) + ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1))))
60		resco 5241	. . . . . . . . . 10 ⊢ ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌))))
61		resundir 5027	. . . . . . . . . . . 12 ⊢ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌))) = ((ℎ ↾ (1...(♯‘𝑌))) ∪ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))))
62		incom 3399	. . . . . . . . . . . . . . . 16 ⊢ ({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)})
63	62, 22	eqtri 2252	. . . . . . . . . . . . . . 15 ⊢ ({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ∅
64		fnsng 5377	. . . . . . . . . . . . . . . . 17 ⊢ ((((♯‘𝑌) + 1) ∈ ℕ0 ∧ 𝑍 ∈ 𝑉) → {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)})
65	18, 9, 64	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)})
66		fnresdisj 5442	. . . . . . . . . . . . . . . 16 ⊢ ({⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)} → (({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ∅ ↔ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))) = ∅))
67	65, 66	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ∅ ↔ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))) = ∅))
68	63, 67	mpbii 148	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))) = ∅)
69	68	uneq2d 3361	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℎ ↾ (1...(♯‘𝑌))) ∪ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌)))) = ((ℎ ↾ (1...(♯‘𝑌))) ∪ ∅))
70		un0 3528	. . . . . . . . . . . . 13 ⊢ ((ℎ ↾ (1...(♯‘𝑌))) ∪ ∅) = (ℎ ↾ (1...(♯‘𝑌)))
71	69, 70	eqtrdi 2280	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℎ ↾ (1...(♯‘𝑌))) ∪ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌)))) = (ℎ ↾ (1...(♯‘𝑌))))
72	61, 71	eqtrid 2276	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌))) = (ℎ ↾ (1...(♯‘𝑌))))
73	72	coeq2d 4892	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌)))) = (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))))
74	60, 73	eqtrid 2276	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))))
75	74	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))))
76		f1ofn 5584	. . . . . . . . . . 11 ⊢ (ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 → ℎ Fn (1...(♯‘𝑌)))
77		fnresdm 5441	. . . . . . . . . . 11 ⊢ (ℎ Fn (1...(♯‘𝑌)) → (ℎ ↾ (1...(♯‘𝑌))) = ℎ)
78	76, 77	syl 14	. . . . . . . . . 10 ⊢ (ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 → (ℎ ↾ (1...(♯‘𝑌))) = ℎ)
79	78	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (ℎ ↾ (1...(♯‘𝑌))) = ℎ)
80	79	coeq2d 4892	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))) = (𝐹 ∘ ℎ))
81	75, 80	eqtrd 2264	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ ℎ))
82		f1of 5583	. . . . . . . . . 10 ⊢ (ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 → ℎ:(1...(♯‘𝑌))⟶𝑌)
83	82	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ℎ:(1...(♯‘𝑌))⟶𝑌)
84	83	frnd 5492	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ran ℎ ⊆ 𝑌)
85		cores 5240	. . . . . . . 8 ⊢ (ran ℎ ⊆ 𝑌 → ((𝐹 ↾ 𝑌) ∘ ℎ) = (𝐹 ∘ ℎ))
86	84, 85	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ↾ 𝑌) ∘ ℎ) = (𝐹 ∘ ℎ))
87	81, 86	eqtr4d 2267	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = ((𝐹 ↾ 𝑌) ∘ ℎ))
88	87	oveq2d 6034	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) = (𝐺 Σg ((𝐹 ↾ 𝑌) ∘ ℎ)))
89		ssun1 3370	. . . . . . . . 9 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
90	89	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ (𝑌 ∪ {𝑍}))
91	7, 90	fssresd 5513	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑌):𝑌⟶𝐵)
92	91	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹 ↾ 𝑌):𝑌⟶𝐵)
93	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝑌 ∈ Fin)
94	4, 6, 92, 93, 14	gfsumval 16683	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgf (𝐹 ↾ 𝑌)) = (𝐺 Σg ((𝐹 ↾ 𝑌) ∘ ℎ)))
95	88, 94	eqtr4d 2267	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) = (𝐺 Σgf (𝐹 ↾ 𝑌)))
96		nn0p1nn 9441	. . . . . . . . . 10 ⊢ ((♯‘𝑌) ∈ ℕ0 → ((♯‘𝑌) + 1) ∈ ℕ)
97	16, 96	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ ℕ)
98		nnuz 9792	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
99	97, 98	eleqtrdi 2324	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ (ℤ≥‘1))
100		eluzfz2 10267	. . . . . . . 8 ⊢ (((♯‘𝑌) + 1) ∈ (ℤ≥‘1) → ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1)))
101	99, 100	syl 14	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1)))
102	101	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1)))
103		fvco3 5717	. . . . . 6 ⊢ (((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}):(1...((♯‘𝑌) + 1))⟶(𝑌 ∪ {𝑍}) ∧ ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1))) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1)) = (𝐹‘((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1))))
104	55, 102, 103	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1)) = (𝐹‘((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1))))
105	76	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ℎ Fn (1...(♯‘𝑌)))
106	65	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)})
107	18	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((♯‘𝑌) + 1) ∈ ℕ0)
108		snidg 3698	. . . . . . . . . 10 ⊢ (((♯‘𝑌) + 1) ∈ ℕ0 → ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)})
109	107, 108	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)})
110	23, 109	jca 306	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅ ∧ ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)}))
111		fvun2 5713	. . . . . . . 8 ⊢ ((ℎ Fn (1...(♯‘𝑌)) ∧ {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)} ∧ (((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅ ∧ ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)})) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1)) = ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)))
112	105, 106, 110, 111	syl3anc 1273	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1)) = ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)))
113	9	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝑍 ∈ 𝑉)
114		fvsng 5850	. . . . . . . 8 ⊢ ((((♯‘𝑌) + 1) ∈ ℕ0 ∧ 𝑍 ∈ 𝑉) → ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)) = 𝑍)
115	107, 113, 114	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)) = 𝑍)
116	112, 115	eqtrd 2264	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1)) = 𝑍)
117	116	fveq2d 5643	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹‘((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1))) = (𝐹‘𝑍))
118	104, 117	eqtrd 2264	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1)) = (𝐹‘𝑍))
119	95, 118	oveq12d 6036	. . 3 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐺 Σg ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) + ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1))) = ((𝐺 Σgf (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
120	59, 119	eqtrd 2264	. 2 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgf 𝐹) = ((𝐺 Σgf (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
121	3, 120	exlimddv 1947	1 ⊢ (𝜑 → (𝐺 Σgf 𝐹) = ((𝐺 Σgf (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  {csn 3669  ⟨cop 3672  ran crn 4726   ↾ cres 4727   ∘ ccom 4729   Fn wfn 5321  ⟶wf 5322  –1-1-onto→wf1o 5325  ‘cfv 5326  (class class class)co 6018  Fincfn 6909  0cc0 8032  1c1 8033   + caddc 8035   − cmin 8350  ℕcn 9143  ℕ₀cn0 9402  ℤcz 9479  ℤ_≥cuz 9755  ...cfz 10243  ♯chash 11037  Basecbs 13083  +_gcplusg 13161   Σ_g cgsu 13341  Mndcmnd 13500  CMndccmn 13872   Σ_gf cgfsu 16681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-seqfrec 10710  df-ihash 11038  df-ndx 13086  df-slot 13087  df-base 13089  df-plusg 13174  df-0g 13342  df-igsum 13343  df-mgm 13440  df-sgrp 13486  df-mnd 13501  df-minusg 13588  df-mulg 13708  df-cmn 13874  df-gfsum 16682

This theorem is referenced by:  gfsumcl  16690