Theorem gsump1 14134

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
gsump1.b	⊢ 𝐵 = (Base‘𝐺)
gsump1.p	⊢ + = (+g‘𝐺)
gsump1.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsump1.f	⊢ (𝜑 → 𝐹:(𝑌 ∪ {𝑍})⟶𝐵)
gsump1.fi	⊢ (𝜑 → 𝑌 ∈ Fin)
gsump1.zv	⊢ (𝜑 → 𝑍 ∈ 𝑉)
gsump1.z	⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)

Assertion

Ref	Expression
gsump1	⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))

Proof of Theorem gsump1

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsump1.fi	. . 3 ⊢ (𝜑 → 𝑌 ∈ Fin)
2		fzf1o 12120	. . 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		gsump1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
5		gsump1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ CMnd)
6	5	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝐺 ∈ CMnd)
7		gsump1.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		gsump1.zv	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
10		gsump1.z	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)
11		unsnfi 7216	. . . . . . 7 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝑌) → (𝑌 ∪ {𝑍}) ∈ Fin)
12	1, 9, 10, 11	syl3anc 1278	. . . . . 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 11198	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ0)
16	1, 15	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ0)
17		peano2nn0 9582	. . . . . . . . . 10 ⊢ ((♯‘𝑌) ∈ ℕ0 → ((♯‘𝑌) + 1) ∈ ℕ0)
18	16, 17	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ ℕ0)
19		f1osng 5677	. . . . . . . . 9 ⊢ ((((♯‘𝑌) + 1) ∈ ℕ0 ∧ 𝑍 ∈ 𝑉) → {⟨((♯‘𝑌) + 1), 𝑍⟩}:{((♯‘𝑌) + 1)}–1-1-onto→{𝑍})
20	18, 9, 19	syl2anc 415	. . . . . . . 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 10465	. . . . . . . 8 ⊢ ((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅
23	22	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅)
24		disjsn 3767	. . . . . . . . 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 5654	. . . . . . 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 1279	. . . . . 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 11226	. . . . . . . . . . 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 6091	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘(𝑌 ∪ {𝑍}))) = (1...((♯‘𝑌) + 1)))
33		1z 9649	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
34		nn0uz 9936	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
35		1m1e0 9352	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
36	35	fveq2i 5693	. . . . . . . . . . . 12 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
37	34, 36	eqtr4i 2262	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
38	16, 37	eleqtrdi 2331	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑌) ∈ (ℤ≥‘(1 − 1)))
39		fzsuc2 10464	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑌) ∈ (ℤ≥‘(1 − 1))) → (1...((♯‘𝑌) + 1)) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
40	33, 38, 39	sylancr 418	. . . . . . . . 9 ⊢ (𝜑 → (1...((♯‘𝑌) + 1)) = ((1...(♯‘𝑌)) ∪ {((♯‘𝑌) + 1)}))
41	32, 40	eqtrd 2271	. . . . . . . 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 5630	. . . . . 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	gsumvalfi 14129	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg 𝐹) = (𝐺 Σgz (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))))
46		gsump1.p	. . . . 5 ⊢ + = (+g‘𝐺)
47	5	cmnmndd 14088	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Mnd)
48	47	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝐺 ∈ Mnd)
49		1zzd 9650	. . . . 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 5630	. . . . . . . 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 5634	. . . . . . 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	8, 55	fcod 5548	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})):(1...((♯‘𝑌) + 1))⟶𝐵)
57	4, 46, 48, 49, 50, 56	gzsumsplit0 14125	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgz (𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))) = ((𝐺 Σgz ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) + ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1))))
58	45, 57	eqtrd 2271	. . 3 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg 𝐹) = ((𝐺 Σgz ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) + ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1))))
59		resco 5287	. . . . . . . . . 10 ⊢ ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌))))
60		resundir 5072	. . . . . . . . . . . 12 ⊢ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌))) = ((ℎ ↾ (1...(♯‘𝑌))) ∪ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))))
61	22	ineqcomi 3423	. . . . . . . . . . . . . . 15 ⊢ ({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ∅
62		fnsng 5423	. . . . . . . . . . . . . . . . 17 ⊢ ((((♯‘𝑌) + 1) ∈ ℕ0 ∧ 𝑍 ∈ 𝑉) → {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)})
63	18, 9, 62	syl2anc 415	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)})
64		fnresdisj 5488	. . . . . . . . . . . . . . . 16 ⊢ ({⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)} → (({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ∅ ↔ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))) = ∅))
65	63, 64	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (({((♯‘𝑌) + 1)} ∩ (1...(♯‘𝑌))) = ∅ ↔ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))) = ∅))
66	61, 65	mpbii 148	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌))) = ∅)
67	66	uneq2d 3383	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℎ ↾ (1...(♯‘𝑌))) ∪ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌)))) = ((ℎ ↾ (1...(♯‘𝑌))) ∪ ∅))
68		un0 3556	. . . . . . . . . . . . 13 ⊢ ((ℎ ↾ (1...(♯‘𝑌))) ∪ ∅) = (ℎ ↾ (1...(♯‘𝑌)))
69	67, 68	eqtrdi 2287	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℎ ↾ (1...(♯‘𝑌))) ∪ ({⟨((♯‘𝑌) + 1), 𝑍⟩} ↾ (1...(♯‘𝑌)))) = (ℎ ↾ (1...(♯‘𝑌))))
70	60, 69	eqtrid 2283	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌))) = (ℎ ↾ (1...(♯‘𝑌))))
71	70	coeq2d 4937	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}) ↾ (1...(♯‘𝑌)))) = (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))))
72	59, 71	eqtrid 2283	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))))
73	72	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))))
74		f1ofn 5635	. . . . . . . . . . 11 ⊢ (ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 → ℎ Fn (1...(♯‘𝑌)))
75		fnresdm 5487	. . . . . . . . . . 11 ⊢ (ℎ Fn (1...(♯‘𝑌)) → (ℎ ↾ (1...(♯‘𝑌))) = ℎ)
76	74, 75	syl 14	. . . . . . . . . 10 ⊢ (ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 → (ℎ ↾ (1...(♯‘𝑌))) = ℎ)
77	76	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (ℎ ↾ (1...(♯‘𝑌))) = ℎ)
78	77	coeq2d 4937	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹 ∘ (ℎ ↾ (1...(♯‘𝑌)))) = (𝐹 ∘ ℎ))
79	73, 78	eqtrd 2271	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = (𝐹 ∘ ℎ))
80		f1of 5634	. . . . . . . . . 10 ⊢ (ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌 → ℎ:(1...(♯‘𝑌))⟶𝑌)
81	80	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ℎ:(1...(♯‘𝑌))⟶𝑌)
82	81	frnd 5538	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ran ℎ ⊆ 𝑌)
83		cores 5286	. . . . . . . 8 ⊢ (ran ℎ ⊆ 𝑌 → ((𝐹 ↾ 𝑌) ∘ ℎ) = (𝐹 ∘ ℎ))
84	82, 83	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ↾ 𝑌) ∘ ℎ) = (𝐹 ∘ ℎ))
85	79, 84	eqtr4d 2274	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌))) = ((𝐹 ↾ 𝑌) ∘ ℎ))
86	85	oveq2d 6091	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgz ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) = (𝐺 Σgz ((𝐹 ↾ 𝑌) ∘ ℎ)))
87		ssun1 3392	. . . . . . . . 9 ⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍})
88	87	a1i 9	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ (𝑌 ∪ {𝑍}))
89	7, 88	fssresd 5561	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝑌):𝑌⟶𝐵)
90	89	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹 ↾ 𝑌):𝑌⟶𝐵)
91	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝑌 ∈ Fin)
92	4, 6, 90, 91, 14	gsumvalfi 14129	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg (𝐹 ↾ 𝑌)) = (𝐺 Σgz ((𝐹 ↾ 𝑌) ∘ ℎ)))
93	86, 92	eqtr4d 2274	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σgz ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) = (𝐺 Σg (𝐹 ↾ 𝑌)))
94		nn0p1nn 9581	. . . . . . . . . 10 ⊢ ((♯‘𝑌) ∈ ℕ0 → ((♯‘𝑌) + 1) ∈ ℕ)
95	16, 94	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ ℕ)
96		nnuz 9937	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
97	95, 96	eleqtrdi 2331	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ (ℤ≥‘1))
98		eluzfz2 10415	. . . . . . . 8 ⊢ (((♯‘𝑌) + 1) ∈ (ℤ≥‘1) → ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1)))
99	97, 98	syl 14	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1)))
100	99	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((♯‘𝑌) + 1) ∈ (1...((♯‘𝑌) + 1)))
101		fvco3 5770	. . . . . 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))))
102	55, 100, 101	syl2anc 415	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1)) = (𝐹‘((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1))))
103	74	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ℎ Fn (1...(♯‘𝑌)))
104	63	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → {⟨((♯‘𝑌) + 1), 𝑍⟩} Fn {((♯‘𝑌) + 1)})
105	18	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((♯‘𝑌) + 1) ∈ ℕ0)
106		snidg 3734	. . . . . . . . . 10 ⊢ (((♯‘𝑌) + 1) ∈ ℕ0 → ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)})
107	105, 106	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)})
108	23, 107	jca 306	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (((1...(♯‘𝑌)) ∩ {((♯‘𝑌) + 1)}) = ∅ ∧ ((♯‘𝑌) + 1) ∈ {((♯‘𝑌) + 1)}))
109		fvun2 5764	. . . . . . . 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)))
110	103, 104, 108, 109	syl3anc 1278	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1)) = ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)))
111	9	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → 𝑍 ∈ 𝑉)
112		fvsng 5902	. . . . . . . 8 ⊢ ((((♯‘𝑌) + 1) ∈ ℕ0 ∧ 𝑍 ∈ 𝑉) → ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)) = 𝑍)
113	105, 111, 112	syl2anc 415	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ({⟨((♯‘𝑌) + 1), 𝑍⟩}‘((♯‘𝑌) + 1)) = 𝑍)
114	110, 113	eqtrd 2271	. . . . . 6 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1)) = 𝑍)
115	114	fveq2d 5694	. . . . 5 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐹‘((ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})‘((♯‘𝑌) + 1))) = (𝐹‘𝑍))
116	102, 115	eqtrd 2271	. . . 4 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1)) = (𝐹‘𝑍))
117	93, 116	oveq12d 6093	. . 3 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → ((𝐺 Σgz ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩})) ↾ (1...(♯‘𝑌)))) + ((𝐹 ∘ (ℎ ∪ {⟨((♯‘𝑌) + 1), 𝑍⟩}))‘((♯‘𝑌) + 1))) = ((𝐺 Σg (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
118	58, 117	eqtrd 2271	. 2 ⊢ ((𝜑 ∧ ℎ:(1...(♯‘𝑌))–1-1-onto→𝑌) → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))
119	3, 118	exlimddv 1954	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝑌)) + (𝐹‘𝑍)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3705  ⟨cop 3708  ran crn 4770   ↾ cres 4771   ∘ ccom 4773   Fn wfn 5367  ⟶wf 5368  –1-1-onto→wf1o 5371  ‘cfv 5372  (class class class)co 6075  Fincfn 7012  0cc0 8169  1c1 8170   + caddc 8172   − cmin 8487  ℕcn 9283  ℕ₀cn0 9542  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  ♯chash 11192  Basecbs 13330  +_gcplusg 13408   Σ_gz cgzsu 13588  Mndcmnd 13706  CMndccmn 14064   Σ_g cgsu 14127

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ihash 11193  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-minusg 13786  df-mulg 13900  df-cmn 14066  df-gsumfi 14128

This theorem is referenced by:  gsumzfi  14135  gsumclfi  14136  gsummptfidmadd  14138  gsumsubmclfi  14140  gsumconstcmn  14143  gsumfsum  14895