Theorem fsum 15747

Description: The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypotheses

Ref	Expression
fsum.1	⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
fsum.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
fsum.3	⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
fsum.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
fsum.5	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)

Assertion

Ref	Expression
fsum	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))

Distinct variable groups:   𝐵,𝑛   𝐶,𝑘   𝑘,𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝑀,𝑛

Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑛)

Proof of Theorem fsum

Dummy variables 𝑓 𝑖 𝑗 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sum 15714	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
2		fvex 6880	. . 3 ⊢ (seq1( + , 𝐺)‘𝑀) ∈ V
3		eleq1w 2845	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴))
4		csbeq1 3855	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
5	3, 4	ifbieq1d 4505	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑗 ∈ 𝐴, ⦋𝑗 / 𝑘⦌𝐵, 0))
6	5	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, ⦋𝑗 / 𝑘⦌𝐵, 0))
7		fsum.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
8	7	ralrimiva 3154	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
9		nfcsb1v 3876	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵
10	9	nfel1 2940	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ
11		csbeq1a 3866	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)
12	11	eleq1d 2847	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ))
13	10, 12	rspc 3569	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ))
14	8, 13	mpan9 514	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ)
15		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → (𝑓‘𝑛) = (𝑓‘𝑖))
16	15	csbeq1d 3856	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑖) / 𝑘⦌𝐵)
17		csbcow 3867	. . . . . . . . . 10 ⊢ ⦋(𝑓‘𝑖) / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑖) / 𝑘⦌𝐵
18	16, 17	eqtr4di 2815	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑖) / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵)
19	18	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑖 ∈ ℕ ↦ ⦋(𝑓‘𝑖) / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵)
20	6, 14, 19	summo 15744	. . . . . . 7 ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
21		fsum.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		fsum.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
23		f1of 6806	. . . . . . . . . . . 12 ⊢ (𝐹:(1...𝑀)–1-1-onto→𝐴 → 𝐹:(1...𝑀)⟶𝐴)
24	22, 23	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝐴)
25		ovex 7429	. . . . . . . . . . 11 ⊢ (1...𝑀) ∈ V
26		fex 7210	. . . . . . . . . . 11 ⊢ ((𝐹:(1...𝑀)⟶𝐴 ∧ (1...𝑀) ∈ V) → 𝐹 ∈ V)
27	24, 25, 26	sylancl 595	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
28		nnuz 12878	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
29	21, 28	eleqtrdi 2872	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
30		fsum.5	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)
31		elfznn 13558	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℕ)
32		fvex 6880	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺‘𝑛) ∈ V
33	30, 32	eqeltrrdi 2871	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → 𝐶 ∈ V)
34		eqid 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ 𝐶) = (𝑛 ∈ ℕ ↦ 𝐶)
35	34	fvmpt2 6987	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝐶 ∈ V) → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = 𝐶)
36	31, 33, 35	syl2an2 696	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = 𝐶)
37	30, 36	eqtr4d 2800	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛))
38	37	ralrimiva 3154	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛))
39		nffvmpt1 6878	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)
40	39	nfeq2 2941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝐺‘𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)
41		fveq2 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
42		fveq2 6867	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘))
43	41, 42	eqeq12d 2778	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → ((𝐺‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) ↔ (𝐺‘𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)))
44	40, 43	rspc 3569	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑀) → (∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑛) → (𝐺‘𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘)))
45	38, 44	mpan9 514	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → (𝐺‘𝑘) = ((𝑛 ∈ ℕ ↦ 𝐶)‘𝑘))
46	29, 45	seqfveq 14039	. . . . . . . . . . 11 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))
47	22, 46	jca 519	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀)))
48		f1oeq1 6794	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐹:(1...𝑀)–1-1-onto→𝐴))
49		fveq1 6866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑛) = (𝐹‘𝑛))
50	49	csbeq1d 3856	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝐹 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
51		fvex 6880	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹‘𝑛) ∈ V
52		fsum.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
53	51, 52	csbie 3887	. . . . . . . . . . . . . . . 16 ⊢ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐶
54	50, 53	eqtrdi 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐹 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = 𝐶)
55	54	mpteq2dv 5194	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ 𝐶))
56	55	seqeq3d 14022	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ 𝐶)))
57	56	fveq1d 6869	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))
58	57	eqeq2d 2773	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀)))
59	48, 58	anbi12d 641	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀)) ↔ (𝐹:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ 𝐶))‘𝑀))))
60	27, 47, 59	spcedv 3557	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑓(𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀)))
61		oveq2 7404	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → (1...𝑚) = (1...𝑀))
62	61	f1oeq2d 6802	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑀)–1-1-onto→𝐴))
63		fveq2 6867	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑀 → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀))
64	63	eqeq2d 2773	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → ((seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀)))
65	62, 64	anbi12d 641	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀))))
66	65	exbidv 1941	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀))))
67	66	rspcev 3581	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ ∃𝑓(𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑀))) → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
68	21, 60, 67	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
69	68	olcd 885	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
70		breq2 5104	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)))
71	70	anbi2d 639	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀))))
72	71	rexbidv 3186	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀))))
73		eqeq1 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) ↔ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
74	73	anbi2d 639	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
75	74	exbidv 1941	. . . . . . . . . . . . 13 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
76	75	rexbidv 3186	. . . . . . . . . . . 12 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
77	72, 76	orbi12d 929	. . . . . . . . . . 11 ⊢ (𝑥 = (seq1( + , 𝐺)‘𝑀) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))))
78	77	moi2 3679	. . . . . . . . . 10 ⊢ ((((seq1( + , 𝐺)‘𝑀) ∈ V ∧ ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) ∧ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))) → 𝑥 = (seq1( + , 𝐺)‘𝑀))
79	2, 78	mpanl1 710	. . . . . . . . 9 ⊢ ((∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ∧ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))) → 𝑥 = (seq1( + , 𝐺)‘𝑀))
80	79	ancom2s 660	. . . . . . . 8 ⊢ ((∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ∧ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))) → 𝑥 = (seq1( + , 𝐺)‘𝑀))
81	80	expr 460	. . . . . . 7 ⊢ ((∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , 𝐺)‘𝑀)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ (seq1( + , 𝐺)‘𝑀) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) → 𝑥 = (seq1( + , 𝐺)‘𝑀)))
82	20, 69, 81	syl2anc 593	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) → 𝑥 = (seq1( + , 𝐺)‘𝑀)))
83	69, 77	syl5ibrcom 249	. . . . . 6 ⊢ (𝜑 → (𝑥 = (seq1( + , 𝐺)‘𝑀) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))))
84	82, 83	impbid 214	. . . . 5 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ 𝑥 = (seq1( + , 𝐺)‘𝑀)))
85	84	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (seq1( + , 𝐺)‘𝑀) ∈ V) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ 𝑥 = (seq1( + , 𝐺)‘𝑀)))
86	85	iota5 6504	. . 3 ⊢ ((𝜑 ∧ (seq1( + , 𝐺)‘𝑀) ∈ V) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) = (seq1( + , 𝐺)‘𝑀))
87	2, 86	mpan2 701	. 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) = (seq1( + , 𝐺)‘𝑀))
88	1, 87	eqtrid 2809	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , 𝐺)‘𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∃*wmo 2564  ∀wral 3076  ∃wrex 3086  Vcvv 3454  ⦋csb 3852   ⊆ wss 3904  ifcif 4480   class class class wbr 5100   ↦ cmpt 5181  ℩cio 6475  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396  ℂcc 11071  0cc0 11073  1c1 11074   + caddc 11076  ℕcn 12210  ℤcz 12568  ℤ_≥cuz 12839  ...cfz 13512  seqcseq 14014   ⇝ cli 15511  Σcsu 15713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-rp 12994  df-fz 13513  df-fzo 13660  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-sum 15714

This theorem is referenced by:  sumz  15749  fsumf1o  15750  fsumcl2lem  15758  fsumadd  15767  sumsnf  15770  fsummulc2  15811  fsumconst  15817  fsumrelem  15835  gsumfsum  21483  sumsnd  45603