Theorem zsum 15771

Description: Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.)

Hypotheses

Ref	Expression
zsum.1	⊢ 𝑍 = (ℤ≥‘𝑀)
zsum.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
zsum.3	⊢ (𝜑 → 𝐴 ⊆ 𝑍)
zsum.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
zsum.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
zsum	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝜑,𝑘   𝑘,𝑍   𝑘,𝑀

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem zsum

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

Step	Hyp	Ref	Expression
1		eleq1w 2846	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
2		csbeq1 3857	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
3	1, 2	ifbieq1d 4513	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 0))
4	3	cbvmptv 5216	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 0))
5		simpll 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝜑)
6		zsum.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
7	6	ralrimiva 3157	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
8		nfcsb1v 3878	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
9	8	nfel1 2941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ
10		csbeq1a 3868	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
11	10	eleq1d 2848	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
12	9, 11	rspc 3570	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
13	7, 12	syl5 35	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐴 → (𝜑 → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
14	5, 13	mpan9 515	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
15		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝑚 ∈ ℤ)
16		zsum.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17	16	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝑀 ∈ ℤ)
18		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝐴 ⊆ (ℤ≥‘𝑚))
19		zsum.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
20		zsum.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ≥‘𝑀)
21	19, 20	sseqtrdi 3978	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
22	21	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝐴 ⊆ (ℤ≥‘𝑀))
23	4, 14, 15, 17, 18, 22	sumrb 15766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
24	23	biimpd 232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
25	24	expimpd 458	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
26	25	rexlimdva 3166	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
27	19	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ⊆ 𝑍)
28		uzssz 12884	. . . . . . . . . . . . . . . 16 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
29	20, 28	eqsstri 3984	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℤ
30		zssre 12599	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℝ
31	29, 30	sstri 3947	. . . . . . . . . . . . . 14 ⊢ 𝑍 ⊆ ℝ
32		ltso 11291	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
33		soss 5591	. . . . . . . . . . . . . 14 ⊢ (𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍))
34	31, 32, 33	mp2 9	. . . . . . . . . . . . 13 ⊢ < Or 𝑍
35		soss 5591	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑍 → ( < Or 𝑍 → < Or 𝐴))
36	27, 34, 35	mpisyl 22	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → < Or 𝐴)
37		fzfi 14010	. . . . . . . . . . . . 13 ⊢ (1...𝑚) ∈ Fin
38		ovex 7445	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ∈ V
39	38	f1oen 8970	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → (1...𝑚) ≈ 𝐴)
40	39	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
41	40	ensymd 9003	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ≈ (1...𝑚))
42		enfii 9171	. . . . . . . . . . . . 13 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
43	37, 41, 42	sylancr 598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ∈ Fin)
44		fz1iso 14501	. . . . . . . . . . . 12 ⊢ (( < Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
45	36, 43, 44	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
46		simpll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
47	46, 13	mpan9 515	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
48		fveq2 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
49	48	csbeq1d 3858	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
50		csbcow 3869	. . . . . . . . . . . . . . . 16 ⊢ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵
51	49, 50	eqtr4di 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
52	51	cbvmptv 5216	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
53		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
54		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
55	16	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
56	21	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑀))
57		simprl 782	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
58		simprr 784	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
59	4, 47, 52, 53, 54, 55, 56, 57, 58	summolem2a 15768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
60	59	expr 461	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
61	60	exlimdv 1963	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
62	45, 61	mpd 16	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
63		breq2 5114	. . . . . . . . . 10 ⊢ (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
64	62, 63	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
65	64	expimpd 458	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
66	65	exlimdv 1963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
67	66	rexlimdva 3166	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
68	26, 67	jaod 872	. . . . 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))) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
69	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
70	21	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ≥‘𝑀))
71		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥)
72		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
73	72	sseq2d 3970	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑀)))
74		seqeq1 14042	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))))
75	74	breq1d 5120	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
76	73, 75	anbi12d 643	. . . . . . . . 9 ⊢ (𝑚 = 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))) ⇝ 𝑥)))
77	76	rspcev 3582	. . . . . . . 8 ⊢ ((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))) ⇝ 𝑥))
78	69, 70, 71, 77	syl12anc 849	. . . . . . 7 ⊢ ((𝜑 ∧ 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))) ⇝ 𝑥))
79	78	orcd 886	. . . . . 6 ⊢ ((𝜑 ∧ 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))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
80	79	ex 417	. . . . 5 ⊢ (𝜑 → (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))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))))
81	68, 80	impbid 215	. . . 4 ⊢ (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))) ↔ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
82		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ (ℤ≥‘𝑀))
83	28, 82	sselid 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℤ)
84	82, 20	eleqtrrdi 2874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ 𝑍)
85		zsum.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
86	85	ralrimiva 3157	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
87	86	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
88		nfcsb1v 3878	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)
89	88	nfeq2 2942	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)
90		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91		csbeq1a 3868	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → if(𝑘 ∈ 𝐴, 𝐵, 0) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0))
92	90, 91	eqeq12d 2779	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0) ↔ (𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)))
93	89, 92	rspc 3570	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0) → (𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)))
94	84, 87, 93	sylc 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0))
95		fvex 6896	. . . . . . . . 9 ⊢ (𝐹‘𝑗) ∈ V
96	94, 95	eqeltrrdi 2872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ V)
97		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 0)
98		nfv 1944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
99		nfcsb1v 3878	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
100		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘0
101	98, 99, 100	nfif 4519	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
102		eleq1w 2846	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
103		csbeq1a 3868	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
104	102, 103	ifbieq1d 4513	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
105	97, 101, 104	cbvmpt 5214	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
106	105	eqcomi 2772	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
107	106	fvmpts 6995	. . . . . . . 8 ⊢ ((𝑗 ∈ ℤ ∧ ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ V) → ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0))
108	83, 96, 107	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0))
109	108, 94	eqtr4d 2801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑗) = (𝐹‘𝑗))
110	16, 109	seqfeq 14065	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑀( + , 𝐹))
111	110	breq1d 5120	. . . 4 ⊢ (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
112	81, 111	bitrd 282	. . 3 ⊢ (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))) ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
113	112	iotabidv 6522	. 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)))) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥))
114		df-sum 15740	. 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))))
115		df-fv 6546	. 2 ⊢ ( ⇝ ‘seq𝑀( + , 𝐹)) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥)
116	113, 114, 115	3eqtr4g 2823	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ⦋csb 3854   ⊆ wss 3906  ifcif 4488   class class class wbr 5110   ↦ cmpt 5193   Or wor 5570  ℩cio 6492  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  (class class class)co 7412   ≈ cen 8941  Fincfn 8944  ℂcc 11099  ℝcr 11100  0cc0 11101  1c1 11102   + caddc 11104   < clt 11244  ℕcn 12234  ℤcz 12592  ℤ_≥cuz 12863  ...cfz 13536  seqcseq 14039  ♯chash 14368   ⇝ cli 15537  Σcsu 15739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-rp 13018  df-fz 13537  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-clim 15541  df-sum 15740

This theorem is referenced by:  isum  15772  sum0  15774  sumz  15775  sumss  15777  fsumsers  15781