Theorem zsum 15732

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 2817	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
2		csbeq1 3877	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
3	1, 2	ifbieq1d 4525	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 0))
4	3	cbvmptv 5225	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑖 ∈ ℤ ↦ if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 0))
5		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝜑)
6		zsum.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
7	6	ralrimiva 3132	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
8		nfcsb1v 3898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
9	8	nfel1 2915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ
10		csbeq1a 3888	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
11	10	eleq1d 2819	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
12	9, 11	rspc 3589	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
13	7, 12	syl5 34	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐴 → (𝜑 → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
14	5, 13	mpan9 506	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
15		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝑚 ∈ ℤ)
16		zsum.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17	16	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝑀 ∈ ℤ)
18		simpr 484	. . . . . . . . . 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 3999	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
22	21	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ≥‘𝑚)) → 𝐴 ⊆ (ℤ≥‘𝑀))
23	4, 14, 15, 17, 18, 22	sumrb 15727	. . . . . . . . 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 229	. . . . . . . 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 453	. . . . . . 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 3141	. . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ⊆ 𝑍)
28		uzssz 12871	. . . . . . . . . . . . . . . 16 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
29	20, 28	eqsstri 4005	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℤ
30		zssre 12593	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℝ
31	29, 30	sstri 3968	. . . . . . . . . . . . . 14 ⊢ 𝑍 ⊆ ℝ
32		ltso 11313	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
33		soss 5581	. . . . . . . . . . . . . 14 ⊢ (𝑍 ⊆ ℝ → ( < Or ℝ → < Or 𝑍))
34	31, 32, 33	mp2 9	. . . . . . . . . . . . 13 ⊢ < Or 𝑍
35		soss 5581	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑍 → ( < Or 𝑍 → < Or 𝐴))
36	27, 34, 35	mpisyl 21	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → < Or 𝐴)
37		fzfi 13988	. . . . . . . . . . . . 13 ⊢ (1...𝑚) ∈ Fin
38		ovex 7436	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ∈ V
39	38	f1oen 8985	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → (1...𝑚) ≈ 𝐴)
40	39	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
41	40	ensymd 9017	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ≈ (1...𝑚))
42		enfii 9198	. . . . . . . . . . . . 13 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
43	37, 41, 42	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ∈ Fin)
44		fz1iso 14478	. . . . . . . . . . . 12 ⊢ (( < Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
45	36, 43, 44	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
46		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
47	46, 13	mpan9 506	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
48		fveq2 6875	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
49	48	csbeq1d 3878	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
50		csbcow 3889	. . . . . . . . . . . . . . . 16 ⊢ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵
51	49, 50	eqtr4di 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
52	51	cbvmptv 5225	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
53		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
54		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
55	16	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
56	21	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑀))
57		simprl 770	. . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . 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 15729	. . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 1933	. . . . . . . . . . 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 15	. . . . . . . . . 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 5123	. . . . . . . . . 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 247	. . . . . . . . 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 453	. . . . . . . 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 1933	. . . . . . 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 3141	. . . . . 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 859	. . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
70	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ≥‘𝑀))
71		simpr 484	. . . . . . . 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 6875	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) = (ℤ≥‘𝑀))
73	72	sseq2d 3991	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑀)))
74		seqeq1 14020	. . . . . . . . . . 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 5129	. . . . . . . . . 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 632	. . . . . . . . 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 3601	. . . . . . . 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 836	. . . . . . 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 873	. . . . . 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 412	. . . . 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 212	. . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ (ℤ≥‘𝑀))
83	28, 82	sselid 3956	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℤ)
84	82, 20	eleqtrrdi 2845	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ 𝑍)
85		zsum.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
86	85	ralrimiva 3132	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
87	86	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
88		nfcsb1v 3898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)
89	88	nfeq2 2916	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)
90		fveq2 6875	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
91		csbeq1a 3888	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → if(𝑘 ∈ 𝐴, 𝐵, 0) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0))
92	90, 91	eqeq12d 2751	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0) ↔ (𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0)))
93	89, 92	rspc 3589	. . . . . . . . . 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 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑗) = ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0))
95		fvex 6888	. . . . . . . . 9 ⊢ (𝐹‘𝑗) ∈ V
96	94, 95	eqeltrrdi 2843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ⦋𝑗 / 𝑘⦌if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ V)
97		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 0)
98		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
99		nfcsb1v 3898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
100		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘0
101	98, 99, 100	nfif 4531	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
102		eleq1w 2817	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
103		csbeq1a 3888	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
104	102, 103	ifbieq1d 4525	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
105	97, 101, 104	cbvmpt 5223	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
106	105	eqcomi 2744	. . . . . . . . 9 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
107	106	fvmpts 6988	. . . . . . . 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 584	. . . . . . 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 2773	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑗) = (𝐹‘𝑗))
110	16, 109	seqfeq 14043	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑀( + , 𝐹))
111	110	breq1d 5129	. . . 4 ⊢ (𝜑 → (seq𝑀( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑀( + , 𝐹) ⇝ 𝑥))
112	81, 111	bitrd 279	. . 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 6514	. 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 15701	. 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 6538	. 2 ⊢ ( ⇝ ‘seq𝑀( + , 𝐹)) = (℩𝑥seq𝑀( + , 𝐹) ⇝ 𝑥)
116	113, 114, 115	3eqtr4g 2795	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( + , 𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  Vcvv 3459  ⦋csb 3874   ⊆ wss 3926  ifcif 4500   class class class wbr 5119   ↦ cmpt 5201   Or wor 5560  ℩cio 6481  –1-1-onto→wf1o 6529  ‘cfv 6530   Isom wiso 6531  (class class class)co 7403   ≈ cen 8954  Fincfn 8957  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130   < clt 11267  ℕcn 12238  ℤcz 12586  ℤ_≥cuz 12850  ...cfz 13522  seqcseq 14017  ♯chash 14346   ⇝ cli 15498  Σcsu 15700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9522  df-card 9951  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-n0 12500  df-z 12587  df-uz 12851  df-rp 13007  df-fz 13523  df-fzo 13670  df-seq 14018  df-exp 14078  df-hash 14347  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15502  df-sum 15701

This theorem is referenced by:  isum  15733  sum0  15735  sumz  15736  sumss  15738  fsumsers  15742