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Theorem zprod 15872

Description: Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)

**Hypotheses**
Ref	Expression
zprod.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
zprod.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
zprod.3	⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
zprod.4	⊢ (𝜑 → 𝐴 ⊆ 𝑍)
zprod.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
zprod.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
zprod	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))

Distinct variable groups: 𝐴,𝑘,𝑛 𝜑,𝑘,𝑛 𝑛,𝑍 𝐵,𝑛,𝑦 𝑘,𝐹 𝜑,𝑦 𝑦,𝐴 𝑘,𝑀,𝑦 Allowed substitution hints: 𝐵(𝑘) 𝐹(𝑦,𝑛) 𝑀(𝑛) 𝑍(𝑦,𝑘)

Proof of Theorem zprod Dummy variables 𝑓 𝑔 𝑖 𝑗 𝑚 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpb 1150	. . . . . . . 8 ⊢ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
2		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖if(𝑘 ∈ 𝐴, 𝐵, 1)
3		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑖 ∈ 𝐴
4		nfcsb1v 3875	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
5		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘1
6	3, 4, 5	nfif 4512	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1)
7		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
8		csbeq1a 3865	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
9	7, 8	ifbieq1d 4506	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1))
10	2, 6, 9	cbvmpt 5202	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑖 ∈ ℤ ↦ if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1))
11		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝜑)
12		zprod.6	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
13	12	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
14	4	nfel1 2916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ
15	8	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
16	14, 15	rspc 3566	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
17	13, 16	syl5 34	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 → (𝜑 → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
18	11, 17	mpan9 506	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
19		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝑚 ∈ ℤ)
20		zprod.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
21	20	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝑀 ∈ ℤ)
22		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝐴 ⊆ (ℤ_≥‘𝑚))
23		zprod.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
24		zprod.1	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑀)
25	23, 24	sseqtrdi 3976	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
26	25	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝐴 ⊆ (ℤ_≥‘𝑀))
27	10, 18, 19, 21, 22, 26	prodrb 15867	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
28	27	biimpd 229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
29	28	expimpd 453	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
30	1, 29	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
31	30	rexlimdva 3139	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
32		uzssz 12784	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
33		zssre 12507	. . . . . . . . . . . . . . . . 17 ⊢ ℤ ⊆ ℝ
34	32, 33	sstri 3945	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ
35	24, 34	eqsstri 3982	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ⊆ ℝ
36	23, 35	sstrdi 3948	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
37	36	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ⊆ ℝ)
38		ltso 11225	. . . . . . . . . . . . 13 ⊢ < Or ℝ
39		soss 5560	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴))
40	37, 38, 39	mpisyl 21	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → < Or 𝐴)
41		fzfi 13907	. . . . . . . . . . . . 13 ⊢ (1...𝑚) ∈ Fin
42		ovex 7401	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑚) ∈ V
43	42	f1oen 8921	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → (1...𝑚) ≈ 𝐴)
44	43	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
45	44	ensymd 8954	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ≈ (1...𝑚))
46		enfii 9122	. . . . . . . . . . . . 13 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
47	41, 45, 46	sylancr 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ∈ Fin)
48		fz1iso 14397	. . . . . . . . . . . 12 ⊢ (( < Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
49	40, 47, 48	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
50		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
51	50, 17	mpan9 506	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
52		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
53	52	csbeq1d 3855	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
54		csbcow 3866	. . . . . . . . . . . . . . . 16 ⊢ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵
55	53, 54	eqtr4di 2790	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
56	55	cbvmptv 5204	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
57		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵)
58		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
59	20	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
60	25	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ_≥‘𝑀))
61		simprl 771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
62		simprr 773	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
63	10, 51, 56, 57, 58, 59, 60, 61, 62	prodmolem2a 15869	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
64	63	expr 456	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
65	64	exlimdv 1935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
66	49, 65	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
67		breq2 5104	. . . . . . . . . 10 ⊢ (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
68	66, 67	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
69	68	expimpd 453	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
70	69	exlimdv 1935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
71	70	rexlimdva 3139	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
72	31, 71	jaod 860	. . . . 5 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
73	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
74	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝐴 ⊆ 𝑍)
75		zprod.3	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
76	24	eleq2i 2829	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
77		eluzelz 12773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
78	77	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℤ)
79		uztrn 12781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ (ℤ_≥‘𝑛) ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑧 ∈ (ℤ_≥‘𝑀))
80	79	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝑧 ∈ (ℤ_≥‘𝑛)) → 𝑧 ∈ (ℤ_≥‘𝑀))
81	24	eleq2i 2829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
82		zprod.5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
83	24, 32	eqsstri 3982	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑍 ⊆ ℤ
84	83	sseli 3931	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
85		iftrue 4487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
86	85	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
87	86, 12	eqeltrd 2837	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
88	87	ex 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ))
89		iffalse 4490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
90		ax-1cn 11096	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℂ
91	89, 90	eqeltrdi 2845	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
92	88, 91	pm2.61d1 180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
93		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
94	93	fvmpt2 6961	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
95	84, 92, 94	syl2anr 598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
96	82, 95	eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘))
97	81, 96	sylan2br 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘))
98	97	ralrimiva 3130	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ_≥‘𝑀)(𝐹‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘))
99		nffvmpt1 6853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧)
100	99	nfeq2 2917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝐹‘𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧)
101		fveq2 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → (𝐹‘𝑘) = (𝐹‘𝑧))
102		fveq2 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧))
103	101, 102	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → ((𝐹‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) ↔ (𝐹‘𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧)))
104	100, 103	rspc 3566	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (ℤ_≥‘𝑀) → (∀𝑘 ∈ (ℤ_≥‘𝑀)(𝐹‘𝑘) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) → (𝐹‘𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧)))
105	98, 104	mpan9 506	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧))
106	80, 105	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝑧 ∈ (ℤ_≥‘𝑛))) → (𝐹‘𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧))
107	106	anassrs 467	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑧 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑧) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧))
108	78, 107	seqfeq 13962	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → seq𝑛( · , 𝐹) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))))
109	108	breq1d 5110	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
110	109	anbi2d 631	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
111	110	exbidv 1923	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
112	76, 111	sylan2b 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
113	112	rexbidva 3160	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
114	75, 113	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
115	114	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
116		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)
117		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (ℤ_≥‘𝑚) = (ℤ_≥‘𝑀))
118	117, 24	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (ℤ_≥‘𝑚) = 𝑍)
119	118	sseq2d 3968	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ_≥‘𝑚) ↔ 𝐴 ⊆ 𝑍))
120	118	rexeqdv 3299	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
121		seqeq1 13939	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))))
122	121	breq1d 5110	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
123	119, 120, 122	3anbi123d 1439	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (𝐴 ⊆ 𝑍 ∧ ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)))
124	123	rspcev 3578	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 ⊆ 𝑍 ∧ ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
125	73, 74, 115, 116, 124	syl13anc 1375	. . . . . . 7 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
126	125	orcd 874	. . . . . 6 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
127	126	ex 412	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))))
128	72, 127	impbid 212	. . . 4 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
129	95, 82	eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = (𝐹‘𝑘))
130	81, 129	sylan2br 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = (𝐹‘𝑘))
131	130	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (ℤ_≥‘𝑀)((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = (𝐹‘𝑘))
132	99	nfeq1 2915	. . . . . . . 8 ⊢ Ⅎ𝑘((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧) = (𝐹‘𝑧)
133	102, 101	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → (((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = (𝐹‘𝑘) ↔ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧) = (𝐹‘𝑧)))
134	132, 133	rspc 3566	. . . . . . 7 ⊢ (𝑧 ∈ (ℤ_≥‘𝑀) → (∀𝑘 ∈ (ℤ_≥‘𝑀)((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑘) = (𝐹‘𝑘) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧) = (𝐹‘𝑧)))
135	131, 134	mpan9 506	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑧) = (𝐹‘𝑧))
136	20, 135	seqfeq 13962	. . . . 5 ⊢ (𝜑 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑀( · , 𝐹))
137	136	breq1d 5110	. . . 4 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
138	128, 137	bitrd 279	. . 3 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
139	138	iotabidv 6484	. 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥))
140		df-prod 15839	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
141		df-fv 6508	. 2 ⊢ ( ⇝ ‘seq𝑀( · , 𝐹)) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥)
142	139, 140, 141	3eqtr4g 2797	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ⦋csb 3851 ⊆ wss 3903 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 Or wor 5539 ℩cio 6454 –1-1-onto→wf1o 6499 ‘cfv 6500 Isom wiso 6501 (class class class)co 7368 ≈ cen 8892 Fincfn 8895 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11178 ℕcn 12157 ℤcz 12500 ℤ_≥cuz 12763 ...cfz 13435 seqcseq 13936 ♯chash 14265 ⇝ cli 15419 ∏cprod 15838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-prod 15839

This theorem is referenced by: iprod 15873 zprodn0 15874 prodss 15882

W3C validator