Theorem prodmo 15892

Description: A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypotheses

Ref	Expression
prodmo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
prodmo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
prodmo.3	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)

Assertion

Ref	Expression
prodmo	⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))))

Distinct variable groups:   𝐴,𝑘,𝑛   𝑘,𝐹,𝑛   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚   𝐵,𝑓,𝑗,𝑚   𝑗,𝐹   𝜑,𝑗,𝑥,𝑚   𝑗,𝐺,𝑥   𝑥,𝑛   𝑓,𝑘,𝑗,𝑚,𝜑,𝑥   𝑥,𝑦   𝑥,𝐴   𝑓,𝐹,𝑚,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦,𝑘,𝑛)   𝐹(𝑦)   𝐺(𝑦,𝑓,𝑘,𝑚,𝑛)

Proof of Theorem prodmo

Dummy variables 𝑎 𝑔 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpb 1150	. . . . . . 7 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
2	1	reximi 3076	. . . . . 6 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
3		3simpb 1150	. . . . . . 7 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) → (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧))
4	3	reximi 3076	. . . . . 6 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧))
5		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑤 → (ℤ≥‘𝑚) = (ℤ≥‘𝑤))
6	5	sseq2d 3955	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑤)))
7		seqeq1 13957	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
8	7	breq1d 5096	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑧 ↔ seq𝑤( · , 𝐹) ⇝ 𝑧))
9	6, 8	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
10	9	cbvrexvw 3217	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))
11	10	anbi2i 624	. . . . . . . 8 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
12		reeanv 3210	. . . . . . . 8 ⊢ (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
13	11, 12	bitr4i 278	. . . . . . 7 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) ↔ ∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
14		simprlr 780	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
16		prodmo.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
17		prodmo.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
18	17	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
19		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑚 ∈ ℤ)
20		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑤 ∈ ℤ)
21		simprll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ≥‘𝑚))
22	21	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ≥‘𝑚))
23		simprrl 781	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ≥‘𝑤))
24	23	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ≥‘𝑤))
25	16, 18, 19, 20, 22, 24	prodrb 15888	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
26	15, 25	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑥)
27		simprrr 782	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
28	27	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
29		climuni 15505	. . . . . . . . . . 11 ⊢ ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ 𝑧) → 𝑥 = 𝑧)
30	26, 28, 29	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑥 = 𝑧)
31	30	expcom 413	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → (𝜑 → 𝑥 = 𝑧))
32	31	ex 412	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧)))
33	32	rexlimivv 3180	. . . . . . 7 ⊢ (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧))
34	13, 33	sylbi 217	. . . . . 6 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧))
35	2, 4, 34	syl2an 597	. . . . 5 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧))
36		prodmo.3	. . . . . . . . . 10 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
37	16, 17, 36	prodmolem2 15891	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) → 𝑧 = 𝑥))
38		equcomi 2019	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑥 = 𝑧)
39	37, 38	syl6 35	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
40	39	expimpd 453	. . . . . . 7 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
41	40	com12 32	. . . . . 6 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) → (𝜑 → 𝑥 = 𝑧))
42	41	ancoms 458	. . . . 5 ⊢ ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧))
43	16, 17, 36	prodmolem2 15891	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
44	43	expimpd 453	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
45	44	com12 32	. . . . 5 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → (𝜑 → 𝑥 = 𝑧))
46		reeanv 3210	. . . . . . . 8 ⊢ (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑤 ∈ ℕ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
47		exdistrv 1957	. . . . . . . . 9 ⊢ (∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) ↔ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
48	47	2rexbii 3114	. . . . . . . 8 ⊢ (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ ∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) ↔ ∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
49		oveq2 7368	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑤 → (1...𝑚) = (1...𝑤))
50	49	f1oeq2d 6770	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑤 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑤)–1-1-onto→𝐴))
51		fveq2 6834	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑤 → (seq1( · , 𝐺)‘𝑚) = (seq1( · , 𝐺)‘𝑤))
52	51	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑤 → (𝑧 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑤)))
53	50, 52	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑤 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑤))))
54	53	exbidv 1923	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑤))))
55		f1oeq1 6762	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓:(1...𝑤)–1-1-onto→𝐴 ↔ 𝑔:(1...𝑤)–1-1-onto→𝐴))
56		fveq1 6833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑗) = (𝑔‘𝑗))
57	56	csbeq1d 3842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑔 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
58	57	mpteq2dv 5180	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))
59	36, 58	eqtrid 2784	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))
60	59	seqeq3d 13962	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → seq1( · , 𝐺) = seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)))
61	60	fveq1d 6836	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (seq1( · , 𝐺)‘𝑤) = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))
62	61	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑧 = (seq1( · , 𝐺)‘𝑤) ↔ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)))
63	55, 62	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑤)) ↔ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
64	63	cbvexvw 2039	. . . . . . . . . . 11 ⊢ (∃𝑓(𝑓:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑤)) ↔ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)))
65	54, 64	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
66	65	cbvrexvw 3217	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑤 ∈ ℕ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)))
67	66	anbi2i 624	. . . . . . . 8 ⊢ ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑤 ∈ ℕ ∃𝑔(𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
68	46, 48, 67	3bitr4i 303	. . . . . . 7 ⊢ (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ ∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))))
69		an4 657	. . . . . . . . . 10 ⊢ (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) ↔ ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴) ∧ (𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))))
70	17	ad4ant14 753	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
71		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑎 → (𝑓‘𝑗) = (𝑓‘𝑎))
72	71	csbeq1d 3842	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑎 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
73	72	cbvmptv 5190	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵) = (𝑎 ∈ ℕ ↦ ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
74	36, 73	eqtri 2760	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑎 ∈ ℕ ↦ ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
75		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑎 → (𝑔‘𝑗) = (𝑔‘𝑎))
76	75	csbeq1d 3842	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑎 → ⦋(𝑔‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑎) / 𝑘⦌𝐵)
77	76	cbvmptv 5190	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵) = (𝑎 ∈ ℕ ↦ ⦋(𝑔‘𝑎) / 𝑘⦌𝐵)
78		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴)) → (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ))
79		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
80		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴)) → 𝑔:(1...𝑤)–1-1-onto→𝐴)
81	16, 70, 74, 77, 78, 79, 80	prodmolem3 15889	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴)) → (seq1( · , 𝐺)‘𝑚) = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))
82		eqeq12 2754	. . . . . . . . . . . 12 ⊢ ((𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)) → (𝑥 = 𝑧 ↔ (seq1( · , 𝐺)‘𝑚) = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)))
83	81, 82	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴)) → ((𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)) → 𝑥 = 𝑧))
84	83	expimpd 453	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑤)–1-1-onto→𝐴) ∧ (𝑥 = (seq1( · , 𝐺)‘𝑚) ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) → 𝑥 = 𝑧))
85	69, 84	biimtrid 242	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) → 𝑥 = 𝑧))
86	85	exlimdvv 1936	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) → 𝑥 = 𝑧))
87	86	rexlimdvva 3195	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑤 ∈ ℕ ∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑤)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))) → 𝑥 = 𝑧))
88	68, 87	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
89	88	com12 32	. . . . 5 ⊢ ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → (𝜑 → 𝑥 = 𝑧))
90	35, 42, 45, 89	ccase 1038	. . . 4 ⊢ (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)))) → (𝜑 → 𝑥 = 𝑧))
91	90	com12 32	. . 3 ⊢ (𝜑 → (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)))) → 𝑥 = 𝑧))
92	91	alrimivv 1930	. 2 ⊢ (𝜑 → ∀𝑥∀𝑧(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)))) → 𝑥 = 𝑧))
93		breq2 5090	. . . . . 6 ⊢ (𝑥 = 𝑧 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑚( · , 𝐹) ⇝ 𝑧))
94	93	3anbi3d 1445	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)))
95	94	rexbidv 3162	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧)))
96		eqeq1 2741	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑚)))
97	96	anbi2d 631	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))))
98	97	exbidv 1923	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))))
99	98	rexbidv 3162	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))))
100	95, 99	orbi12d 919	. . 3 ⊢ (𝑥 = 𝑧 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)))))
101	100	mo4 2567	. 2 ⊢ (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) ↔ ∀𝑥∀𝑧(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)))) → 𝑥 = 𝑧))
102	92, 101	sylibr 234	1 ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , 𝐺)‘𝑚))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538   ≠ wne 2933  ∃wrex 3062  ⦋csb 3838   ⊆ wss 3890  ifcif 4467   class class class wbr 5086   ↦ cmpt 5167  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  0cc0 11029  1c1 11030   · cmul 11034  ℕcn 12165  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  seqcseq 13954   ⇝ cli 15437

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441

This theorem is referenced by:  fprod  15897