Theorem prodmo 16108

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 1167	. . . . . . 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 3100	. . . . . 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 1167	. . . . . . 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 3100	. . . . . 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 6885	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑤 → (ℤ≥‘𝑚) = (ℤ≥‘𝑤))
6	5	sseq2d 3940	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑤)))
7		seqeq1 14150	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
8	7	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑧 ↔ seq𝑤( · , 𝐹) ⇝ 𝑧))
9	6, 8	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))
10	9	cbvrexvw 3241	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑧) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))
11	10	anbi2i 635	. . . . . . . 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 3234	. . . . . . . 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 281	. . . . . . 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 792	. . . . . . . . . . . . 13 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑚( · , 𝐹) ⇝ 𝑥)
15	14	adantl 487	. . . . . . . . . . . 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 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
19		simprll 791	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑚 ∈ ℤ)
20		simprlr 792	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑤 ∈ ℤ)
21		simprll 791	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ≥‘𝑚))
22	21	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝐴 ⊆ (ℤ≥‘𝑚))
23		simprrl 793	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → 𝐴 ⊆ (ℤ≥‘𝑤))
24	23	adantl 487	. . . . . . . . . . . . 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 16104	. . . . . . . . . . . 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 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑥)
27		simprrr 794	. . . . . . . . . . . 12 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
28	27	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → seq𝑤( · , 𝐹) ⇝ 𝑧)
29		climuni 15724	. . . . . . . . . . 11 ⊢ ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ 𝑧) → 𝑥 = 𝑧)
30	26, 28, 29	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)))) → 𝑥 = 𝑧)
31	30	expcom 419	. . . . . . . . 9 ⊢ (((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) ∧ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧))) → (𝜑 → 𝑥 = 𝑧))
32	31	ex 418	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧)))
33	32	rexlimivv 3204	. . . . . . 7 ⊢ (∃𝑚 ∈ ℤ ∃𝑤 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑧)) → (𝜑 → 𝑥 = 𝑧))
34	13, 33	sylbi 220	. . . . . 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 608	. . . . 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 16107	. . . . . . . . 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 2049	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑥 = 𝑧)
39	37, 38	syl6 36	. . . . . . . 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 459	. . . . . . 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 33	. . . . . 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 464	. . . . 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 16107	. . . . . . 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 459	. . . . . 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 33	. . . . 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 3234	. . . . . . . 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 1987	. . . . . . . . 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 3138	. . . . . . . 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 7428	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑤 → (1...𝑚) = (1...𝑤))
50	49	f1oeq2d 6820	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑤 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑤)–1-1-onto→𝐴))
51		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑤 → (seq1( · , 𝐺)‘𝑚) = (seq1( · , 𝐺)‘𝑤))
52	51	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑤 → (𝑧 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑤)))
53	50, 52	anbi12d 644	. . . . . . . . . . . 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 1953	. . . . . . . . . . 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 6812	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓:(1...𝑤)–1-1-onto→𝐴 ↔ 𝑔:(1...𝑤)–1-1-onto→𝐴))
56		fveq1 6884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑗) = (𝑔‘𝑗))
57	56	csbeq1d 3850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑔 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
58	57	mpteq2dv 5198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))
59	36, 58	eqtrid 2807	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))
60	59	seqeq3d 14155	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → seq1( · , 𝐺) = seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)))
61	60	fveq1d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (seq1( · , 𝐺)‘𝑤) = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤))
62	61	eqeq2d 2771	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑧 = (seq1( · , 𝐺)‘𝑤) ↔ 𝑧 = (seq1( · , (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵))‘𝑤)))
63	55, 62	anbi12d 644	. . . . . . . . . . . 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 2069	. . . . . . . . . . 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 290	. . . . . . . . . 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 3241	. . . . . . . . 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 635	. . . . . . . 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 306	. . . . . . 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 669	. . . . . . . . . 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 765	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑎 → (𝑓‘𝑗) = (𝑓‘𝑎))
72	71	csbeq1d 3850	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑎 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
73	72	cbvmptv 5208	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵) = (𝑎 ∈ ℕ ↦ ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
74	36, 73	eqtri 2783	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑎 ∈ ℕ ↦ ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
75		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑎 → (𝑔‘𝑗) = (𝑔‘𝑎))
76	75	csbeq1d 3850	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑎 → ⦋(𝑔‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑎) / 𝑘⦌𝐵)
77	76	cbvmptv 5208	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵) = (𝑎 ∈ ℕ ↦ ⦋(𝑔‘𝑎) / 𝑘⦌𝐵)
78		simplr 781	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . 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 16105	. . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 250	. . . . . . . . . . 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 459	. . . . . . . . . 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 245	. . . . . . . . 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 1966	. . . . . . . 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 3219	. . . . . . 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 246	. . . . . 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 33	. . . . 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 1053	. . . 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 33	. . 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 1960	. 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 5106	. . . . . 6 ⊢ (𝑥 = 𝑧 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑚( · , 𝐹) ⇝ 𝑧))
94	93	3anbi3d 1470	. . . . 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 3186	. . . 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 2764	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = (seq1( · , 𝐺)‘𝑚) ↔ 𝑧 = (seq1( · , 𝐺)‘𝑚)))
97	96	anbi2d 642	. . . . . 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 1953	. . . . 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 3186	. . . 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 932	. . 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 2591	. 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 237	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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562   ≠ wne 2955  ∃wrex 3086  ⦋csb 3846   ⊆ wss 3895  ifcif 4481   class class class wbr 5102   ↦ cmpt 5185  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  ℂcc 11201  0cc0 11203  1c1 11204   · cmul 11208  ℕcn 12338  ℤcz 12696  ℤ_≥cuz 12968  ...cfz 13642  seqcseq 14147   ⇝ cli 15656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-sup 9437  df-oi 9507  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-z 12697  df-uz 12969  df-rp 13124  df-fz 13643  df-fzo 13792  df-seq 14148  df-exp 14208  df-hash 14478  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-clim 15660

This theorem is used by:  fprod  16113