Theorem enq0tr 6624

Description: The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6625. (Contributed by Jim Kingdon, 14-Nov-2019.)

Assertion

Ref	Expression
enq0tr	⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ)

Proof of Theorem enq0tr

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑠 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
2		vex 2604	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
3		eleq1 2141	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
4	3	anbi1d 452	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5		eqeq1 2087	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
6	5	anbi1d 452	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
7	6	anbi1d 452	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
8	7	4exbidv 1791	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → (∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
9	4, 8	anbi12d 456	. . . . . . . . . 10 ⊢ (𝑥 = 𝑓 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))))
10		eleq1 2141	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
11	10	anbi2d 451	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12		eqeq1 2087	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
13	12	anbi2d 451	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
14	13	anbi1d 452	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
15	14	4exbidv 1791	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
16	11, 15	anbi12d 456	. . . . . . . . . 10 ⊢ (𝑦 = 𝑔 → (((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))))
17		df-enq0 6614	. . . . . . . . . 10 ⊢ ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)))}
18	1, 2, 9, 16, 17	brab 4027	. . . . . . . . 9 ⊢ (𝑓 ~Q0 𝑔 ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))))
19		vex 2604	. . . . . . . . . 10 ⊢ ℎ ∈ V
20		eleq1 2141	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
21	20	anbi1d 452	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
22		eqeq1 2087	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑔 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑔 = ⟨𝑎, 𝑏⟩))
23	22	anbi1d 452	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑔 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩)))
24	23	anbi1d 452	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑔 → (((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)) ↔ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))))
25	24	4exbidv 1791	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → (∃𝑎∃𝑏∃𝑠∃𝑡((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)) ↔ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))))
26	21, 25	anbi12d 456	. . . . . . . . . 10 ⊢ (𝑥 = 𝑔 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
27		eleq1 2141	. . . . . . . . . . . 12 ⊢ (𝑦 = ℎ → (𝑦 ∈ (ω × N) ↔ ℎ ∈ (ω × N)))
28	27	anbi2d 451	. . . . . . . . . . 11 ⊢ (𝑦 = ℎ → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
29		eqeq1 2087	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (𝑦 = ⟨𝑠, 𝑡⟩ ↔ ℎ = ⟨𝑠, 𝑡⟩))
30	29	anbi2d 451	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩)))
31	30	anbi1d 452	. . . . . . . . . . . 12 ⊢ (𝑦 = ℎ → (((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)) ↔ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))))
32	31	4exbidv 1791	. . . . . . . . . . 11 ⊢ (𝑦 = ℎ → (∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)) ↔ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))))
33	28, 32	anbi12d 456	. . . . . . . . . 10 ⊢ (𝑦 = ℎ → (((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) ↔ ((𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
34		df-enq0 6614	. . . . . . . . . 10 ⊢ ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))}
35	2, 19, 26, 33, 34	brab 4027	. . . . . . . . 9 ⊢ (𝑔 ~Q0 ℎ ↔ ((𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))))
36	18, 35	anbi12i 447	. . . . . . . 8 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) ↔ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ ((𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
37	36	biimpi 118	. . . . . . 7 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ ((𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
38		an4 550	. . . . . . 7 ⊢ ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))) ∧ ((𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))) ∧ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
39	37, 38	sylib 120	. . . . . 6 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))) ∧ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
40		3anass 923	. . . . . . . 8 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
41		anass 393	. . . . . . . . 9 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))))
42		anass 393	. . . . . . . . . 10 ⊢ (((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ℎ ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
43	42	anbi2i 444	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ω × N) ∧ ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ℎ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))))
44		anidm 388	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ↔ 𝑔 ∈ (ω × N))
45	44	anbi1i 445	. . . . . . . . . 10 ⊢ (((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ℎ ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))
46	45	anbi2i 444	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ω × N) ∧ ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ℎ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
47	41, 43, 46	3bitr2i 206	. . . . . . . 8 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
48	40, 47	bitr4i 185	. . . . . . 7 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
49	48	anbi1i 445	. . . . . 6 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))) ∧ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
50	39, 49	sylibr 132	. . . . 5 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
51		ee8anv 1851	. . . . . 6 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) ↔ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))))
52	51	anbi2i 444	. . . . 5 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
53	50, 52	sylibr 132	. . . 4 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
54		19.42vvvv 1831	. . . . . . 7 ⊢ (∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
55	54	2exbii 1537	. . . . . 6 ⊢ (∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ ∃𝑣∃𝑢((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
56	55	2exbii 1537	. . . . 5 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
57		19.42vvvv 1831	. . . . 5 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
58	56, 57	bitri 182	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡(((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
59	53, 58	sylibr 132	. . 3 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → ∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))))
60		3simpb 936	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) → (𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))
61	60	adantr 270	. . . . . . . 8 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))
62		simplll 499	. . . . . . . . . 10 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → 𝑓 = ⟨𝑧, 𝑤⟩)
63		simprlr 504	. . . . . . . . . 10 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → ℎ = ⟨𝑠, 𝑡⟩)
64	62, 63	jca 300	. . . . . . . . 9 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → (𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩))
65	64	adantl 271	. . . . . . . 8 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩))
66		oveq1 5539	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ∅ → (𝑣 ·𝑜 𝑡) = (∅ ·𝑜 𝑡))
67	63	adantl 271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ℎ = ⟨𝑠, 𝑡⟩)
68		simpl3 943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ℎ ∈ (ω × N))
69	67, 68	eqeltrrd 2156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ⟨𝑠, 𝑡⟩ ∈ (ω × N))
70		opelxp 4392	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑠, 𝑡⟩ ∈ (ω × N) ↔ (𝑠 ∈ ω ∧ 𝑡 ∈ N))
71	69, 70	sylib 120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑠 ∈ ω ∧ 𝑡 ∈ N))
72	71	simprd 112	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑡 ∈ N)
73		pinn 6499	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ N → 𝑡 ∈ ω)
74	72, 73	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑡 ∈ ω)
75		nnm0r 6081	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ ω → (∅ ·𝑜 𝑡) = ∅)
76	74, 75	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (∅ ·𝑜 𝑡) = ∅)
77	76	eqeq2d 2092	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑣 ·𝑜 𝑡) = (∅ ·𝑜 𝑡) ↔ (𝑣 ·𝑜 𝑡) = ∅))
78	66, 77	syl5ib 152	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑣 ·𝑜 𝑡) = ∅))
79		simprr 498	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))
80		eqtr2 2099	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
81		vex 2604	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑣 ∈ V
82		vex 2604	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑢 ∈ V
83	81, 82	opth 3992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑣 = 𝑎 ∧ 𝑢 = 𝑏))
84	80, 83	sylib 120	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → (𝑣 = 𝑎 ∧ 𝑢 = 𝑏))
85		oveq1 5539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑎 → (𝑣 ·𝑜 𝑡) = (𝑎 ·𝑜 𝑡))
86		oveq1 5539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑏 → (𝑢 ·𝑜 𝑠) = (𝑏 ·𝑜 𝑠))
87	85, 86	eqeqan12d 2096	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ((𝑣 ·𝑜 𝑡) = (𝑢 ·𝑜 𝑠) ↔ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))
88	84, 87	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → ((𝑣 ·𝑜 𝑡) = (𝑢 ·𝑜 𝑠) ↔ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))
89	88	ad2ant2lr 493	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩)) → ((𝑣 ·𝑜 𝑡) = (𝑢 ·𝑜 𝑠) ↔ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))
90	89	ad2ant2r 492	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → ((𝑣 ·𝑜 𝑡) = (𝑢 ·𝑜 𝑠) ↔ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))
91	79, 90	mpbird 165	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → (𝑣 ·𝑜 𝑡) = (𝑢 ·𝑜 𝑠))
92	91	eqeq1d 2089	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → ((𝑣 ·𝑜 𝑡) = ∅ ↔ (𝑢 ·𝑜 𝑠) = ∅))
93	92	adantl 271	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑣 ·𝑜 𝑡) = ∅ ↔ (𝑢 ·𝑜 𝑠) = ∅))
94	78, 93	sylibd 147	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑢 ·𝑜 𝑠) = ∅))
95		simpllr 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → 𝑔 = ⟨𝑣, 𝑢⟩)
96	95	adantl 271	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑔 = ⟨𝑣, 𝑢⟩)
97		simpl2 942	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑔 ∈ (ω × N))
98	96, 97	eqeltrrd 2156	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ⟨𝑣, 𝑢⟩ ∈ (ω × N))
99		opelxp 4392	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑣, 𝑢⟩ ∈ (ω × N) ↔ (𝑣 ∈ ω ∧ 𝑢 ∈ N))
100	98, 99	sylib 120	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 ∈ ω ∧ 𝑢 ∈ N))
101	100	simprd 112	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑢 ∈ N)
102		pinn 6499	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ N → 𝑢 ∈ ω)
103	101, 102	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑢 ∈ ω)
104	71	simpld 110	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑠 ∈ ω)
105		nnm00 6125	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ω ∧ 𝑠 ∈ ω) → ((𝑢 ·𝑜 𝑠) = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
106	103, 104, 105	syl2anc 403	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑢 ·𝑜 𝑠) = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
107	94, 106	sylibd 147	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑢 = ∅ ∨ 𝑠 = ∅)))
108		elni2 6504	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ N ↔ (𝑢 ∈ ω ∧ ∅ ∈ 𝑢))
109	108	simprbi 269	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ N → ∅ ∈ 𝑢)
110	101, 109	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ∅ ∈ 𝑢)
111		n0i 3256	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ 𝑢 → ¬ 𝑢 = ∅)
112		biorf 695	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑢 = ∅ → (𝑠 = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
113	110, 111, 112	3syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑠 = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
114	107, 113	sylibrd 167	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → 𝑠 = ∅))
115	62	adantl 271	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑓 = ⟨𝑧, 𝑤⟩)
116		simpl1 941	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑓 ∈ (ω × N))
117	115, 116	eqeltrrd 2156	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ⟨𝑧, 𝑤⟩ ∈ (ω × N))
118		opelxp 4392	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ (ω × N) ↔ (𝑧 ∈ ω ∧ 𝑤 ∈ N))
119	117, 118	sylib 120	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ∈ ω ∧ 𝑤 ∈ N))
120	119	simprd 112	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑤 ∈ N)
121		pinn 6499	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ N → 𝑤 ∈ ω)
122	120, 121	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑤 ∈ ω)
123		nnm0 6077	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ω → (𝑤 ·𝑜 ∅) = ∅)
124	122, 123	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑤 ·𝑜 ∅) = ∅)
125		oveq2 5540	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ∅ → (𝑤 ·𝑜 𝑠) = (𝑤 ·𝑜 ∅))
126	125	eqeq1d 2089	. . . . . . . . . . . . 13 ⊢ (𝑠 = ∅ → ((𝑤 ·𝑜 𝑠) = ∅ ↔ (𝑤 ·𝑜 ∅) = ∅))
127	124, 126	syl5ibrcom 155	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑠 = ∅ → (𝑤 ·𝑜 𝑠) = ∅))
128	114, 127	syld 44	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑤 ·𝑜 𝑠) = ∅))
129		oveq2 5540	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ∅ → (𝑤 ·𝑜 𝑣) = (𝑤 ·𝑜 ∅))
130	124	eqeq2d 2092	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑤 ·𝑜 𝑣) = (𝑤 ·𝑜 ∅) ↔ (𝑤 ·𝑜 𝑣) = ∅))
131	129, 130	syl5ib 152	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑤 ·𝑜 𝑣) = ∅))
132		simprlr 504	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
133	132	eqeq1d 2089	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑧 ·𝑜 𝑢) = ∅ ↔ (𝑤 ·𝑜 𝑣) = ∅))
134	131, 133	sylibrd 167	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑧 ·𝑜 𝑢) = ∅))
135	119	simpld 110	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑧 ∈ ω)
136		nnm00 6125	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → ((𝑧 ·𝑜 𝑢) = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
137	135, 103, 136	syl2anc 403	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑧 ·𝑜 𝑢) = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
138	134, 137	sylibd 147	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑧 = ∅ ∨ 𝑢 = ∅)))
139		biorf 695	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑢 = ∅ → (𝑧 = ∅ ↔ (𝑢 = ∅ ∨ 𝑧 = ∅)))
140		orcom 679	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = ∅ ∨ 𝑧 = ∅) ↔ (𝑧 = ∅ ∨ 𝑢 = ∅))
141	139, 140	syl6bb 194	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑢 = ∅ → (𝑧 = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
142	110, 111, 141	3syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
143	138, 142	sylibrd 167	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → 𝑧 = ∅))
144		oveq1 5539	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝑧 ·𝑜 𝑡) = (∅ ·𝑜 𝑡))
145	144	eqeq1d 2089	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ((𝑧 ·𝑜 𝑡) = ∅ ↔ (∅ ·𝑜 𝑡) = ∅))
146	76, 145	syl5ibrcom 155	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 = ∅ → (𝑧 ·𝑜 𝑡) = ∅))
147	143, 146	syld 44	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑧 ·𝑜 𝑡) = ∅))
148	128, 147	jcad 301	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → ((𝑤 ·𝑜 𝑠) = ∅ ∧ (𝑧 ·𝑜 𝑡) = ∅)))
149		eqtr3 2100	. . . . . . . . . . 11 ⊢ (((𝑤 ·𝑜 𝑠) = ∅ ∧ (𝑧 ·𝑜 𝑡) = ∅) → (𝑤 ·𝑜 𝑠) = (𝑧 ·𝑜 𝑡))
150	149	eqcomd 2086	. . . . . . . . . 10 ⊢ (((𝑤 ·𝑜 𝑠) = ∅ ∧ (𝑧 ·𝑜 𝑡) = ∅) → (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))
151	148, 150	syl6 33	. . . . . . . . 9 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ → (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))
152		simplr 496	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣))
153	91, 152	oveq12d 5550	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠))) → ((𝑣 ·𝑜 𝑡) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑢 ·𝑜 𝑠) ·𝑜 (𝑤 ·𝑜 𝑣)))
154	153	adantl 271	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑣 ·𝑜 𝑡) ·𝑜 (𝑧 ·𝑜 𝑢)) = ((𝑢 ·𝑜 𝑠) ·𝑜 (𝑤 ·𝑜 𝑣)))
155	100	simpld 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → 𝑣 ∈ ω)
156		nnmcl 6083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ω ∧ 𝑡 ∈ ω) → (𝑣 ·𝑜 𝑡) ∈ ω)
157	155, 74, 156	syl2anc 403	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 ·𝑜 𝑡) ∈ ω)
158		nnmcom 6091	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ ω) → (𝑐 ·𝑜 𝑑) = (𝑑 ·𝑜 𝑐))
159	158	adantl 271	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ ω)) → (𝑐 ·𝑜 𝑑) = (𝑑 ·𝑜 𝑐))
160		nnmass 6089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ ω ∧ 𝑒 ∈ ω) → ((𝑐 ·𝑜 𝑑) ·𝑜 𝑒) = (𝑐 ·𝑜 (𝑑 ·𝑜 𝑒)))
161	160	adantl 271	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ ω ∧ 𝑒 ∈ ω)) → ((𝑐 ·𝑜 𝑑) ·𝑜 𝑒) = (𝑐 ·𝑜 (𝑑 ·𝑜 𝑒)))
162	157, 135, 103, 159, 161	caov13d 5704	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑣 ·𝑜 𝑡) ·𝑜 (𝑧 ·𝑜 𝑢)) = (𝑢 ·𝑜 (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡))))
163		nnmcl 6083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·𝑜 𝑣) ∈ ω)
164	122, 155, 163	syl2anc 403	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑤 ·𝑜 𝑣) ∈ ω)
165	161, 103, 104, 164	caovassd 5680	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑢 ·𝑜 𝑠) ·𝑜 (𝑤 ·𝑜 𝑣)) = (𝑢 ·𝑜 (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣))))
166	154, 162, 165	3eqtr3d 2121	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑢 ·𝑜 (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡))) = (𝑢 ·𝑜 (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣))))
167		nnmcl 6083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ω ∧ (𝑣 ·𝑜 𝑡) ∈ ω) → (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) ∈ ω)
168	135, 157, 167	syl2anc 403	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) ∈ ω)
169		nnmcl 6083	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ω ∧ (𝑤 ·𝑜 𝑣) ∈ ω) → (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
170	104, 164, 169	syl2anc 403	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω)
171		nnmcan 6115	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ ω ∧ (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) ∈ ω ∧ (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣)) ∈ ω) ∧ ∅ ∈ 𝑢) → ((𝑢 ·𝑜 (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡))) = (𝑢 ·𝑜 (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣))) ↔ (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) = (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣))))
172	103, 168, 170, 110, 171	syl31anc 1172	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑢 ·𝑜 (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡))) = (𝑢 ·𝑜 (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣))) ↔ (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) = (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣))))
173	166, 172	mpbid 145	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) = (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣)))
174	135, 155, 74, 159, 161	caov12d 5702	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ·𝑜 (𝑣 ·𝑜 𝑡)) = (𝑣 ·𝑜 (𝑧 ·𝑜 𝑡)))
175	104, 122, 155, 159, 161	caov13d 5704	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑠 ·𝑜 (𝑤 ·𝑜 𝑣)) = (𝑣 ·𝑜 (𝑤 ·𝑜 𝑠)))
176	173, 174, 175	3eqtr3d 2121	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 ·𝑜 (𝑧 ·𝑜 𝑡)) = (𝑣 ·𝑜 (𝑤 ·𝑜 𝑠)))
177	176	adantr 270	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ∧ ∅ ∈ 𝑣) → (𝑣 ·𝑜 (𝑧 ·𝑜 𝑡)) = (𝑣 ·𝑜 (𝑤 ·𝑜 𝑠)))
178		nnmcl 6083	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ω ∧ 𝑡 ∈ ω) → (𝑧 ·𝑜 𝑡) ∈ ω)
179	135, 74, 178	syl2anc 403	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ·𝑜 𝑡) ∈ ω)
180		nnmcl 6083	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ω ∧ 𝑠 ∈ ω) → (𝑤 ·𝑜 𝑠) ∈ ω)
181	122, 104, 180	syl2anc 403	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑤 ·𝑜 𝑠) ∈ ω)
182	155, 179, 181	3jca 1118	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 ∈ ω ∧ (𝑧 ·𝑜 𝑡) ∈ ω ∧ (𝑤 ·𝑜 𝑠) ∈ ω))
183		nnmcan 6115	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ω ∧ (𝑧 ·𝑜 𝑡) ∈ ω ∧ (𝑤 ·𝑜 𝑠) ∈ ω) ∧ ∅ ∈ 𝑣) → ((𝑣 ·𝑜 (𝑧 ·𝑜 𝑡)) = (𝑣 ·𝑜 (𝑤 ·𝑜 𝑠)) ↔ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))
184	182, 183	sylan 277	. . . . . . . . . . 11 ⊢ ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ∧ ∅ ∈ 𝑣) → ((𝑣 ·𝑜 (𝑧 ·𝑜 𝑡)) = (𝑣 ·𝑜 (𝑤 ·𝑜 𝑠)) ↔ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))
185	177, 184	mpbid 145	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) ∧ ∅ ∈ 𝑣) → (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))
186	185	ex 113	. . . . . . . . 9 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (∅ ∈ 𝑣 → (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))
187		0elnn 4358	. . . . . . . . . 10 ⊢ (𝑣 ∈ ω → (𝑣 = ∅ ∨ ∅ ∈ 𝑣))
188	155, 187	syl 14	. . . . . . . . 9 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑣 = ∅ ∨ ∅ ∈ 𝑣))
189	151, 186, 188	mpjaod 670	. . . . . . . 8 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))
190	61, 65, 189	jca32 303	. . . . . . 7 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
191	190	2eximi 1532	. . . . . 6 ⊢ (∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
192	191	exlimivv 1817	. . . . 5 ⊢ (∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
193	192	exlimivv 1817	. . . 4 ⊢ (∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
194	193	2eximi 1532	. . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑎∃𝑏∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·𝑜 𝑢) = (𝑤 ·𝑜 𝑣)) ∧ ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑎 ·𝑜 𝑡) = (𝑏 ·𝑜 𝑠)))) → ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
195	59, 194	syl 14	. 2 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
196		19.42vvvv 1831	. . 3 ⊢ (∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))) ↔ ((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
197	5	anbi1d 452	. . . . . . 7 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩)))
198	197	anbi1d 452	. . . . . 6 ⊢ (𝑥 = 𝑓 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
199	198	4exbidv 1791	. . . . 5 ⊢ (𝑥 = 𝑓 → (∃𝑧∃𝑤∃𝑠∃𝑡((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)) ↔ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
200	4, 199	anbi12d 456	. . . 4 ⊢ (𝑥 = 𝑓 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))))
201	27	anbi2d 451	. . . . 5 ⊢ (𝑦 = ℎ → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
202	29	anbi2d 451	. . . . . . 7 ⊢ (𝑦 = ℎ → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩)))
203	202	anbi1d 452	. . . . . 6 ⊢ (𝑦 = ℎ → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)) ↔ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
204	203	4exbidv 1791	. . . . 5 ⊢ (𝑦 = ℎ → (∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)) ↔ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
205	201, 204	anbi12d 456	. . . 4 ⊢ (𝑦 = ℎ → (((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))) ↔ ((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))))
206		df-enq0 6614	. . . 4 ⊢ ~Q0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠)))}
207	1, 19, 200, 205, 206	brab 4027	. . 3 ⊢ (𝑓 ~Q0 ℎ ↔ ((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))))
208	196, 207	bitr4i 185	. 2 ⊢ (∃𝑧∃𝑤∃𝑠∃𝑡((𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩) ∧ (𝑧 ·𝑜 𝑡) = (𝑤 ·𝑜 𝑠))) ↔ 𝑓 ~Q0 ℎ)
209	195, 208	sylib 120	1 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∅c0 3251  ⟨cop 3401   class class class wbr 3785  ωcom 4331   × cxp 4361  (class class class)co 5532   ·_𝑜 comu 6022  Ncnpi 6462   ~_Q0 ceq0 6476

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-ni 6494  df-enq0 6614

This theorem is referenced by:  enq0er  6625