Theorem enq0tr 7375

Description: The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7376. (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 2729	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
2		vex 2729	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
3		eleq1 2229	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
4	3	anbi1d 461	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5		eqeq1 2172	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
6	5	anbi1d 461	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
7	6	anbi1d 461	. . . . . . . . . . . 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 1858	. . . . . . . . . . 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 465	. . . . . . . . . 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 2229	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
11	10	anbi2d 460	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12		eqeq1 2172	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
13	12	anbi2d 460	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
14	13	anbi1d 461	. . . . . . . . . . . 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 1858	. . . . . . . . . . 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 465	. . . . . . . . . 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 7365	. . . . . . . . . 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 4250	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ ℎ ∈ V
20		eleq1 2229	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
21	20	anbi1d 461	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
22		eqeq1 2172	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑔 → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑔 = ⟨𝑎, 𝑏⟩))
23	22	anbi1d 461	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑔 → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩)))
24	23	anbi1d 461	. . . . . . . . . . . 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 1858	. . . . . . . . . . 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 465	. . . . . . . . . 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 2229	. . . . . . . . . . . 12 ⊢ (𝑦 = ℎ → (𝑦 ∈ (ω × N) ↔ ℎ ∈ (ω × N)))
28	27	anbi2d 460	. . . . . . . . . . 11 ⊢ (𝑦 = ℎ → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
29		eqeq1 2172	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (𝑦 = ⟨𝑠, 𝑡⟩ ↔ ℎ = ⟨𝑠, 𝑡⟩))
30	29	anbi2d 460	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((𝑔 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑔 = ⟨𝑎, 𝑏⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩)))
31	30	anbi1d 461	. . . . . . . . . . . 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 1858	. . . . . . . . . . 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 465	. . . . . . . . . 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 7365	. . . . . . . . . 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 4250	. . . . . . . . 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 456	. . . . . . . 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 119	. . . . . . 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 576	. . . . . . 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 121	. . . . . 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 972	. . . . . . . 8 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
41		anass 399	. . . . . . . . 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 399	. . . . . . . . . 10 ⊢ (((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ℎ ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
43	42	anbi2i 453	. . . . . . . . 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 394	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ↔ 𝑔 ∈ (ω × N))
45	44	anbi1i 454	. . . . . . . . . 10 ⊢ (((𝑔 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ℎ ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))
46	45	anbi2i 453	. . . . . . . . 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 207	. . . . . . . 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 186	. . . . . . 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 454	. . . . . 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 133	. . . . 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 1923	. . . . . 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 453	. . . . 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 133	. . . 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 1901	. . . . . . 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 1594	. . . . . 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 1594	. . . . 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 1901	. . . . 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 183	. . . 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 133	. . 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 985	. . . . . . . . 9 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N) ∧ ℎ ∈ (ω × N)) → (𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N)))
61	60	adantr 274	. . . . . . . 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 523	. . . . . . . . . 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 528	. . . . . . . . . 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 304	. . . . . . . . 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 275	. . . . . . . 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 5849	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ∅ → (𝑣 ·o 𝑡) = (∅ ·o 𝑡))
67	63	adantl 275	. . . . . . . . . . . . . . . . . . . . . 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 992	. . . . . . . . . . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . . . . . . . . . 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 4634	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑠, 𝑡⟩ ∈ (ω × N) ↔ (𝑠 ∈ ω ∧ 𝑡 ∈ N))
71	69, 70	sylib 121	. . . . . . . . . . . . . . . . . . . 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 113	. . . . . . . . . . . . . . . . . . 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 7250	. . . . . . . . . . . . . . . . . . 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 6447	. . . . . . . . . . . . . . . . . 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 2177	. . . . . . . . . . . . . . . 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 153	. . . . . . . . . . . . . . 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 522	. . . . . . . . . . . . . . . . . 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 2184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
81		vex 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑣 ∈ V
82		vex 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑢 ∈ V
83	81, 82	opth 4215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩ ↔ (𝑣 = 𝑎 ∧ 𝑢 = 𝑏))
84	80, 83	sylib 121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 = ⟨𝑣, 𝑢⟩ ∧ 𝑔 = ⟨𝑎, 𝑏⟩) → (𝑣 = 𝑎 ∧ 𝑢 = 𝑏))
85		oveq1 5849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑎 → (𝑣 ·o 𝑡) = (𝑎 ·o 𝑡))
86		oveq1 5849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑏 → (𝑢 ·o 𝑠) = (𝑏 ·o 𝑠))
87	85, 86	eqeqan12d 2181	. . . . . . . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . 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 166	. . . . . . . . . . . . . . . . 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 2174	. . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . 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 148	. . . . . . . . . . . . . 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 524	. . . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . . 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 991	. . . . . . . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . . . . . . 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 4634	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑣, 𝑢⟩ ∈ (ω × N) ↔ (𝑣 ∈ ω ∧ 𝑢 ∈ N))
100	98, 99	sylib 121	. . . . . . . . . . . . . . . . 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 113	. . . . . . . . . . . . . . . 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 7250	. . . . . . . . . . . . . . . 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 111	. . . . . . . . . . . . . . 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 6497	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ ω ∧ 𝑠 ∈ ω) → ((𝑢 ·o 𝑠) = ∅ ↔ (𝑢 = ∅ ∨ 𝑠 = ∅)))
106	103, 104, 105	syl2anc 409	. . . . . . . . . . . . . 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 148	. . . . . . . . . . . . 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 7255	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ N ↔ (𝑢 ∈ ω ∧ ∅ ∈ 𝑢))
109	108	simprbi 273	. . . . . . . . . . . . . . 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 3414	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ 𝑢 → ¬ 𝑢 = ∅)
112		biorf 734	. . . . . . . . . . . . . 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 168	. . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . 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 990	. . . . . . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . . . . . 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 4634	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑤⟩ ∈ (ω × N) ↔ (𝑧 ∈ ω ∧ 𝑤 ∈ N))
119	117, 118	sylib 121	. . . . . . . . . . . . . . . 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 113	. . . . . . . . . . . . . . 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 7250	. . . . . . . . . . . . . . 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 6443	. . . . . . . . . . . . . 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 5850	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ∅ → (𝑤 ·o 𝑠) = (𝑤 ·o ∅))
126	125	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑠 = ∅ → ((𝑤 ·o 𝑠) = ∅ ↔ (𝑤 ·o ∅) = ∅))
127	124, 126	syl5ibrcom 156	. . . . . . . . . . . 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 45	. . . . . . . . . . 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 5850	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = ∅ → (𝑤 ·o 𝑣) = (𝑤 ·o ∅))
130	124	eqeq2d 2177	. . . . . . . . . . . . . . . 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 153	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . 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 2174	. . . . . . . . . . . . . . 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 168	. . . . . . . . . . . . . 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 111	. . . . . . . . . . . . . . 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 6497	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ω ∧ 𝑢 ∈ ω) → ((𝑧 ·o 𝑢) = ∅ ↔ (𝑧 = ∅ ∨ 𝑢 = ∅)))
137	135, 103, 136	syl2anc 409	. . . . . . . . . . . . . 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 148	. . . . . . . . . . . . 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 734	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑢 = ∅ → (𝑧 = ∅ ↔ (𝑢 = ∅ ∨ 𝑧 = ∅)))
140		orcom 718	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 = ∅ ∨ 𝑧 = ∅) ↔ (𝑧 = ∅ ∨ 𝑢 = ∅))
141	139, 140	bitrdi 195	. . . . . . . . . . . . . 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 168	. . . . . . . . . . . 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 5849	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ∅ → (𝑧 ·o 𝑡) = (∅ ·o 𝑡))
145	144	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∅ → ((𝑧 ·o 𝑡) = ∅ ↔ (∅ ·o 𝑡) = ∅))
146	76, 145	syl5ibrcom 156	. . . . . . . . . . . 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 45	. . . . . . . . . . 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 305	. . . . . . . . . 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 2185	. . . . . . . . . . 11 ⊢ (((𝑤 ·o 𝑠) = ∅ ∧ (𝑧 ·o 𝑡) = ∅) → (𝑤 ·o 𝑠) = (𝑧 ·o 𝑡))
150	149	eqcomd 2171	. . . . . . . . . 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 520	. . . . . . . . . . . . . . . . 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 5860	. . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . 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 111	. . . . . . . . . . . . . . . . 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 6449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ω ∧ 𝑡 ∈ ω) → (𝑣 ·o 𝑡) ∈ ω)
157	155, 74, 156	syl2anc 409	. . . . . . . . . . . . . . . 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 6457	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ ω) → (𝑐 ·o 𝑑) = (𝑑 ·o 𝑐))
159	158	adantl 275	. . . . . . . . . . . . . . . 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 6455	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ ω ∧ 𝑒 ∈ ω) → ((𝑐 ·o 𝑑) ·o 𝑒) = (𝑐 ·o (𝑑 ·o 𝑒)))
161	160	adantl 275	. . . . . . . . . . . . . . . 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 6025	. . . . . . . . . . . . . . 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 6449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ω ∧ 𝑣 ∈ ω) → (𝑤 ·o 𝑣) ∈ ω)
164	122, 155, 163	syl2anc 409	. . . . . . . . . . . . . . . 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 6001	. . . . . . . . . . . . . . 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 2206	. . . . . . . . . . . . . 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 6449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ω ∧ (𝑣 ·o 𝑡) ∈ ω) → (𝑧 ·o (𝑣 ·o 𝑡)) ∈ ω)
168	135, 157, 167	syl2anc 409	. . . . . . . . . . . . . . 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 6449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ω ∧ (𝑤 ·o 𝑣) ∈ ω) → (𝑠 ·o (𝑤 ·o 𝑣)) ∈ ω)
170	104, 164, 169	syl2anc 409	. . . . . . . . . . . . . . 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 6487	. . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . 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 146	. . . . . . . . . . . . 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 6023	. . . . . . . . . . . . 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 6025	. . . . . . . . . . . . 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 2206	. . . . . . . . . . . 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 274	. . . . . . . . . . 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 6449	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ω ∧ 𝑡 ∈ ω) → (𝑧 ·o 𝑡) ∈ ω)
179	135, 74, 178	syl2anc 409	. . . . . . . . . . . . 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 6449	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ω ∧ 𝑠 ∈ ω) → (𝑤 ·o 𝑠) ∈ ω)
181	122, 104, 180	syl2anc 409	. . . . . . . . . . . . 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 1167	. . . . . . . . . . . 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 6487	. . . . . . . . . . . 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 281	. . . . . . . . . . 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 146	. . . . . . . . . 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 114	. . . . . . . . 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 4596	. . . . . . . . . 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 708	. . . . . . . 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 308	. . . . . . 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 1589	. . . . . 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 1884	. . . . 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 1884	. . . 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 1589	. . 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 1901	. . 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 461	. . . . . . 7 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩)))
198	197	anbi1d 461	. . . . . 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 1858	. . . . 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 465	. . . 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 460	. . . . 5 ⊢ (𝑦 = ℎ → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ ℎ ∈ (ω × N))))
202	29	anbi2d 460	. . . . . . 7 ⊢ (𝑦 = ℎ → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑠, 𝑡⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ ℎ = ⟨𝑠, 𝑡⟩)))
203	202	anbi1d 461	. . . . . 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 1858	. . . . 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 465	. . . 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 7365	. . . 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 4250	. . 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 186	. 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 121	1 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∅c0 3409  ⟨cop 3579   class class class wbr 3982  ωcom 4567   × cxp 4602  (class class class)co 5842   ·_o comu 6382  Ncnpi 7213   ~_Q0 ceq0 7227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388  df-omul 6389  df-ni 7245  df-enq0 7365

This theorem is referenced by:  enq0er  7376