Theorem enq0sym 7182

Description: The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7185. (Contributed by Jim Kingdon, 14-Nov-2019.)

Assertion

Ref	Expression
enq0sym	⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓)

Proof of Theorem enq0sym

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

Step	Hyp	Ref	Expression
1		vex 2658	. . . . . . . 8 ⊢ 𝑓 ∈ V
2		vex 2658	. . . . . . . 8 ⊢ 𝑔 ∈ V
3		eleq1 2175	. . . . . . . . . 10 ⊢ (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
4	3	anbi1d 458	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5		eqeq1 2119	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
6	5	anbi1d 458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
7	6	anbi1d 458	. . . . . . . . . 10 ⊢ (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 1822	. . . . . . . . 9 ⊢ (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 462	. . . . . . . 8 ⊢ (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 2175	. . . . . . . . . 10 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
11	10	anbi2d 457	. . . . . . . . 9 ⊢ (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12		eqeq1 2119	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
13	12	anbi2d 457	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
14	13	anbi1d 458	. . . . . . . . . 10 ⊢ (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 1822	. . . . . . . . 9 ⊢ (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 462	. . . . . . . 8 ⊢ (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 7174	. . . . . . . 8 ⊢ ~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 4152	. . . . . . 7 ⊢ (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	18	biimpi 119	. . . . . 6 ⊢ (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))))
20		opeq12 3671	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → ⟨𝑧, 𝑤⟩ = ⟨𝑎, 𝑏⟩)
21	20	eqeq2d 2124	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (𝑓 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
22	21	anbi1d 458	. . . . . . . . 9 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
23		simpl 108	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → 𝑧 = 𝑎)
24	23	oveq1d 5741	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (𝑧 ·o 𝑢) = (𝑎 ·o 𝑢))
25		simpr 109	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → 𝑤 = 𝑏)
26	25	oveq1d 5741	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (𝑤 ·o 𝑣) = (𝑏 ·o 𝑣))
27	24, 26	eqeq12d 2127	. . . . . . . . 9 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣)))
28	22, 27	anbi12d 462	. . . . . . . 8 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣))))
29		opeq12 3671	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → ⟨𝑣, 𝑢⟩ = ⟨𝑐, 𝑑⟩)
30	29	eqeq2d 2124	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (𝑔 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
31	30	anbi2d 457	. . . . . . . . 9 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)))
32		simpr 109	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → 𝑢 = 𝑑)
33	32	oveq2d 5742	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (𝑎 ·o 𝑢) = (𝑎 ·o 𝑑))
34		simpl 108	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → 𝑣 = 𝑐)
35	34	oveq2d 5742	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (𝑏 ·o 𝑣) = (𝑏 ·o 𝑐))
36	33, 35	eqeq12d 2127	. . . . . . . . 9 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → ((𝑎 ·o 𝑢) = (𝑏 ·o 𝑣) ↔ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐)))
37	31, 36	anbi12d 462	. . . . . . . 8 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣)) ↔ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
38	28, 37	cbvex4v 1878	. . . . . . 7 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐)))
39	38	anbi2i 450	. . . . . 6 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
40	19, 39	sylib 121	. . . . 5 ⊢ (𝑓 ~Q0 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
41		19.42vv 1861	. . . . 5 ⊢ (∃𝑎∃𝑏((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
42	40, 41	sylibr 133	. . . 4 ⊢ (𝑓 ~Q0 𝑔 → ∃𝑎∃𝑏((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
43		19.42vv 1861	. . . . 5 ⊢ (∃𝑐∃𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) ↔ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
44	43	2exbii 1566	. . . 4 ⊢ (∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) ↔ ∃𝑎∃𝑏((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ∃𝑐∃𝑑((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
45	42, 44	sylibr 133	. . 3 ⊢ (𝑓 ~Q0 𝑔 → ∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))))
46		pm3.22 263	. . . . . . 7 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)))
47	46	adantr 272	. . . . . 6 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)))
48		pm3.22 263	. . . . . . 7 ⊢ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩))
49	48	ad2antrl 479	. . . . . 6 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩))
50		simprr 504	. . . . . . . 8 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))
51		eleq1 2175	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ ⟨𝑎, 𝑏⟩ ∈ (ω × N)))
52		opelxp 4527	. . . . . . . . . . . . . 14 ⊢ (⟨𝑎, 𝑏⟩ ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ N))
53	51, 52	syl6bb 195	. . . . . . . . . . . . 13 ⊢ (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ N)))
54	53	biimpcd 158	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ω × N) → (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑎 ∈ ω ∧ 𝑏 ∈ N)))
55		eleq1 2175	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ ⟨𝑐, 𝑑⟩ ∈ (ω × N)))
56		opelxp 4527	. . . . . . . . . . . . . 14 ⊢ (⟨𝑐, 𝑑⟩ ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑 ∈ N))
57	55, 56	syl6bb 195	. . . . . . . . . . . . 13 ⊢ (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
58	57	biimpcd 158	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ω × N) → (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
59	54, 58	im2anan9 570	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → ((𝑎 ∈ ω ∧ 𝑏 ∈ N) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ N))))
60	59	imp 123	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)) → ((𝑎 ∈ ω ∧ 𝑏 ∈ N) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
61	60	adantrr 468	. . . . . . . . 9 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ((𝑎 ∈ ω ∧ 𝑏 ∈ N) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
62		pinn 7059	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ N → 𝑑 ∈ ω)
63		nnmcom 6337	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ω ∧ 𝑑 ∈ ω) → (𝑎 ·o 𝑑) = (𝑑 ·o 𝑎))
64	62, 63	sylan2 282	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ω ∧ 𝑑 ∈ N) → (𝑎 ·o 𝑑) = (𝑑 ·o 𝑎))
65		pinn 7059	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ N → 𝑏 ∈ ω)
66		nnmcom 6337	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ·o 𝑐) = (𝑐 ·o 𝑏))
67	65, 66	sylan 279	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ N ∧ 𝑐 ∈ ω) → (𝑏 ·o 𝑐) = (𝑐 ·o 𝑏))
68	64, 67	eqeqan12d 2128	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ω ∧ 𝑑 ∈ N) ∧ (𝑏 ∈ N ∧ 𝑐 ∈ ω)) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
69	68	an42s 561	. . . . . . . . 9 ⊢ (((𝑎 ∈ ω ∧ 𝑏 ∈ N) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ N)) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
70	61, 69	syl 14	. . . . . . . 8 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
71	50, 70	mpbid 146	. . . . . . 7 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏))
72	71	eqcomd 2118	. . . . . 6 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))
73	47, 49, 72	jca32 306	. . . . 5 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
74	73	2eximi 1561	. . . 4 ⊢ (∃𝑐∃𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
75	74	2eximi 1561	. . 3 ⊢ (∃𝑎∃𝑏∃𝑐∃𝑑((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐))) → ∃𝑎∃𝑏∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
76	45, 75	syl 14	. 2 ⊢ (𝑓 ~Q0 𝑔 → ∃𝑎∃𝑏∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
77		exrot4 1650	. . 3 ⊢ (∃𝑎∃𝑏∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ∃𝑐∃𝑑∃𝑎∃𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
78		19.42vv 1861	. . . . 5 ⊢ (∃𝑎∃𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
79	78	2exbii 1566	. . . 4 ⊢ (∃𝑐∃𝑑∃𝑎∃𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
80		19.42vv 1861	. . . . 5 ⊢ (∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑐∃𝑑∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
81		opeq12 3671	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ⟨𝑧, 𝑤⟩ = ⟨𝑐, 𝑑⟩)
82	81	eqeq2d 2124	. . . . . . . . 9 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑔 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
83	82	anbi1d 458	. . . . . . . 8 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
84		simpl 108	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
85	84	oveq1d 5741	. . . . . . . . 9 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧 ·o 𝑢) = (𝑐 ·o 𝑢))
86		simpr 109	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
87	86	oveq1d 5741	. . . . . . . . 9 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑤 ·o 𝑣) = (𝑑 ·o 𝑣))
88	85, 87	eqeq12d 2127	. . . . . . . 8 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣)))
89	83, 88	anbi12d 462	. . . . . . 7 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣))))
90		opeq12 3671	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
91	90	eqeq2d 2124	. . . . . . . . 9 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (𝑓 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
92	91	anbi2d 457	. . . . . . . 8 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩)))
93		simpr 109	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → 𝑢 = 𝑏)
94	93	oveq2d 5742	. . . . . . . . 9 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (𝑐 ·o 𝑢) = (𝑐 ·o 𝑏))
95		simpl 108	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → 𝑣 = 𝑎)
96	95	oveq2d 5742	. . . . . . . . 9 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (𝑑 ·o 𝑣) = (𝑑 ·o 𝑎))
97	94, 96	eqeq12d 2127	. . . . . . . 8 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ((𝑐 ·o 𝑢) = (𝑑 ·o 𝑣) ↔ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎)))
98	92, 97	anbi12d 462	. . . . . . 7 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))))
99	89, 98	cbvex4v 1878	. . . . . 6 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑐∃𝑑∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎)))
100		eleq1 2175	. . . . . . . . . 10 ⊢ (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
101	100	anbi1d 458	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
102		eqeq1 2119	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑔 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑧, 𝑤⟩))
103	102	anbi1d 458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
104	103	anbi1d 458	. . . . . . . . . 10 ⊢ (𝑥 = 𝑔 → (((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
105	104	4exbidv 1822	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → (∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
106	101, 105	anbi12d 462	. . . . . . . 8 ⊢ (𝑥 = 𝑔 → (((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
107		eleq1 2175	. . . . . . . . . 10 ⊢ (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
108	107	anbi2d 457	. . . . . . . . 9 ⊢ (𝑦 = 𝑓 → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
109		eqeq1 2119	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
110	109	anbi2d 457	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑓 → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
111	110	anbi1d 458	. . . . . . . . . 10 ⊢ (𝑦 = 𝑓 → (((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
112	111	4exbidv 1822	. . . . . . . . 9 ⊢ (𝑦 = 𝑓 → (∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
113	108, 112	anbi12d 462	. . . . . . . 8 ⊢ (𝑦 = 𝑓 → (((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))))
114	2, 1, 106, 113, 17	brab 4152	. . . . . . 7 ⊢ (𝑔 ~Q0 𝑓 ↔ ((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))))
115	114	biimpri 132	. . . . . 6 ⊢ (((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣))) → 𝑔 ~Q0 𝑓)
116	99, 115	sylan2br 284	. . . . 5 ⊢ (((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑐∃𝑑∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
117	80, 116	sylbi 120	. . . 4 ⊢ (∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ∃𝑎∃𝑏((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
118	79, 117	sylbi 120	. . 3 ⊢ (∃𝑐∃𝑑∃𝑎∃𝑏((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
119	77, 118	sylbi 120	. 2 ⊢ (∃𝑎∃𝑏∃𝑐∃𝑑((𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)) ∧ ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩) ∧ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎))) → 𝑔 ~Q0 𝑓)
120	76, 119	syl 14	1 ⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1312  ∃wex 1449   ∈ wcel 1461  ⟨cop 3494   class class class wbr 3893  ωcom 4462   × cxp 4495  (class class class)co 5726   ·_o comu 6263  Ncnpi 7022   ~_Q0 ceq0 7036

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-oadd 6269  df-omul 6270  df-ni 7054  df-enq0 7174

This theorem is referenced by:  enq0er  7185