Theorem enq0sym 7449

Description: The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7452. (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 2755	. . . . . . . 8 ⊢ 𝑓 ∈ V
2		vex 2755	. . . . . . . 8 ⊢ 𝑔 ∈ V
3		eleq1 2252	. . . . . . . . . 10 ⊢ (𝑥 = 𝑓 → (𝑥 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
4	3	anbi1d 465	. . . . . . . . 9 ⊢ (𝑥 = 𝑓 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
5		eqeq1 2196	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑓 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑧, 𝑤⟩))
6	5	anbi1d 465	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑓 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
7	6	anbi1d 465	. . . . . . . . . 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 1881	. . . . . . . . 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 473	. . . . . . . 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 2252	. . . . . . . . . 10 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
11	10	anbi2d 464	. . . . . . . . 9 ⊢ (𝑦 = 𝑔 → ((𝑓 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N))))
12		eqeq1 2196	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑣, 𝑢⟩))
13	12	anbi2d 464	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
14	13	anbi1d 465	. . . . . . . . . 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 1881	. . . . . . . . 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 473	. . . . . . . 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 7441	. . . . . . . 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 4287	. . . . . . 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 120	. . . . . 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 3795	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → ⟨𝑧, 𝑤⟩ = ⟨𝑎, 𝑏⟩)
21	20	eqeq2d 2201	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (𝑓 = ⟨𝑧, 𝑤⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
22	21	anbi1d 465	. . . . . . . . 9 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → ((𝑓 = ⟨𝑧, 𝑤⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩)))
23		simpl 109	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → 𝑧 = 𝑎)
24	23	oveq1d 5906	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (𝑧 ·o 𝑢) = (𝑎 ·o 𝑢))
25		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → 𝑤 = 𝑏)
26	25	oveq1d 5906	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → (𝑤 ·o 𝑣) = (𝑏 ·o 𝑣))
27	24, 26	eqeq12d 2204	. . . . . . . . 9 ⊢ ((𝑧 = 𝑎 ∧ 𝑤 = 𝑏) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝑎 ·o 𝑢) = (𝑏 ·o 𝑣)))
28	22, 27	anbi12d 473	. . . . . . . 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 3795	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → ⟨𝑣, 𝑢⟩ = ⟨𝑐, 𝑑⟩)
30	29	eqeq2d 2201	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (𝑔 = ⟨𝑣, 𝑢⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
31	30	anbi2d 464	. . . . . . . . 9 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑣, 𝑢⟩) ↔ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)))
32		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → 𝑢 = 𝑑)
33	32	oveq2d 5907	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (𝑎 ·o 𝑢) = (𝑎 ·o 𝑑))
34		simpl 109	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → 𝑣 = 𝑐)
35	34	oveq2d 5907	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → (𝑏 ·o 𝑣) = (𝑏 ·o 𝑐))
36	33, 35	eqeq12d 2204	. . . . . . . . 9 ⊢ ((𝑣 = 𝑐 ∧ 𝑢 = 𝑑) → ((𝑎 ·o 𝑢) = (𝑏 ·o 𝑣) ↔ (𝑎 ·o 𝑑) = (𝑏 ·o 𝑐)))
37	31, 36	anbi12d 473	. . . . . . . 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 1942	. . . . . . 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 457	. . . . . 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 122	. . . . 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 1923	. . . . 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 134	. . . 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 1923	. . . . 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 1617	. . . 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 134	. . 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		pm3.22 265	. . . . . . 7 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N)))
47	46	adantr 276	. . . . . 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		pm3.22 265	. . . . . . 7 ⊢ ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩))
49	48	ad2antrl 490	. . . . . 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 531	. . . . . . . 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 2252	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ ⟨𝑎, 𝑏⟩ ∈ (ω × N)))
52		opelxp 4671	. . . . . . . . . . . . . 14 ⊢ (⟨𝑎, 𝑏⟩ ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ N))
53	51, 52	bitrdi 196	. . . . . . . . . . . . 13 ⊢ (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑓 ∈ (ω × N) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ N)))
54	53	biimpcd 159	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (ω × N) → (𝑓 = ⟨𝑎, 𝑏⟩ → (𝑎 ∈ ω ∧ 𝑏 ∈ N)))
55		eleq1 2252	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ ⟨𝑐, 𝑑⟩ ∈ (ω × N)))
56		opelxp 4671	. . . . . . . . . . . . . 14 ⊢ (⟨𝑐, 𝑑⟩ ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑 ∈ N))
57	55, 56	bitrdi 196	. . . . . . . . . . . . 13 ⊢ (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑔 ∈ (ω × N) ↔ (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
58	57	biimpcd 159	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ω × N) → (𝑔 = ⟨𝑐, 𝑑⟩ → (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
59	54, 58	im2anan9 598	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) → ((𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩) → ((𝑎 ∈ ω ∧ 𝑏 ∈ N) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ N))))
60	59	imp 124	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (ω × N) ∧ 𝑔 ∈ (ω × N)) ∧ (𝑓 = ⟨𝑎, 𝑏⟩ ∧ 𝑔 = ⟨𝑐, 𝑑⟩)) → ((𝑎 ∈ ω ∧ 𝑏 ∈ N) ∧ (𝑐 ∈ ω ∧ 𝑑 ∈ N)))
61	60	adantrr 479	. . . . . . . . 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 7326	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ N → 𝑑 ∈ ω)
63		nnmcom 6508	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ω ∧ 𝑑 ∈ ω) → (𝑎 ·o 𝑑) = (𝑑 ·o 𝑎))
64	62, 63	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ω ∧ 𝑑 ∈ N) → (𝑎 ·o 𝑑) = (𝑑 ·o 𝑎))
65		pinn 7326	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ N → 𝑏 ∈ ω)
66		nnmcom 6508	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ω ∧ 𝑐 ∈ ω) → (𝑏 ·o 𝑐) = (𝑐 ·o 𝑏))
67	65, 66	sylan 283	. . . . . . . . . . 11 ⊢ ((𝑏 ∈ N ∧ 𝑐 ∈ ω) → (𝑏 ·o 𝑐) = (𝑐 ·o 𝑏))
68	64, 67	eqeqan12d 2205	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ω ∧ 𝑑 ∈ N) ∧ (𝑏 ∈ N ∧ 𝑐 ∈ ω)) → ((𝑎 ·o 𝑑) = (𝑏 ·o 𝑐) ↔ (𝑑 ·o 𝑎) = (𝑐 ·o 𝑏)))
69	68	an42s 589	. . . . . . . . 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 147	. . . . . . 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 2195	. . . . . 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 310	. . . . 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 1612	. . . 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 1612	. . 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 1702	. . 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 1923	. . . . 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 1617	. . . 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 1923	. . . . 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 3795	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ⟨𝑧, 𝑤⟩ = ⟨𝑐, 𝑑⟩)
82	81	eqeq2d 2201	. . . . . . . . 9 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑔 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑐, 𝑑⟩))
83	82	anbi1d 465	. . . . . . . 8 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
84		simpl 109	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑧 = 𝑐)
85	84	oveq1d 5906	. . . . . . . . 9 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑧 ·o 𝑢) = (𝑐 ·o 𝑢))
86		simpr 110	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → 𝑤 = 𝑑)
87	86	oveq1d 5906	. . . . . . . . 9 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → (𝑤 ·o 𝑣) = (𝑑 ·o 𝑣))
88	85, 87	eqeq12d 2204	. . . . . . . 8 ⊢ ((𝑧 = 𝑐 ∧ 𝑤 = 𝑑) → ((𝑧 ·o 𝑢) = (𝑤 ·o 𝑣) ↔ (𝑐 ·o 𝑢) = (𝑑 ·o 𝑣)))
89	83, 88	anbi12d 473	. . . . . . 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 3795	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ⟨𝑣, 𝑢⟩ = ⟨𝑎, 𝑏⟩)
91	90	eqeq2d 2201	. . . . . . . . 9 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (𝑓 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑎, 𝑏⟩))
92	91	anbi2d 464	. . . . . . . 8 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ((𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑐, 𝑑⟩ ∧ 𝑓 = ⟨𝑎, 𝑏⟩)))
93		simpr 110	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → 𝑢 = 𝑏)
94	93	oveq2d 5907	. . . . . . . . 9 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (𝑐 ·o 𝑢) = (𝑐 ·o 𝑏))
95		simpl 109	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → 𝑣 = 𝑎)
96	95	oveq2d 5907	. . . . . . . . 9 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → (𝑑 ·o 𝑣) = (𝑑 ·o 𝑎))
97	94, 96	eqeq12d 2204	. . . . . . . 8 ⊢ ((𝑣 = 𝑎 ∧ 𝑢 = 𝑏) → ((𝑐 ·o 𝑢) = (𝑑 ·o 𝑣) ↔ (𝑐 ·o 𝑏) = (𝑑 ·o 𝑎)))
98	92, 97	anbi12d 473	. . . . . . 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 1942	. . . . . 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 2252	. . . . . . . . . 10 ⊢ (𝑥 = 𝑔 → (𝑥 ∈ (ω × N) ↔ 𝑔 ∈ (ω × N)))
101	100	anbi1d 465	. . . . . . . . 9 ⊢ (𝑥 = 𝑔 → ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N))))
102		eqeq1 2196	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑔 → (𝑥 = ⟨𝑧, 𝑤⟩ ↔ 𝑔 = ⟨𝑧, 𝑤⟩))
103	102	anbi1d 465	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑔 → ((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩)))
104	103	anbi1d 465	. . . . . . . . . 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 1881	. . . . . . . . 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 473	. . . . . . . 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 2252	. . . . . . . . . 10 ⊢ (𝑦 = 𝑓 → (𝑦 ∈ (ω × N) ↔ 𝑓 ∈ (ω × N)))
108	107	anbi2d 464	. . . . . . . . 9 ⊢ (𝑦 = 𝑓 → ((𝑔 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ↔ (𝑔 ∈ (ω × N) ∧ 𝑓 ∈ (ω × N))))
109		eqeq1 2196	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑓 → (𝑦 = ⟨𝑣, 𝑢⟩ ↔ 𝑓 = ⟨𝑣, 𝑢⟩))
110	109	anbi2d 464	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑓 → ((𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ↔ (𝑔 = ⟨𝑧, 𝑤⟩ ∧ 𝑓 = ⟨𝑣, 𝑢⟩)))
111	110	anbi1d 465	. . . . . . . . . 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 1881	. . . . . . . . 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 473	. . . . . . . 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 4287	. . . . . . 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 133	. . . . . 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 288	. . . . 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 121	. . . 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 121	. . 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 121	. 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 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2160  ⟨cop 3610   class class class wbr 4018  ωcom 4604   × cxp 4639  (class class class)co 5891   ·_o comu 6433  Ncnpi 7289   ~_Q0 ceq0 7303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-oadd 6439  df-omul 6440  df-ni 7321  df-enq0 7441

This theorem is referenced by:  enq0er  7452